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(LOGIC AS INFORMATION) 155.33 520 T
(COMPRESSION BY MULTIPLE) 129.98 494 T
(ALIGNMENT, UNIFICATION AND) 113.68 468 T
(SEARCH) 256 442 T
0 18 Q
(J Gerard Wolff) 246.51 400 T
1 12 Q
(October 1996) 273.17 374 T
1 14 Q
([Running title: \322Logic and Multiple Alignment\323]) 168.92 348.67 T
-0.28 (Dr J G Wolff, School of Electronic Engineering and Computer Systems, University) 72.11 78.67 P
0.05 (of Wales, Dean Street, Bangor, Gwynedd, LL57 1UT, UK. Tel: +44 1248 382691.) 74.78 62.67 P
(E-mail: gerry@sees.bangor.ac.uk. Fax: +44 1248 361429. Web: http://) 108.18 46.67 T
(www.sees.bangor.ac.uk/~gerry/.) 215.32 30.67 T
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(Abstract) 72 712 T
1 F
3.89 (This article considers the proposal that logic may usefully be understood as information) 72 688 P
0.94 (compression by) 72 672 P
2 F
0.94 (multiple alignment) 152.54 672 P
1 F
0.94 (, with \324unification\325 and \324search\325, where \324multiple alignment\325) 243.82 672 P
1.79 (has a meaning which is close to its meaning in bio-informatics, \324unification\325 means a simple) 72 656 P
-0.24 (merging of matching patterns, and \324search\325 has its normal meaning in the context of computing. A) 72 640 P
0.24 (motivation for developing these ideas, not considered in this article, is the possible integration of) 72 624 P
2.94 (concepts in logic with other concepts in computing including learning, pattern recognition,) 72 608 P
(information retrieval and others.) 72 592 T
2.43 (A theme throughout this article is that these concepts may model \324standard\325 forms of logic) 72 566 P
2.12 (exemplified by propositional logic, first order logic and related ideas but they also offer the) 72 550 P
(possibility of some rationalisation and simplification of concepts in logic.) 72 534 T
-0.19 (The way in which basic concepts in logic \050AND, OR, NOT etc.\051 may be modelled in the proposed) 72 508 P
(framework is discussed and some possible rationalisations are proposed.) 72 492 T
2.34 (The way in which the multiple alignment framework may accommodate the composition of) 72 466 P
(complex forms of logic from simpler constituents is described.) 72 450 T
(Logical deduction is considered both in standard forms and in proposed revisions.) 72 424 T
0.41 (Finally, there is an outline description of how the multiple alignment framework may be applied) 72 398 P
-0.22 (to a selection of problems associated with nonmonotonic and uncertain reasoning: inheritance and) 72 382 P
-0.35 (default reasoning \050including the negation of a default attribute\051, abductive reasoning and inductive) 72 366 P
(reasoning.) 72 350 T
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(1.) 72 712 T
(INTRODUCTION) 108 712 T
1 F
3.89 (This article considers the proposal that logic may usefully be understood as information) 72 683 P
0.65 (compression and, more specifically, that logic may usefully be understood a process of) 72 667 P
2 F
0.65 (multiple) 500.66 667 P
1.28 (alignment) 72 651 P
1 F
1.28 (, with \324unification\325 and \324search\325, where \324multiple alignment\325 has a meaning which is) 120 651 P
3.03 (close to its meaning in bio-informatics, \324unification\325 means a simple merging of matching) 72 635 P
(patterns,) 72 619 T
1 10 Q
(1) 112.99 623.8 T
1 12 Q
( and \324search\325 has its normal meaning in the context of computing.) 117.99 619 T
0.53 (That logic may have any significant relation to information compression \050IC\051 may, at first sight,) 72 593 P
0.6 (seem an outlandish idea. But consider an elementary form of \324deduction\325 which is illustrated by) 72 577 P
-0.4 (how we perceive the scene shown in Fig. 1. We interpret this scene as a building with a car in front) 72 561 P
0.45 (of it and with a tree in front of the car and the building. Although some parts of the building are) 72 545 P
0.31 (obscured by the car and the tree, and some parts of the car are obscured by the tree, we naturally) 72 529 P
(assume that the unseen parts are \324really\325 present.) 72 513 T
(-------------------------------------------------------------------------) 108 487 T
(Please insert Fig. 1 about here) 144 472 T
(-------------------------------------------------------------------------) 108 457 T
(Fig. 1) 108 433 T
(. A scene in which some objects are partially obscured by other objects.) 136.01 433 T
0.38 (This process of inference of the unseen parts of a scene from those parts which have been seen -) 72 409 P
0.73 (which is so prominent in our everyday perceptions of the world - may be viewed as a primitive) 72 393 P
1.28 (form of \324deduction\325. From the parts of each object or pattern which we can see, and from our) 72 377 P
0.32 (knowledge of what the whole object or pattern looks like, we deduce what parts are hidden from) 72 361 P
(view.) 72 345 T
0.91 (Making the connection between patterns of information on the retina and patterns stored in our) 72 319 P
0.82 (brains is a process of recognition. It has been understood for some time that recognition can be) 72 303 P
2.01 (understood as a process of IC \050Watanabe, 1972, 1985\051. Thus inference or \324deduction\325 which) 72 287 P
-0.14 (depends on the recognition of a whole pattern from a subset of its parts may also be understood as) 72 271 P
(IC.) 72 255 T
1.79 (It is not hard to see how recognition may be understood as IC: we suppose that the external) 72 229 P
0.25 (information in the patterns which are fully or partly seen is merged \050\324unified\325\051 with the matching) 72 213 P
1.3 (patterns which we have already stored in our brains; in each case where an external pattern is) 72 197 P
0.12 (unified with a stored pattern, there is an overall compression of information because two patterns) 72 181 P
-0.25 (are reduced to one. If an external pattern can be successfully unified with a stored pattern, we may) 72 165 P
-0.19 (communicate an occurrence of the pattern using a short identifier like \324building\325, \324car\325 or \324tree\325, in) 72 149 P
2.34 (essentially the same way that relatively short \324codes\325 are used to identify the occurrence of) 72 133 P
72 114 540 128.98 C
72 114 540 128.98 R
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0 0 612 792 C
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0.48 (1. In logic, \324unification\325 means a process of making substitutions for variables in each of) 90 106 P
2.29 (two formulae so that the two formulae become identical and may thus \050implicitly or) 90 90 P
-0.3 (explicitly\051 be merged to form a single instance. The meaning of \324unification\325 in the present) 90 74 P
-0.28 (context is the simpler but obviously related meaning of merging two identical patterns into) 90 58 P
1.59 (one. Throughout this article, the term will normally be used with its simpler meaning.) 90 42 P
(Where the \324logic\325 meaning of the term is intended, this will be clearly marked.) 90 26 T
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1.34 (relatively large patterns in standard methods for IC \050see, for example, Storer \0501988\051\051. Another) 72 712 P
0.34 (example of the use of a short code to represent a larger pattern is the use of \324IC\325 in this article to) 72 696 P
(represent the phrase \324information compression\325.) 72 680 T
-0.37 (In case the example of scene analysis, just described, seems too remote from logic in its traditional) 72 654 P
(sense, consider the following simple syllogism:) 72 638 T
2 F
(If today is Tuesday, then tomorrow will be Wednesday.) 108 612 T
(Today is Tuesday.) 108 595 T
(Therefore, tomorrow will be Wednesday.) 108 578 T
1 F
0.3 (The fact that Wednesday always follows Tuesday is a familiar pattern of events described by the) 72 554 P
0.31 (first proposition. The observation that today is Tuesday leads us to the conclusion that tomorrow) 72 538 P
0.58 (will be Wednesday because we can unify this observation with the larger pattern represented by) 72 522 P
0.75 (the first proposition. \050No doubt this larger pattern is itself part of an even larger pattern that we) 72 506 P
(learned as children listing all seven days of the week in their correct order\051.) 72 490 T
0.25 (The concept of \324unification\325 as a simple merging of identical patterns is simpler than the concept) 72 464 P
0.63 (of \324unification\325 in logic but it is a prominent part of that concept. Given that \324unification\325 in the) 72 448 P
1.27 (sense of logic is a key concept in that field, and given that it embraces a concept with a clear) 72 432 P
0.2 (connection with IC, there is further support for the idea that there may be a more than superficial) 72 416 P
(relation between logic and IC.) 72 400 T
0 F
(1.1.) 72 374 T
(Multiple alignment, unification and search) 108 374 T
1 F
0.34 (In the rest of this article, the phrase \324information compression by multiple alignment, unification) 72 350 P
(and search\325 will normally be abbreviated as \324ICMAUS\325.) 72 334 T
-0.22 (As described later in this article, the ICMAUS framework may be capable of modelling a range of) 72 308 P
0.21 (established concepts in propositional logic, first order logic and related areas. If this were all that) 72 292 P
1.54 (it could do, readers might reasonably ask \322Has anything been gained by representing existing) 72 276 P
0.24 (concepts in a new form?\323 At least two possible justifications for developing these proposals may) 72 260 P
(be offered:) 72 244 T
(\245) 72 218 T
3.67 (ICMAUS may facilitate the integration of concepts in logic with other aspects of) 108 218 P
2.57 (computing such as learning \050Wolff, 1996\051, parsing \050Wolff, submitted, b\051, best-match) 108 202 P
1.34 (information retrieval \050Wolff, 1994\051, fuzzy pattern recognition, and others. The possible) 108 186 P
0.64 (application of ICMAUS to aspects of computing other than logic has been considered in) 108 170 P
(the articles cited, in Wolff \050submitted, a\051 and elsewhere.) 108 150.53 T
1 10 Q
(2) 377.6 155.33 T
72 98 540 112.98 C
72 98 540 112.98 R
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1.85 (2. The application of ICMAUS concepts to aspects of computing other than logic are) 90 90 P
-0.17 (discussed in articles, some of which are not yet published but which are available from the) 90 74 P
0.94 (author on request. It would take too much space and take us too far afield to attempt to) 90 58 P
0.14 (summarise this work here. As far as the author is aware, there are no other articles on this) 90 42 P
(subject by other authors which can be cited at the time of writing.) 90 26 T
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(\245) 72 712 T
2.53 (Within the proposed framework, there seems to be some scope for rationalising and) 108 712 P
(simplifying concepts in logic and integrating different areas within the field.) 108 696 T
1.05 (The possibility of rationalising some aspects of logic will be a recurring theme throughout this) 72 670 P
2.39 (article which will be considered alongside discussion of possible ways in which established) 72 654 P
0.72 (concepts may be modelled as ICMAUS. For brevity, \324LS\325 is used as a shorthand for \324standard\325,) 72 638 P
-0.35 (established concepts in propositional logic, first order logic and related areas \050as described in Copi) 72 622 P
0.3 (\0501986\051, Klop \0501992\051 and Davis \0501993\051\051. \324LR\325 is used to mark concepts which are, in some sense,) 72 606 P
(\324rationalised\325 or \324revised\325.) 72 590 T
0.31 (Although ICMAUS provides a framework which may accommodate LR concepts, some of these) 72 564 P
1.5 (ideas are similar to ideas which are already well-established and can be seen, for example, in) 72 548 P
0.75 (Prolog. However, it should be emphasised that the ICMAUS proposals are not merely syntactic) 72 532 P
-0.53 (variations on Prolog - they extend into the realms of fuzzy and probabilistic matching, well beyond) 72 516 P
-0.17 (the clockwork operations of Prolog. The LR proposals outlined in Section 7 appear to be radically) 72 500 P
(new.) 72 484 T
0 F
(1.2.) 72 458 T
(The status of these proposals) 108 458 T
1 F
0.43 (These proposals are by no means a polished and \324complete\325 set of ideas. They are the result of a) 72 434 P
1.11 (fairly long period of study and reflection, driven by a strong intuition that this line of thinking) 72 418 P
1.43 (offers potential dividends if answers can be found to the many questions and problems which) 72 402 P
0.05 (naturally arise in any attempt to develop a new conceptual framework which will accommodate a) 72 386 P
(wide range of established concepts, including concepts in logic.) 72 370 T
0.01 (A computer model is under development which incorporates several of the ideas described in this) 72 344 P
0.03 (article. A description of the model and its application to parsing is presented in Wolff \050submitted,) 72 328 P
2.22 (b\051. Some of the thinking in the current model is outlined briefly in Section 3.1 but a fuller) 72 312 P
2.16 (description of the model would be inappropriate in this article because the model is not yet) 72 296 P
(sufficiently mature to demonstrate all the concepts described below.) 72 280 T
0.12 (I believe that the overall framework of ideas is already sufficiently clear and sufficiently novel to) 72 254 P
0.54 (be of interest to readers now. I hope that readers will carry the thinking further, either to expose) 72 238 P
0.85 (weaknesses which have not yet been seen or to develop and refine the ideas into a more robust) 72 222 P
(form.) 72 206 T
0 F
(1.3.) 72 180 T
(Presentation) 108 180 T
1 F
1.03 (In what follows, Section 2 presents some preliminary ideas as background and context to what) 72 156 P
0.31 (follows. Section 3 describes what \324multiple alignment\325 means in this programme of research and) 72 140 P
0.6 (describes in outline how it relates to IC, probabilities and search. Section 4 describes how some) 72 124 P
0.03 (basic concepts in logic may be modelled in the ICMAUS framework. Section 5 describes how, in) 72 108 P
2.42 (this framework, simpler structures in logic may be assembled or \324composed\325 to make more) 72 92 P
0.67 (complex structure. Section 6 considers how deductive reasoning may be modelled as ICMAUS.) 72 76 P
0.69 (And Section 7 examines the possibility that nonmonotonic and uncertain reasoning may also be) 72 60 P
(understood in these terms.) 72 44 T
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(2.) 72 712 T
(PRELIMINARIES) 108 712 T
(2.1.) 72 683 T
(Background) 108 683 T
1 F
(The proposals in this article have their origins in three main strands of thinking:) 72 659 T
(\245) 72 633 T
0.37 (From at least as long ago as the writings of Ernst Mach in the last century there has been) 108 633 P
0.74 (recognition of a close connection between human thinking and information compression) 108 617 P
-0.66 (\050see, for example, Zipf \0501949\051, Attneave \0501954\051, Von B\216k\216sy \0501967\051, Barlow \0501969\051, Wolff) 108 601 P
(\0501991\051\051) 108 585 T
(\245) 72 559 T
1.28 (Since Solomonoff\325s two seminal papers on inductive inference \0501964\051 there has been a) 108 559 P
-0.58 (stream of research developing the idea that inductive inference and some kinds of inductive) 108 543 P
-0.32 (problem solving are closely related to information compression \050see, for example, Wallace) 108 527 P
0.37 (and Boulton \0501968\051, Cook and Rosenfeld \0501976\051, Wallace and Freeman \0501987\051, Rissanen) 108 511 P
(\0501987\051, Wolff \0501991\051, Gammerman \0501991\051\051.) 108 495 T
(\245) 72 469 T
-0.26 (Concepts of inductive inference are closely related to concepts in Algorithmic Information) 108 469 P
0.77 (Theory \050see, for example, Li. and Vit\207nyi \0501993\051\051, based on the idea that randomness in) 108 453 P
2 (information can be defined as incompressibility of information. Solomonoff\325s original) 108 437 P
(work is regarded as formally equivalent to AIT.) 108 421 T
(Some of this thinking is reviewed in Wolff \0501993, 1995a\051.) 72 395 T
1.7 (More immediately, these proposals originate in a programme of research developing the \324SP\325) 72 369 P
2.44 (conjecture that) 72 353 P
2 F
2.44 (all kinds of computing and formal reasoning may usefully be understood as) 150.86 353 P
2.67 (information compression by pattern matching, unification and search) 72 337 P
1 F
2.67 ( \050Wolff 1990-96\051. The) 424.69 337 P
(background and motivation for the research is described most fully in Wolff \0501993\051.) 72 321 T
0.25 (The main difference between the SP programme and other work related to inductive inference or) 72 295 P
(AIT are:) 72 279 T
(\245) 72 253 T
0.13 (The SP programme seeks to integrate) 108 253 P
2 F
0.13 (all) 291.43 253 P
1 F
0.13 ( kinds of computing and formal reasoning within) 304.1 253 P
4.11 (a framework of information compression. The goal is broader than understanding) 108 237 P
-0.16 (\324inductive inference\325, unless that term is taken to mean \324all kinds of computing and formal) 108 221 P
(reasoning\325.) 108 205 T
(\245) 72 179 T
2.14 (The SP programme is based on the hypothesis that) 108 179 P
2 F
2.14 (all) 372.95 179 P
1 F
2.14 ( kinds of compression may be) 385.62 179 P
-0.22 (understood in terms of pattern matching, unification and search. All existing and projected) 108 163 P
0.14 (SP models are restricted to pattern matching, unification and search and avoid \324arithmetic) 108 147 P
0.79 (coding\325 and related techniques which are used in some other research related to IC. The) 108 131 P
3.43 (restriction has been imposed in the interests of simplicity in the SP theory. If, as) 108 115 P
0.88 (conjectured, arithmetic may be understood in terms of pattern matching, unification and) 108 99 P
0.91 (search, then arithmetic coding may also be understood in those terms. But until this has) 108 83 P
0.25 (been demonstrated, the adoption of arithmetic coding or any similar technique would add) 108 67 P
(unwanted complexity to the SP model.) 108 51 T
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(2.2.) 72 712 T
(Representation of knowledge) 108 712 T
1 F
0.05 (A working hypothesis in this programme of research is that, with the aid of established principles) 72 688 P
0.4 (for IC, it should be possible to model any kind of knowledge \050including any kind of language or) 72 672 P
0.34 (notational system\051 with patterns) 72 656 P
1 10 Q
0.28 (3) 226.67 660.8 P
1 12 Q
0.34 ( of atomic symbols in one or more dimensions,) 231.67 656 P
2 F
0.34 (where there are) 463.35 656 P
0.72 (no \324meta\325 symbols with special meanings) 72 640 P
1 F
0.72 ( \050for further discussion, see Wolff \0501995a\051, Section 3\051.) 273.27 640 P
0.02 (For the time being, the focus of attention is one-dimensional patterns but, at some stage, the ideas) 72 624 P
(may be generalised to patterns in two or more dimensions.) 72 608 T
1.22 (This hypothesis is motivated in part by the thought that any good theory of knowledge should) 72 582 P
0.53 (acknowledge the fact that much of our empirical knowledge is derived from \324raw\325 data received) 72 566 P
0.85 (by our senses and should describe how such raw data may become \324knowledge\325. There is good) 72 550 P
2.43 (empirical evidence, some of which is reviewed in Wolff \0501993\051, that the path from data to) 72 534 P
2.96 (knowledge is a process of information compression. Given a \324principle of least effort\325, an) 72 518 P
-0.47 (implications of this view of data, knowledge and learning is that the forms of stored knowledge are) 72 502 P
-0.51 (likely to have recognisable similarities to the patterns of raw data which are received by the senses.) 72 486 P
0.05 (As emphasised in the first paragraph above, a key part of the hypothesis is that there should be no) 72 460 P
-0.18 (\324meta\325 symbols with hidden meanings. By hypothesis, all symbols should enter into matching and) 72 444 P
-0.66 (unification in the same way and all meanings should derive from the processes of pattern matching,) 72 428 P
-0.16 (unification and search. Given this \324principle of transparency\325, one of the tasks of the research is to) 72 412 P
0.02 (see whether and how, within the framework which is to be described, it may be possible to model) 72 396 P
0.26 (the semantics of meta symbols as they appear in established formalisms. Examples are presented) 72 380 P
(below.) 72 364 T
1.11 (Since the aim is to analyse established concepts into more primitive elements, it should not be) 72 338 P
2 (surprising if the analysis sometimes appears more complex than what is being analysed. For) 72 322 P
-0.51 (example, a fragment of logic like \324) 72 306 P
3 F
-0.51 (\330) 235.24 306 P
2 F
-0.51 (p) 243.8 306 P
1 F
-0.51 (\325 appears much less simple in terms of ICMAUS \050see Section) 249.8 306 P
1.1 (5.2, below\051. The suggestion here is that this is a matter of appearance, not reality, and that the) 72 290 P
3.41 (ICMAUS analysis has not added anything which was not already implicit in the original) 72 274 P
(representation.) 72 258 T
0.06 (The significance of IC in the hypothesis is that, within the scope of established principles of IC, it) 72 232 P
0.36 (ensures that) 72 216 P
2 F
0.36 (any) 132.7 216 P
1 F
0.36 ( kind of knowledge can be represented in a succinct way. It is a curious fact that) 150.03 216 P
2.13 (although much discussion of how knowledge can or should be represented \050in, for example,) 72 200 P
3.67 (theoretical linguistics\051 places emphasis on the idea that knowledge should be represented) 72 184 P
3.29 (succinctly, and although many of the techniques which are used for the representation of) 72 168 P
-0.14 (knowledge can be seen as examples of techniques for IC, the connection between the two areas of) 72 152 P
(research is rarely acknowledged.) 72 136 T
72 114 540 128.98 C
72 114 540 128.98 R
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0.27 (3. The term) 90 106 P
2 F
0.27 (pattern) 149.48 106 P
1 F
0.27 ( in this article means any array of symbols in one or more dimensions) 184.15 106 P
1.64 (including a sequence, string, substring or subsequence of symbols. The term) 90 90 P
2 F
1.64 (substring) 477.32 90 P
1 F
0.29 (means a sequence of symbols within a larger sequence where the symbols are contiguous) 90 74 P
0.42 (within the larger sequence. The term) 90 58 P
2 F
0.42 (subsequence) 271.46 58 P
1 F
0.42 ( means a sequence of symbols within a) 332.11 58 P
3.07 (larger sequence where the symbols are not necessarily contiguous within the larger) 90 42 P
(sequence.) 90 26 T
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(2.2.1.) 72 712 T
(Representation of atomic symbols) 108 712 T
1 F
0.41 (For the sake of clarity and readability of the examples presented below, the convention has been) 72 688 P
-0.73 (adopted that an atomic symbol is any string of one or more characters bounded by spaces. Although) 72 672 P
0.2 (this convention means that the space character is treated as a meta symbol, it should be clear that) 72 656 P
0.51 (this is purely a cosmetic matter and that any set of patterns in this form may be translated into a) 72 640 P
(form which does not depend on a special character to separate one atomic symbol from the next.) 72 624 T
0 F
(2.3.) 72 598 T
(Logic and uncertainty) 108 598 T
1 F
-0.44 (As described below, the ICMAUS framework in its full generality is one in which probabilities are) 72 574 P
0.16 (the rule and where it is normally impossible to prove that the best possible result has been found.) 72 558 P
1.67 (A working hypothesis is that the all-or-nothing character of traditional kinds of logic may be) 72 542 P
(modelled with probabilities which are close to 0 or 1.) 72 526 T
-0.28 (A probabilistic framework has the attraction that it suggests the possibility of developing a unified) 72 500 P
-0.64 (treatment for both traditional kinds of logic and concepts of nonmonotonic and uncertain reasoning) 72 484 P
-0.43 (\050discussed in Section 7\051. It also sits well with some other aspects of computing which we may seek) 72 468 P
-0.25 (to integrate with logic such as unsupervised learning \050Wolff, 1996\051, parsing \050Wolff, submitted, b\051,) 72 452 P
4.38 (best-match information retrieval \050Wolff, 1994\051, fuzzy pattern recognition, and others \050as) 72 436 P
(previously noted, these matters are considered in the articles cited and elsewhere\051.) 72 420 T
0.25 (This \324probabilistic\325 approach to logic differs from \324fuzzy logic\325 \050see, for example, Klir and Yuan) 72 394 P
0.93 (\0501995\051, Kosko \0501994\051\051 because it is not a \324fuzzy\325 superset of logic. It seeks to develop a single) 72 378 P
3.07 (conceptual framework within which deductive and probabilistic kinds of inference may be) 72 362 P
0.11 (integrated and simplified. There are many other differences between the concepts to be described) 72 346 P
(and concepts in the area known as fuzzy logic.) 72 330 T
0 F
(3.) 72 294 T
(MULTIPLE ALIGNMENT, COMPRESSION, PROBABILITIES AND SEARCH) 108 294 T
1 F
0.72 (The term \324multiple alignment\325 is borrowed from the field of bio-informatics where it means the) 72 265 P
2.65 (arrangement of \050symbolic representations of\051 two or more DNA sequences or two or more) 72 249 P
0.46 (sequences of amino acid residues so that matching symbols are aligned vertically in columns. In) 72 233 P
0.16 (biochemistry, this kind of analysis is used in the process of determining the structure, function or) 72 217 P
(evolution of the corresponding molecules.) 72 201 T
0.23 (In this area of research on multiple alignment, it is generally recognised that there is normally an) 72 175 P
0.15 (astronomically large number of alternative ways in which symbols may be aligned and that some) 72 159 P
0.03 (are better than others. This raises two questions: what one might mean by a \324good\325 alignment and) 72 143 P
-0.46 (how to find \324good\325 alignments amongst the many possibilities. These two questions are considered) 72 127 P
(briefly in the next two sub-sections.) 72 111 T
0 F
(3.1.) 72 85 T
(Compression and probabilities) 108 85 T
1 F
2.22 (Intuitively, a \324good\325 alignment is one with many positive matches \050\324hits\325\051 between symbols,) 72 61 P
-0.41 (relatively few \324gaps\325 between hits \050where there are unmatched symbols\051 and any gaps should be as) 72 45 P
1.98 (short as possible. In these intuitive terms, the best alignment of a pair of strings with a low) 72 29 P
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7 X
0 K
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1 12 Q
0 X
(- 9 -) 296 757 T
72 18 540 720 R
7 X
V
0 X
0.92 (\324Hamming distance\325 is better than the best alignment of a pair of strings with a high Hamming) 72 712 P
(distance.) 72 696 T
-0.57 (Since compression may be achieved by unification of matching patterns, it should be clear in broad) 72 670 P
0.05 (terms that a \324good\325 alignment \050with many hits and few unmatched symbols\051 is one in which there) 72 654 P
-0.33 (is a relatively large scope for IC. In the extreme case, two strings of symbols are identical and may) 72 638 P
1.51 (be aligned without any gaps between hits. In this case, the two strings can be reduced to one) 72 622 P
0.78 (without loss of non-redundant information \050provided a record is kept of the fact that there were) 72 606 P
(originally two copies of the information\051.) 72 590 T
-0.42 (In other cases, there may be only substrings that match. In cases like this, one of a pair of matching) 72 564 P
0.2 (substrings is given a \324tag\325 or \324code\325, the other substring is deleted and its position is marked with) 72 548 P
1.6 (a copy of the tag or code. As previously noted, the use of \324IC\325 as a shorthand for the phrase) 72 532 P
-0.42 (\324information compression\325 is an example of this technique. Of course, it only makes sense to make) 72 516 P
-0.44 (this substitution if the substrings are sufficiently large to yield a net saving after the cost of the tags) 72 500 P
(is counted.) 72 484 T
2 F
(3.1.1.) 72 458 T
(Probabilities) 108 458 T
1 F
-0.17 (The connection between concepts of information and probability can be seen in Shannon\325s classic) 72 434 P
(formula for the average information provided by the occurrence of an \324event\325:) 72 418 T
1.81 (where) 72 346 P
2 F
1.81 (p) 106.13 346 P
2 10 Q
1.51 (i) 112.13 343 P
1 12 Q
1.81 ( is the probability of the) 114.91 346 P
2 F
1.81 (i) 243.77 346 P
1 F
1.81 (th event in a set of) 247.11 346 P
2 F
1.81 (n) 349.3 346 P
1 F
1.81 ( alternative events. More simply, the) 355.3 346 P
(information \050in bits\051 provided by the occurrence of any specific event is:) 72 330 T
2.46 (In general, the occurrence of an event which is probable conveys less information than the) 72 259 P
(occurrence of an event which is rare.) 72 243 T
2.08 (For a sequence of events, e) 72 217 P
1 10 Q
1.73 (1) 212.34 214 P
1 12 Q
2.08 ( ... e) 217.34 217 P
2 10 Q
1.73 (m) 241.82 214 P
1 12 Q
2.08 ( \050e.g., the characters in an ordinary text\051, the information) 249.04 217 P
(conveyed by the sequence is:) 72 201 T
1.63 (One can either compute the number of bits for each of the several events \050using the absolute) 72 132 P
0.34 (probability of the event in each case\051 and then add them up or one can compute a probability for) 72 116 P
0.04 (the whole sequence \050normally, a very small number\051 and derive the number of bits from that. The) 72 100 P
(former is normally more convenient than the latter.) 72 84 T
0.63 (An approximation to these values can be obtained if each event is coded in accordance with the) 72 58 P
0.14 (Huffman principle \0501952\051 so that commonly occurring events receive short codes and rare events) 72 42 P
-0.13 (receive long codes. The use of some kind of \324prefix\325 coding scheme \050e.g., 0, 01, 011, 0111, 01111) 72 26 P
72 18 540 720 C
72 368 540 414 C
2 12 Q
0 X
0 K
(H) 112 383 T
1 F
(=) 127 383.01 T
3 F
(\345) 140 383.11 T
2 F
(n) 141.28 396 T
(p) 153 383 T
2 10 Q
(i) 159 380 T
1 12 Q
( log) 161.78 383 T
2 F
(p) 183.12 383 T
2 10 Q
(i) 189.12 380 T
72 18 540 720 C
0 0 612 792 C
72 18 540 720 C
72 281 540 326 C
2 12 Q
0 X
0 K
(h) 109 301.96 T
2 10 Q
(i) 115 298.96 T
1 12 Q
( = log) 117.78 301.96 T
1 10 Q
(2) 145.88 298.96 T
1 12 Q
(1) 155.39 309 T
154.39 305 162.39 305 2 L
0.5 H
2 Z
N
2 F
(p) 154 294 T
2 10 Q
(i) 160 291 T
72 18 540 720 C
0 0 612 792 C
72 18 540 720 C
72 154 540 197 C
2 12 Q
0 X
0 K
(H) 106 167 T
1 10 Q
(s) 114.66 164 T
1 12 Q
( =) 118.55 167 T
3 F
(\345) 136 167.11 T
2 F
(m) 137 180 T
(h) 148 167 T
2 10 Q
(j) 154 164 T
1 12 Q
( = log) 156.78 167 T
1 10 Q
(2) 184.88 164 T
1 12 Q
(\0501) 192.88 167 T
3 F
(/ \325) 205.88 167 T
2 F
(p) 225.09 167 T
2 10 Q
(j) 231.09 164 T
1 12 Q
(\051) 233.87 167 T
2 F
(m) 212 180 T
72 18 540 720 C
0 0 612 792 C
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(- 10 -) 293 757 T
72 18 540 720 R
7 X
V
0 X
1.92 (...\051 ensures that codes can be discriminated unambiguously even when they are concatonated) 72 712 P
(without breaks between successive codes.) 72 696 T
1.67 (In any body of information in which there are relatively large repeating patterns or relatively) 72 670 P
0.68 (frequent repeating patterns, or both, the above formulae are inaccurate because they do not take) 72 654 P
-0.1 (account of the redundancy represented by the repeating patterns. However, the Huffman principle) 72 638 P
-0.35 (can also be applied to the tags or codes mentioned earlier used as markers for repeating substrings.) 72 622 P
1.52 (Substrings which repeat frequently \050and thus have a high probability\051 should be given shorter) 72 606 P
0.09 (codes than substrings which occur rarely. In this way, the redundancy represented by the repeated) 72 590 P
(substrings can be coded and the size of a compressed body of information can be minimised.) 72 574 T
-0.74 (Thus, compression as well as information is closely related to concepts of probability: an alignment) 72 548 P
-0.45 (which gives good compression is also one with a relatively high probability. For further discussion) 72 532 P
-0.14 (see Allison and Yee \0501990\051, Allison, Wallace) 72 516 P
2 F
-0.14 (et al.) 291.96 516 P
1 F
-0.14 ( \0501992\051, Chan, Wong) 315.82 516 P
2 F
-0.14 (et al.) 419.91 516 P
1 F
-0.14 (\0501992\051, and Allison) 446.62 516 P
(and Wallace \0501994\051.) 72 500 T
2 F
(3.1.2.) 72 474 T
(Towards a new method for evaluation of alignments) 108 474 T
1 F
2.58 (A fuller development of the ideas presented in this article will require precise methods for) 72 450 P
0.48 (calculating the compressions and probabilities that may be achieved in the proposed framework.) 72 434 P
0.86 (Methods already exist for evaluating alignments in these terms \050see the articles cited\051 but those) 72 418 P
0.74 (which have been examined are not fully satisfactory in terms of the principles which have been) 72 402 P
0.03 (adopted as working hypotheses in this programme of research. In particular, it seems that none of) 72 386 P
0.43 (the existing methods can be applied directly to the generalised version of the multiple alignment) 72 370 P
0.54 (problem described in Section 3.3 and none of them stay strictly within the framework of pattern) 72 354 P
(matching, unification and search.) 72 338 T
0.05 (Steps towards a new method are described in Wolff \050submitted, b\051. In summary, this new method) 72 312 P
-0.11 (treats the alignment problem as one of finding an economical encoding for a sequence of symbols) 72 296 P
0.74 (designated as New in terms of other sequences designated as Old. Each sequence in Old has an) 72 280 P
0.29 (associated cost which is the number of bits required to discriminate the beginning and end of the) 72 264 P
0 (sequence together with the number of bits required to identify the sequence uniquely amongst the) 72 248 P
3.03 (set of Old sequences. The \324compression score\325 of any alignment is the number of bits of) 72 232 P
-0.57 (information from New which it encodes, minus the information cost \050in bits\051 of the sequences from) 72 216 P
0.13 (Old which are used in the encoding. The latter can be less than the sum of the bits for each of the) 72 200 P
0.83 (Old sequence in the alignment because some sequences in Old can encode sequences of two or) 72 184 P
0.02 (more other sequences in Old and this recursively through arbitrarily many levels in the manner of) 72 168 P
0.32 (hierarchical grammars. When some sequences in Old are encoded in terms of other sequences as) 72 152 P
(just described, the number of bits required overall for the encoding can be reduced.) 72 136 T
0 F
(3.2.) 72 110 T
(Search) 108 110 T
1 F
-0.56 (Given that the abstract space of possible alignments is normally extremely large, exhaustive search) 72 86 P
0.98 (is normally impossible. It is generally recognised that any practical search for good alignments) 72 70 P
1.22 (means constraining the search so that parts of the search space \050often very large parts\051 are not) 72 54 P
2.42 (entered at all. In broad terms, this may be done in two ways \050which need not be mutually) 72 38 P
(exclusive\051:) 72 22 T
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(- 11 -) 293 757 T
72 18 540 720 R
7 X
V
0 X
(\245) 72 712 T
2.16 (The space of possible alignments may be searched in stages, selecting only the most) 108 712 P
2.88 (promising path or paths at each stage and using some measure of \324goodness\325 \050e.g.,) 108 696 P
1.03 (compression or probability\051 to guide the selections made at each stage. Versions of this) 108 680 P
-0.49 (kind of \324metrics-guided\325 search include \324hill climbing\325, \324beam search\325, \324genetic algorithm\325,) 108 664 P
1.26 (\324simulated annealing\325 and others. With this kind of constraint, large parts of the search) 108 648 P
-0.25 (space may be pruned away but no part of the search space is ruled out) 108 632 P
2 F
-0.25 (a priori) 442.48 632 P
1 F
-0.25 (. Potentially,) 479.24 632 P
(any kind of alignment can be reached.) 108 616 T
(\245) 72 590 T
-0.26 (Arbitrary constraints may be applied so that parts of the search space can never be entered.) 108 590 P
1.03 (For example, some arbitrary ceiling may be set on the maximum number of unmatched) 108 574 P
-0.31 (symbols between hits. This means that it is not necessary to examine any of the alignments) 108 558 P
-0.16 (containing one or more gaps between hits which exceed the maximum which has been set.) 108 542 P
0.94 (With this kind of constraint, a lot of searching can be saved because parts of the search) 108 526 P
0.76 (space are ruled out from the beginning. The penalty, of course, is that solutions in those) 108 510 P
(areas - some of which may be good solutions - can never be found.) 108 494 T
0.4 (With either kind of constraint, it is not possible to guarantee that the best possible alignment has) 72 468 P
-0.38 (been found unless the sequences to be aligned are very few and small. There is a trade-off between) 72 452 P
0.15 (the amount of searching required and the \324quality\325 of the results obtained. It is always possible to) 72 436 P
-0.32 (make the search manageable by discarding larger parts of the search space. That said, it seems that) 72 420 P
-0.52 (acceptably good results can often be obtained with relatively modest amounts of searching: a small) 72 404 P
(sacrifice in quality can often mean dramatic savings in the amount of searching required.) 72 388 T
1 (There is now a number of methods of searching which give quite reasonably good results with) 72 362 P
1.37 (biochemical sequences. Some of these methods are reviewed in Taylor \0501988\051, Barton \0501990\051,) 72 346 P
1.38 (Chan, Wong) 72 330 P
2 F
1.38 (et al.) 138.42 330 P
1 F
1.38 ( \0501992\051 and Day and McMorris \0501992\051. However, it seems that none of the) 163.8 330 P
2.89 (existing methods are adapted to the generalised version of the multiple alignment problem) 72 314 P
0.23 (described in Section 3.3 and none of the methods are suitable for finding the kinds of alignments) 72 298 P
(described later in this article. A new search method is described in Wolff \050submitted, b\051.) 72 282 T
0 F
(3.3.) 72 256 T
(A generalised version of the multiple alignment problem) 108 256 T
1 F
0 (By contrast with most work in bio-informatics, \324multiple alignment\325 in the present context covers) 72 232 P
0.74 (cases where a given sequence may appear two or more times in different parts of an alignment.) 72 216 P
-0.73 (This includes cases where any one sequence may be aligned with itself \050with the obvious restriction) 72 200 P
0.16 (that any one symbol should not be aligned with itself\051. These points are illustrated in some of the) 72 184 P
(examples below.) 72 168 T
0 F
(4.) 72 132 T
(SOME BASIC CONCEPTS IN LOGIC) 108 132 T
1 F
-0.17 (This section considers how some of the simpler LS concepts may be understood as ICMAUS and,) 72 103 P
(in Section 4.6, describes some possible LR versions of these ideas.) 72 87 T
0 F
(4.1.) 72 61 T
(Logical operators) 108 61 T
1 F
-0.59 (Each of the basic logical operators, NOT \050) 72 37 P
3 F
-0.59 (\330) 271.16 37 P
1 F
-0.59 (\051, IMPLIES \050) 279.72 37 P
3 F
-0.59 (\311) 342.2 37 P
1 F
-0.59 (\051, AND \050) 350.76 37 P
3 F
-0.59 (\331) 392.57 37 P
1 F
-0.59 (\051, inclusive OR \050) 399.8 37 P
3 F
-0.59 (\332) 478.03 37 P
1 F
-0.59 (\051, exclusive) 485.27 37 P
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(- 12 -) 293 757 T
72 18 540 720 R
7 X
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0 X
-0.1 (OR, and EQUIVALENT \050) 72 712 P
3 F
-0.1 (\253) 198.32 712 P
1 F
-0.1 (\051 is defined in LS by a truth table as illustrated by the three examples) 210.83 712 P
(in Fig. 2.) 72 696 T
(-------------------------------------------------------------------------) 108 670 T
(Please insert Fig. 2 about here) 144 655 T
(-------------------------------------------------------------------------) 108 640 T
(Fig. 2. Examples of truth table definitions of logical operators.) 108 616 T
-0.22 (For any given table, a logical operation is performed by searching for an exact match between one) 72 592 P
0.23 (or more \324input\325 values and the values in the \324input\325 cell or cells of one of the rows in the table. If) 72 576 P
0.3 (such a match is found, then the \324output\325 value may be read from the output cell of the same row.) 72 560 P
-0.43 (For example, if the input values \3240\325 and \3241\325 \050in that order\051 are applied to the table for IMPLIES, the) 72 544 P
(third row of the table is selected and the output is \3241\325.) 72 528 T
-0.68 (This process of \324table lookup\325 may be seen as partial pattern matching with unification as described) 72 502 P
0.54 (in the Introduction. For example, the IMPLIES operator, may be represented as a set of patterns) 72 486 P
(like this:) 72 470 T
4 F
(1 1 ; 1 ;) 108 444 T
(1 0 ; 0 ;) 108 429 T
(0 1 ; 1 ;) 108 414 T
(0 0 ; 1 ;) 108 399 T
1 F
-0.64 (Given this definition and an \324input\325 pattern like this: \324) 72 375 P
4 F
-0.71 (0 1 ; ;) 323.92 375 P
1 F
-0.64 (\325, the best alignment of the input pattern) 351.81 375 P
-0.14 (with one of the patterns in the definition appears to be what is shown in Fig. 3. The corresponding) 72 359 P
(unification is also shown.) 72 343 T
(-------------------------------------------------------------------------) 108 317 T
(Please insert Fig. 3 about here) 144 302 T
(-------------------------------------------------------------------------) 108 287 T
-0.12 (Fig. 3. Alignment and unification of an \324input\325 pattern with part of the LS definition of the) 108 263 P
(IMPLIES operation. Symbols which are the result of unification are shown in bold type.) 108 249 T
2.07 (In this and later examples, symbols which are the result of the unification of two \050or more\051) 72 225 P
0.46 (matching symbols are shown in bold type. In the unification shown in this figure, the \324output\325 is) 72 209 P
0.65 (the symbol in plain type which originated in the \324output\325 part of the definition and has not been) 72 193 P
(unified with any symbol from the \324input\325.) 72 177 T
0.56 (In accordance with the remarks in Section 2.2, the semi-colon symbol \050\324) 72 151 P
4 F
0.62 (;) 424.45 151 P
1 F
0.56 (\325\051 which appears in the) 427.79 151 P
0.65 (example is) 72 135 P
2 F
0.65 (not) 127.95 135 P
1 F
0.65 ( a meta symbol with special meaning: it enters into matching and unification like) 143.29 135 P
-0.59 (any other symbol. That said, symbols like this have a role to play which is comparable with the role) 72 119 P
-0.19 (of brackets and other punctuation marks in conventional systems. In an ICMAUS framework, it is) 72 103 P
2.21 (necessary to include instances of symbols like \324) 72 87 P
4 F
2.45 (;) 315.74 87 P
1 F
2.21 (\325 in the sequences to be matched to reduce) 319.08 87 P
-0.68 (ambiguity and ensure that the \324correct\325 alignment will give more compression than any of the many) 72 71 P
(incorrect alternatives.) 72 55 T
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7 X
V
0 F
0 X
(4.2.) 72 712 T
(Simple functions and simple relations) 108 712 T
1 F
1.38 (In its simplest form, a \324function\325 is a table which relates input values to corresponding output) 72 688 P
0.26 (values. In its simplest form, a \324relation\325 is the same except that output values are restricted to the) 72 672 P
-0.46 (set {1, 0}. Any of the logical operators considered in the last subsection may be regarded as simple) 72 656 P
(relations.) 72 640 T
0.02 (Any simple function, simple relation or logical operator which is represented in tabular form may) 72 614 P
0.07 (be evaluated by table lookup. It should be clear from the example presented in the last subsection) 72 598 P
(that any such case may be interpreted as alignment and unification of patterns.) 72 582 T
0 F
(4.3.) 72 556 T
(Simple rewrite rules) 108 556 T
1 F
(In their simplest form, rewrite rules like this:) 72 532 T
4 F
(green light) 108 506 T
3 F
(\256) 167.36 506 T
4 F
( go) 179.21 506 T
1 F
0.59 (are very much like the rows in a table representing a function, relation or logical operation. The) 72 482 P
-0.41 (symbol \050or symbols\051 to the left of the arrow represents potential input and the symbol \050or symbols\051) 72 466 P
(to the right of the arrow represents potential output.) 72 450 T
0.08 (As before, actual output is obtained by reading the symbol\050s\051 to the right of the arrow when there) 72 424 P
0.46 (is a match between actual input symbol\050s\051 and symbol\050s\051 to the left of the arrow. As before, this) 72 408 P
(effect may be seen in terms of the alignment and unification of patterns.) 72 392 T
2 F
(4.3.1.) 72 366 T
(Rewriting) 108 366 T
1 F
0.39 (Rules like these are described as \324rewrite\325 rules because each rule is commonly interpreted as an) 72 342 P
0.03 (instruction to delete a pattern of one or more symbols in the input which matches the left side and) 72 326 P
(replace it with a copy of the symbol\050s\051 on the right.) 72 310 T
1.52 (At first sight, this appears to be rather different from alignment and unification of patterns as) 72 284 P
-0.17 (described above. But if a pattern in the input is deleted and the matching pattern on the left side of) 72 268 P
0.07 (a rule is retained, this is equivalent to the merging or unification of the two patterns. And it seems) 72 252 P
0.14 (that explicit copying of the symbol\050s\051 on the right side of any rule is not necessary: an equivalent) 72 236 P
-0.2 (effect may be achieved by reading these \324output\325 symbols) 72 220 P
2 F
-0.2 (as if) 350.45 220 P
1 F
-0.2 (they had been copied into different) 373.38 220 P
(contexts.) 72 204 T
2.03 (A relatively full account of how the workings of simple rewrite rules may be understood as) 72 178 P
0.02 (ICMAUS is given in Wolff \050submitted, b\051. Examples also appear later in this article \050see Sections) 72 162 P
(4.5.1, 4.6.2 and 5.1\051.) 72 146 T
2 F
(4.3.2.) 72 120 T
(Reduction) 108 120 T
1 F
1.42 (Rewrite rules are sometimes also called \324reduction\325 rules \050Klop, 1992\051, apparently because, in) 72 96 P
1.36 (some applications, rules are applied that have more symbols on the left than on the right. If a) 72 80 P
-0.13 (structure is processed by these rules so that patterns which match the left sides are replaced by the) 72 64 P
(copies of the corresponding right sides, the size of the structure will be reduced.) 72 48 T
0.29 (This alternative name for rewrite rules seems to be somewhat misleading because there are other) 72 22 P
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(- 14 -) 293 757 T
72 18 540 720 R
7 X
V
0 X
3.14 (applications, especially in theoretical linguistics, where rewrite rules commonly have more) 72 712 P
0.15 (symbols on the right than on the left. In cases like these, the application of rewrite rules can have) 72 696 P
(the effect of converting a relatively small structure into a larger one.) 72 680 T
-0.35 (Notwithstanding the use of the alternative name \324reduction\325, it seems that the possible connections) 72 654 P
0.08 (between these kinds of rules and concepts of information theory, redundancy and IC as presented) 72 638 P
(in this article \050Section 3.1 and elsewhere\051 are not generally recognised.) 72 622 T
0 F
(4.4.) 72 596 T
(Variables) 108 596 T
1 F
1.23 (A variable may be regarded as a receptacle for information which may be empty. Normally, a) 72 572 P
0.61 (variable has a name but in some systems, e.g., Prolog, it is possible to specify variables without) 72 556 P
(names which function merely as place markers for missing information.) 72 540 T
1.71 (It seems that the effect of a variable may be modelled with ICMAUS by providing a pair of) 72 514 P
-0.46 (\324marker\325 symbols which may be aligned with matching symbols at the start and end of information) 72 498 P
2.44 (representing a possible value for the variable. For example, an empty variable, \324) 72 482 P
2 F
2.44 (x) 484.81 482 P
1 F
2.44 (\325, may be) 490.14 482 P
-0.48 (represented by the pair of contiguous symbols, \324) 72 466 P
4 F
-0.53 (x ;) 299.94 466 P
1 F
-0.48 (\325, within a larger sequence; possible \324values\325 for) 312.08 466 P
0.59 (this \324variable\325 may be flanked by matching symbols as, for example, in \324) 72 450 P
4 F
0.65 (x 1 ;) 427.67 450 P
1 F
0.59 (\325 or \324) 451.66 450 P
4 F
0.65 (x 0 ;) 476.83 450 P
1 F
0.59 (\325. Fig. 4) 500.82 450 P
(shows how such a \324variable\325 may acquire a \324value\325 by alignment and unification of patterns.) 72 434 T
(-------------------------------------------------------------------------) 108 408 T
(Please insert Fig. 4 about here) 144 393 T
(-------------------------------------------------------------------------) 108 378 T
-0.64 (Fig. 4. Alignment and unification of a \324variable\325 with a \324value\325. The ellipses \050\324) 108 354 P
4 F
-0.71 (...) 475.66 354 P
1 F
-0.64 (\325\051 represent) 485.67 354 P
(other symbols which may lie to the left and right of \324) 108 340 T
4 F
(x ;) 361.64 340 T
1 F
(\325.) 374.32 340 T
0.81 (The similarity between Fig. 4 and Fig. 3 will not have escaped the reader. In Fig. 3, the pair of) 72 316 P
0.02 (symbols \324) 72 300 P
4 F
0.02 (; ;) 119.02 300 P
1 F
0.02 (\325 in \324) 129.05 300 P
4 F
0.02 (0 1 ; ;) 152.41 300 P
1 F
0.02 (\325 may be seen to function as an unnamed variable which receives \324) 182.5 300 P
4 F
0.02 (1) 501.96 300 P
1 F
0.02 (\325 as its) 508.63 300 P
(\324value\325.) 72 284 T
0 F
(4.5.) 72 258 T
(Types for variables) 108 258 T
1 F
1.55 (In the example just considered, it seems that there are two alternative \324best\325 alignments \050with) 72 234 P
-0.64 (corresponding unifications\051: the one shown in Fig. 4 and one where \324) 72 218 P
4 F
-0.71 (x ;) 396.25 218 P
1 F
-0.64 (\325 is aligned and unified with) 408.21 218 P
0.76 (\324) 72 202 P
4 F
0.85 (x 0 ;) 76 202 P
1 F
0.76 (\325. Thus the patterns \324) 100.37 202 P
4 F
0.85 (x 1 ;) 203.07 202 P
1 F
0.76 (\325 and \324) 227.44 202 P
4 F
0.85 (x 0 ;) 260.28 202 P
1 F
0.76 (\325 may be seen as providing a definition of the set of) 284.66 202 P
0.77 (possible \324values\325 for the \324variable\325 \324) 72 186 P
4 F
0.86 (x ;) 248.14 186 P
1 F
0.77 (\325. In other words, they provide a \324type definition\325 for the) 261.67 186 P
(\324variable\325 which, in this case, is a definition of the Boolean type.) 72 170 T
2 F
(4.5.1.) 72 144 T
(Types with Infinite Range) 108 144 T
1 F
0.36 (In many cases, of course, the type of a variable covers an infinite range of values, e.g., \324integer\325,) 72 120 P
-0.29 (\324real\325, \324string\325 and others. Often, a type of this kind may be defined using simple rewrite rules. For) 72 104 P
(example, strings of alphabet characters may be defined with rules like these:) 72 88 T
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72 747 540 765 R
7 X
0 K
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(- 15 -) 293 757 T
72 18 540 720 R
7 X
V
0 X
(\244) 108 712 T
3 F
(\256) 117.34 712 T
1 F
(\244) 108 697 T
3 F
(\256) 117.34 697 T
4 F
( a) 129.18 697 T
1 F
(\244) 142.52 697 T
4 F
(...) 108 682 T
1 F
(\244) 108 667 T
3 F
(\256) 117.34 667 T
4 F
( z) 129.18 667 T
1 F
(\244) 141.85 667 T
(\244) 108 652 T
3 F
(\256) 117.34 652 T
4 F
( A) 129.18 652 T
1 F
(\244) 143.86 652 T
4 F
(...) 108 637 T
1 F
(\244) 108 622 T
3 F
(\256) 117.34 622 T
4 F
( Z) 129.18 622 T
1 F
(\244) 143.18 622 T
0.27 (The first rule \050\324\244) 72 598 P
3 F
0.27 (\256) 156.08 598 P
1 F
0.27 (\325\051 expresses the idea that a string may be empty.) 167.93 598 P
0.27 (The subsequent rules define) 404.89 598 P
(recursively the infinite set of alphabetic strings containing one or more letters.) 72 582 T
0.07 (In keeping with the ideas described above, the rewrite rules just shown may be translated into the) 72 556 P
(following patterns:) 72 540 T
(\244) 108 514 T
4 F
( ;) 114 514 T
1 F
(\244) 108 499 T
4 F
( a) 114 499 T
1 F
(\244) 127.34 499 T
4 F
(...) 108 484 T
1 F
(\244) 108 469 T
4 F
( z) 114 469 T
1 F
(\244) 126.67 469 T
(\244) 108 454 T
4 F
( A) 114 454 T
1 F
(\244) 128.68 454 T
4 F
(...) 108 439 T
1 F
(\244) 108 424 T
4 F
( Z) 114 424 T
1 F
(\244) 128 424 T
1.02 (Given these Old patterns, together with an Old pattern, \324) 72 400 P
4 F
1.13 (1 a b c) 351.47 400 P
1 F
1.02 (\244) 395.37 400 P
4 F
1.13 ( ; d e f) 401.37 400 P
1 F
1.02 ( #\325 which contains a) 439.27 400 P
0.05 (\324variable\325, and a New \324input\325 string \050\324) 72 384 P
4 F
0.05 (a b c) 253.22 384 P
0.05 (x y z d e f) 282.39 384 P
1 F
0.05 (\325\051, the best alignment appears to be what is) 334 384 P
(shown in Fig. 5.) 72 368 T
(-------------------------------------------------------------------------) 108 342 T
(Please insert Fig. 5 about here) 144 327 T
(-------------------------------------------------------------------------) 108 312 T
0.49 (Fig. 5. An alignment amongst patterns comprising an alphabetic string \050\324) 108 288 P
4 F
0.54 (a b c x y z d e) 462.17 288 P
0.16 (f) 108 274 P
1 F
0.14 (\325\051, a pattern containing a \324variable\325 \050\324) 111.34 274 P
4 F
0.16 (1 a b c) 290.48 274 P
1 F
0.14 (\244) 330.49 274 P
4 F
0.16 ( ; d e f) 336.49 274 P
1 F
0.14 ( #\325\051 and patterns which constitute a) 370.49 274 P
(\324type definition\325 of the variable.) 108 260 T
(The corresponding unification is:) 72 236 T
4 F
(1) 108 210 T
5 F
(a b c) 118.01 210 T
0 F
(\244) 148.69 210 T
5 F
( x) 154.69 210 T
0 F
(\244) 168.04 210 T
5 F
( y) 174.04 210 T
0 F
(\244) 187.38 210 T
5 F
( z) 193.38 210 T
0 F
(\244) 206.05 210 T
5 F
( ; d e f) 212.05 210 T
4 F
(#) 250.73 210 T
1 F
2.18 (Notwithstanding the inclusion of instances of the \324service\325 symbols,\325) 72 186 P
4 F
2.43 (1) 419.76 186 P
1 F
2.18 (\325, \324\244\325, \324) 426.44 186 P
4 F
2.43 (;) 464.79 186 P
1 F
2.18 (\325 and \324) 468.12 186 P
4 F
2.43 (#) 503.81 186 P
1 F
2.18 (\325, this) 510.48 186 P
(unification achieves the effect of an \324assignment\325 of the alphabetic string to the \324variable\325.) 72 170 T
1.1 (Compression of the \324New\325 input string \324) 72 144 P
4 F
1.23 (a b c) 271.27 144 P
1.23 (x y z d e f) 303.84 144 P
1 F
1.1 (\325 is achieved because it is, in effect,) 361.33 144 P
2.08 (recognised as an instance of a pre-established \324Old\325 string, \324) 72 128 P
4 F
2.31 (1 a b c) 378.6 128 P
1 F
2.08 (\244) 427.19 128 P
4 F
2.31 ( ; d e f) 433.19 128 P
1 F
2.08 ( #\325, with the) 475.78 128 P
0.14 (interpolation of \324) 72 112 P
4 F
0.16 (x y z) 153.61 112 P
1 F
0.14 (\325 in the position of the variable. Thus the New string may be encoded as \324) 178.59 112 P
4 F
0.16 (1) 533.33 112 P
1.26 (x y z #) 72 96 P
1 F
1.14 (\325. Assuming that the identifier \3241\325 is encoded with fewer bits than \324) 110.47 96 P
4 F
1.26 (a b c) 445.06 96 P
1 F
1.14 (\325 and that the) 473.6 96 P
0.56 (termination symbol, \324) 72 80 P
4 F
0.62 (#) 176.79 80 P
1 F
0.56 (\325, requires fewer bits than \324) 183.46 80 P
4 F
0.62 (x y z) 316.21 80 P
1 F
0.56 (\325, the encoding of the New string clearly) 342.13 80 P
(requires fewer bits than the original.) 72 64 T
-0.47 (Why is it necessary to include the symbols \324\244) 72 38 P
4 F
-0.52 ( ;) 286.22 38 P
1 F
-0.47 (\325 in the middle of \324) 292.37 38 P
4 F
-0.52 (1 a b c) 380.35 38 P
1 F
-0.47 (\244) 417.62 38 P
4 F
-0.52 ( ; d e f) 423.62 38 P
1 F
-0.47 ( #\325, and to include) 454.89 38 P
1.19 (patterns like \324\244) 72 22 P
4 F
1.33 ( a) 146.37 22 P
1 F
1.19 (\244\325, \324\244) 162.36 22 P
4 F
1.33 ( b) 189.55 22 P
1 F
1.19 (\244\325 and so on in the set of Old patterns? Without these symbols and) 205.54 22 P
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72 18 540 720 R
7 X
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0 X
-0.31 (patterns, it would not be possible to encode \324) 72 712 P
4 F
-0.34 (1 a b c) 284.54 712 P
1 F
-0.31 (\244) 322.54 712 P
4 F
-0.34 ( ; d e f) 328.54 712 P
1 F
-0.31 ( #\325 as \324) 360.55 712 P
4 F
-0.34 (1 x y z #) 392.62 712 P
1 F
-0.31 (\325 because there would) 435.95 712 P
0.08 (be ambiguity about the relative positions of the symbols \324) 72 696 P
4 F
0.09 (a) 349.02 696 P
1 F
0.08 (\325, \324) 355.69 696 P
4 F
0.09 (b) 369.76 696 P
1 F
0.08 (\325, \324) 376.43 696 P
4 F
0.09 (c) 390.5 696 P
1 F
0.08 (\325, \324) 396.5 696 P
4 F
0.09 (x) 410.57 696 P
1 F
0.08 (\325, \324) 416.57 696 P
4 F
0.09 (y) 430.64 696 P
1 F
0.08 (\325, \324) 436.64 696 P
4 F
0.09 (z) 450.71 696 P
1 F
0.08 (\325, \324) 456.71 696 P
4 F
0.09 (d) 470.78 696 P
1 F
0.08 (\325, \324) 477.45 696 P
4 F
0.09 (e) 491.52 696 P
1 F
0.08 (\325 and \324) 498.19 696 P
4 F
0.09 (f) 529.67 696 P
1 F
0.08 (\325.) 533 696 P
0.48 (The \324service\325 symbols and patterns serve as \324anchors\325 which ensure that the relative positions of) 72 680 P
(the alphabetic symbols are preserved.) 72 664 T
2 F
(4.5.2.) 72 638 T
(Separating the Declarations of Types and Variables) 108 638 T
1 F
0.72 (The two examples of \324type definitions\325 given so far differ from type definitions in conventional) 72 614 P
-0.13 (systems because, in each case, the definition contains the name of the variable whose type is to be) 72 598 P
1.34 (specified. The inflexibility of an arrangement like this is avoided, in conventional systems, by) 72 582 P
0.69 (defining each type independently of any specific variable and then connecting it to one or more) 72 566 P
(variables in one or more declarations like \324) 72 550 T
2 F
(int i) 277.62 550 T
1 F
(\325 or \324) 296.63 550 T
2 F
(float r) 320.62 550 T
1 F
(\325.) 350.29 550 T
-0.39 (To see how this idea may be modelled with ICMAUS, consider a function whose name and formal) 72 524 P
0.41 (parameter is declared as \324) 72 508 P
2 F
0.41 (g) 196.26 508 P
1 F
0.41 (\050) 202.26 508 P
2 F
0.41 (boolean x) 206.26 508 P
1 F
0.41 (\051\325. This declaration, and the definition of the Boolean type,) 253.66 508 P
(may be represented with the following patterns:) 72 492 T
4 F
(g \050 boolean x # ; \051) 108 466 T
(boolean # 1 ;) 108 436 T
(boolean # 0 ;) 108 421 T
1 F
0.08 (In this case, the definition of the Boolean type provided by the second and third patterns does not) 72 397 P
2.12 (contain the name, \324) 72 381 P
4 F
2.36 (x) 170.35 381 P
1 F
2.12 (\325, of the variable. The symbol \324) 176.35 381 P
4 F
2.36 (boolean) 338.38 381 P
1 F
2.12 (\325 provides the link between the) 381.08 381 P
-0.57 (variable declaration and the type definition. Any other variable may be linked to the type definition) 72 365 P
(in the same way.) 72 349 T
0.09 (If) 72 323 P
2 F
0.09 (g) 82.9 323 P
1 F
0.09 (\050) 88.9 323 P
2 F
0.09 (x) 92.9 323 P
1 F
0.09 (\051 is \324called\325 from some context like this: \324...) 98.23 323 P
2 F
0.09 (g) 308.76 323 P
1 F
0.09 (\0501\051 ...\325, the actual parameter \050\3241\325\051 is assigned to) 314.76 323 P
0.23 (the formal parameter \050\324) 72 307 P
2 F
0.23 (x) 184.31 307 P
1 F
0.23 (\325\051. The way in which this may be modelled by alignment and unification) 189.64 307 P
(of patterns is shown in Fig. 6.) 72 291 T
(-------------------------------------------------------------------------) 108 265 T
(Please insert Fig. 6 about here) 144 250 T
(-------------------------------------------------------------------------) 108 235 T
-0.33 (Fig. 6. Alignment and unification of patterns for the assignment of an \324actual parameter\325 to) 108 211 P
(a \324formal parameter\325.) 108 197 T
1.3 (In this example, \324) 72 173 P
4 F
1.45 (#) 159.89 173 P
1 F
1.3 (\325 may be regarded as a \324service\325 symbol whose function is to ensure that a) 166.56 173 P
0.46 (\324correct\325 alignment is achieved. It seems that its interpolation between the left and right markers) 72 157 P
0.28 (of the \324variable\325, \324) 72 141 P
4 F
0.31 (x ;) 160.13 141 P
1 F
0.28 (\325, does not invalidate the assignment and would not prevent the value of the) 173.11 141 P
(variable being \324transferred\325 to other contexts as described in Section 5, below.) 72 125 T
2 F
(4.5.3.) 72 99 T
(Universal and Existential Quantification) 108 99 T
1 F
-0.19 (In many cases, a formula in first order logic which contains one or more instances of the universal) 72 75 P
1.5 (quantifier \050\324) 72 59 P
3 F
1.5 (") 131.14 59 P
1 F
1.5 (\325\051 or the existential quantifier \050\324) 139.7 59 P
3 F
1.5 ($) 298.82 59 P
1 F
1.5 (\325\051 or both may be converted into an equivalent) 305.4 59 P
-0.48 (\324clausal\325 form in which the quantifier\050s\051 have been omitted \050Eisinger and Ohlbach, 1993\051. Systems) 72 43 P
-0.17 (like Prolog appear to provide much of the \324power\325 of first order logic without the need for explicit) 72 27 P
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(- 17 -) 293 757 T
72 18 540 720 R
7 X
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0 X
(quantification of variables.) 72 712 T
0.01 (Accordingly, no attempt has been made in this programme of research to model the universal and) 72 686 P
4.14 (existential quantifiers explicitly. It is assumed as a working hypothesis that concepts of) 72 670 P
0.79 (quantification may be accommodated in the ICMAUS framework in a manner comparable with) 72 654 P
-0.33 (their treatment in Prolog. That said, it should be emphasised again that the ICMAUS framework is) 72 638 P
2 F
0.11 (not) 72 622 P
1 F
0.11 ( merely a syntactic variation on Prolog, that it lends itself to tasks requiring fuzzy matching in) 87.34 622 P
(ways which would be cumbersome to achieve with Prolog.) 72 606 T
0 F
(4.6.) 72 580 T
(Possible rationalisations of basic concepts) 108 580 T
1 F
2.85 (This section considers how, within the proposed framework, there may be some scope for) 72 556 P
0 (replacing some of the basic LS concepts with simpler alternatives. As previously noted, while the) 72 540 P
-0.48 (ICMAUS framework provides a home for these LR concepts, some of them are similar to concepts) 72 524 P
0.16 (which are already recognised in, for example, Prolog. The following subsections discuss some of) 72 508 P
(these possibilities.) 72 492 T
2 F
(4.6.1.) 72 466 T
(AND) 108 466 T
1 F
-0.35 (Within the concept of a string, sequence or \324pattern\325 of symbols \050which is part of the foundation of) 72 442 P
1.53 (the ICMAUS proposals\051 the relationship between a symbol and any of its neighbours may be) 72 426 P
0.08 (regarded as an example of the concept \324AND\325. In accordance with the truth-table definition of the) 72 410 P
-0.1 (concept, the AND relationship between any two symbols applies only if they are both present in a) 72 394 P
(given pattern.) 72 378 T
0.17 (If it is accepted that AND is an implicit primitive within the concept of a sequence of symbols, it) 72 352 P
0.47 (seems unnecessarily cumbersome to interpret the concept always in terms of its truth table or an) 72 336 P
1.21 (equivalent set of patterns. Any conjunction of propositions like \324) 72 320 P
2 F
1.21 (p) 392.84 320 P
3 F
1.21 (\331) 403.75 320 P
2 F
1.21 (q) 415.19 320 P
1 F
1.21 (\325 may be converted into) 421.19 320 P
2.16 (something like \324) 72 304 P
2 F
2.16 (p) 154.32 304 P
2.16 (q) 165.48 304 P
1 F
2.16 (\325, where the space between the two propositions represents the primitive) 171.48 304 P
(concept of AND, replacing the LS concept represented by \324) 72 288 T
3 F
(\331) 357.25 288 T
1 F
(\325.) 364.49 288 T
0.6 (An example of how this more primitive concept of conjunction may be applied in the ICMAUS) 72 262 P
(framework is presented in Section 6.2.) 72 246 T
2 F
(4.6.2.) 72 220 T
(Exclusive OR) 108 220 T
1 F
0.43 (Some concept of disjunction is embodied in the idea that, within the ICMAUS framework, there) 72 196 P
0.05 (may be two or more) 72 180 P
2 F
0.05 (alternative) 171.91 180 P
1 F
0.05 ( alignments \050with corresponding unifications\051 for any given set of) 223.9 180 P
1.08 (patterns and, more specifically, that amongst these alternative alignments, there may be two or) 72 164 P
(more alternative \324best\325 alignments.) 72 148 T
-0.22 (This idea appeared in Section 4.5, above, where it was suggested that the two alternative values of) 72 122 P
2.34 (a variable with a Boolean type might be modelled by two alternative \324best\325 alignments and) 72 106 P
(unifications of patterns.) 72 90 T
0.57 (In a similar way, it seems that disjunction in the operation of rewrite rules, may be modelled by) 72 64 P
(alternative alignments of patterns. Consider, for example, some \324linguistic\325 rules like these:) 72 48 T
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(- 18 -) 293 757 T
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7 X
V
4 F
0 X
(NP) 108 712 T
3 F
(\256) 128 712 T
4 F
( D N) 139.85 712 T
(D) 108 697 T
3 F
(\256) 120 697 T
4 F
( the) 131.84 697 T
3 F
(\332) 155.2 697 T
4 F
( a) 162.43 697 T
3 F
(\332) 175.78 697 T
4 F
( some) 183.01 697 T
3 F
(\332) 219.02 697 T
4 F
( one) 226.26 697 T
3 F
(\332) 252.95 697 T
4 F
( ...) 260.18 697 T
(...) 108 682 T
1 F
0.18 (In the context of the first rule, the second rule may be read as a statement that the \324determiner\325 at) 72 658 P
(the beginning of a \324noun phrase\325 may be realised as \324the\325 or \324a\325 or \324some\325 or \324one\325 etc.) 72 642 T
(These rules may also be written like this:) 72 616 T
4 F
(NP) 108 590 T
3 F
(\256) 128 590 T
4 F
( D N) 139.85 590 T
(D) 108 575 T
3 F
(\256) 120 575 T
4 F
( the) 131.84 575 T
(D) 108 560 T
3 F
(\256) 120 560 T
4 F
( a) 131.84 560 T
(D) 108 545 T
3 F
(\256) 120 545 T
4 F
( some) 131.84 545 T
(D) 108 530 T
3 F
(\256) 120 530 T
4 F
( one) 131.84 530 T
(...) 108 515 T
1 F
(And then translated into patterns like this:) 72 491 T
4 F
(NP D ; N ; ;) 108 465 T
(D the ;) 108 450 T
(D a ;) 108 435 T
(D some ;) 108 420 T
(D one ;) 108 405 T
(...) 108 390 T
1 F
-0.15 (It seems that the alternatives in the rewrite rules may be modelled by alternative \324best\325 alignments) 72 366 P
-0.18 (and unifications between the pattern \324D ;\325 within the pattern \324) 72 350 P
4 F
-0.2 (NP D ; N ; ;) 364.8 350 P
1 F
-0.18 (\325and each of the \324) 424.48 350 P
4 F
-0.2 (D ... ;) 507.73 350 P
1 F
-0.18 (\325) 536 350 P
0.72 (patterns representing specific words. In keeping with the discussion in Sections 4.4 and 4.5, the) 72 334 P
1.12 (former may be regarded as a \324variable\325 and the latter may be seen as a \324type definition\325 of the) 72 318 P
(\324variable\325.) 72 302 T
1.16 (Modelling disjunction in this way is similar to the way in which two or more clauses within a) 72 276 P
(Prolog procedure represent alternative realisations of that procedure.) 72 260 T
0.38 (These examples seem to correspond to the \324exclusive\325 version of the OR relation because one of) 72 234 P
-0.41 (the alternatives is normally chosen, and only one. The way in which an LR version of the inclusive) 72 218 P
(OR concept might be modelled in terms of ICMAUS is not, at present, clear.) 72 202 T
2 F
(4.6.3.) 72 176 T
(IMPLIES) 108 176 T
1 F
0.03 (It is generally understood that the LS concept of \324material implication\325 \050) 72 152 P
2 F
0.03 (p) 417.93 152 P
3 F
0.03 (\311) 426.9 152 P
2 F
0.03 (q) 438.42 152 P
1 F
0.03 (\051, otherwise) 444.42 152 P
0.03 (defined) 504.01 152 P
0.2 (as) 72 136 P
3 F
0.2 (\330) 85.19 136 P
1 F
0.2 (\050) 93.75 136 P
2 F
0.2 (p) 97.75 136 P
3 F
0.2 (\331) 106.94 136 P
0.2 (\330) 117.38 136 P
2 F
0.2 (q) 125.93 136 P
1 F
0.2 (\051, is not the only meaning of \324implication\325 \050Copi, 1986\051. And, of the several possible) 131.93 136 P
-0.22 (meanings, it appears to be one of the least \324natural\325. Certainly, it can lead to such counter-intuitive) 72 120 P
0.88 (statements as: \322If a statement is false then it materially implies any statement whatever\323 \050Copi,) 72 104 P
(1986\051.) 72 88 T
-0.59 (In the Introduction, it was suggested that human capabilities for recognising patterns when they are) 72 62 P
3.25 (partially obscured or fragmented supports a kind of \324naturalistic\325 inference or implication:) 72 46 P
-0.44 (recognition of a pattern from a subset of its parts means that the unseen parts are, in effect, inferred) 72 30 P
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7 X
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0 X
(or implied.) 72 712 T
0.71 (One of the strengths of the ICMAUS framework is that it can model the recognition of patterns) 72 686 P
0.13 (from subsets of their parts. Thus it may also provide a means of modelling a form of inference or) 72 670 P
0.21 (implication which appears to be more natural than material implication. An example is presented) 72 654 P
(in Section 6, below.) 72 638 T
-0.29 (This alternative view of implication as partial pattern matching suggests that the LR version of the) 72 612 P
-0.33 (IMPLIES relation is essentially the same as the LR version of the AND relation: if two symbols or) 72 596 P
-0.15 (patterns of symbols are found together within a larger pattern then there is mutual implication and) 72 580 P
(a mutual AND relationship between them.) 72 564 T
2 F
(4.6.4.) 72 538 T
(TRUE and FALSE) 108 538 T
1 F
-0.23 (In standard treatments of logic, propositions normally have explicit values for TRUE and FALSE.) 72 514 P
1.58 (This kind of explicit marking of positive and negative values may also be used in computing) 72 498 P
-0.41 (applications like commercial databases. For example, a database for railway bookings, may record) 72 482 P
0.96 (whether a seat faces forwards or backwards with \322Seat faces forward: yes/no\323 or, equivalently,) 72 466 P
(\322Seat faces: F/B\323.) 72 450 T
-0.13 (However, some compression can normally be achieved if a default value is assumed in every case) 72 424 P
-0.35 (which is not marked otherwise. Thus seats may be assumed to face forwards unless marked \322B\323 or) 72 408 P
0.94 (they may be assumed to be second class unless marked \322First\323. When this kind of technique is) 72 392 P
0.38 (used, most compression is achieved if the default value is the one which occurs most frequently,) 72 376 P
-0.72 (i.e., is most probable. In the case of seats for smokers or non-smokers, the latter is the most frequent) 72 360 P
-0.21 (in UK railway carriages today and should be chosen as the default value. Ten or twenty years ago,) 72 344 P
1.55 (the opposite was true. When Nixon said \322I am not a crook\323, he was acknowledging the prior) 72 328 P
(expectation that he was.) 72 312 T
0.16 (In many applications the imbalance between TRUE and FALSE cases is so extreme and it would) 72 286 P
0.47 (be very inefficient to use explicit values for TRUE and FALSE. For example, any travel agent\325s) 72 270 P
0 (database of flights between cities would normally store only those which are available, not all the) 72 254 P
-0.68 (multitude of possible flights which are) 72 238 P
2 F
-0.68 (not) 255.9 238 P
1 F
-0.68 ( available. If the database is queried for a particular flight) 271.23 238 P
-0.47 (and no match is found, this is normally interpreted to mean that the flight is not available \050although) 72 222 P
(it could mean that some mistake has been made in entering the available flights\051.) 72 206 T
0.61 (The practice of assigning the value TRUE to any query which finds a match and FALSE to any) 72 180 P
0.36 (query which fails to find a match is, of course, the principle of \324negation as failure\325, which has a) 72 164 P
0.22 (key role in the operation of Prolog. In the light of the foregoing remarks, negation as failure may) 72 148 P
-0.11 (be seen as an extreme example of how compression may be achieved by the use of default values.) 72 132 P
1.38 (In general, there seems to be a closer connection than is normally recognised between IC and) 72 106 P
(concepts associated with TRUE and FALSE.) 72 90 T
0.07 (ICMAUS may be used in a \324negation as failure\325 mode when that is appropriate. And, like Prolog,) 72 64 P
-0.43 (ICMAUS will accommodate explicit marking of TRUE and FALSE if that is required \050see Section) 72 48 P
-0.2 (5, below\051. As discussed in Section 7.1, the framework seems also to allow the provision of default) 72 32 P
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72 747 540 765 R
7 X
0 K
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0 X
(- 20 -) 293 757 T
72 18 540 720 R
7 X
V
0 X
(values which can be overridden with alternative values if that is required.) 72 712 T
0 F
(5.) 72 676 T
(COMPOSITION OF LOGICAL STRUCTURES) 108 676 T
1 F
-0.43 (Logical formulae may be composed, recursively, from simpler formulae as shown by these rewrite) 72 647 P
(rules:) 72 631 T
4 F
(f) 108 605 T
3 F
(\256) 114.67 605 T
1 F
(\324atomic formula\325) 129.85 605 T
4 F
(f) 108 590 T
3 F
(\256) 114.67 590 T
(\050\330) 129.85 590 T
4 F
(f\051) 142.4 590 T
(f) 108 575 T
3 F
(\256) 114.67 575 T
4 F
( \050f) 126.52 575 T
3 F
(\311) 140.52 575 T
4 F
(f) 152.08 575 T
3 F
(\051) 155.41 575 T
4 F
(f) 108 560 T
3 F
(\256) 114.67 560 T
4 F
( \050f) 126.52 560 T
3 F
(\331) 140.52 560 T
4 F
(f\051) 150.76 560 T
(f) 108 545 T
3 F
(\256) 114.67 545 T
4 F
( \050f) 126.52 545 T
3 F
(\332) 140.52 545 T
4 F
(f\051) 150.76 545 T
(f) 108 530 T
3 F
(\256) 114.67 530 T
4 F
( \050f) 126.52 530 T
3 F
(\253) 140.52 530 T
4 F
(f\051) 156.02 530 T
1 F
1.49 (In a similar way, an atomic formula is composed of a relation symbol \050\324) 72 506 P
2 F
1.49 (r) 436.04 506 P
1 F
1.49 (\325\051 with one or more) 440.71 506 P
(\324terms\325, arranged like this:) 72 490 T
2 F
(r) 108 464 T
1 F
(\050) 112.67 464 T
2 F
(t) 116.66 464 T
1 10 Q
(1) 120 461 T
3 12 Q
(,) 125 464 T
2 F
(t) 130.5 464 T
1 10 Q
(2) 133.84 461 T
3 12 Q
(, ...,) 138.84 464 T
2 F
(t) 159.84 464 T
2 10 Q
(n) 163.17 461 T
3 12 Q
(\051.) 168.17 464 T
1 F
0.04 (And, amongst the terms, there may be \324functions\325 which may be composed recursively of simpler) 72 437.67 P
(structures.) 72 421.67 T
0 F
(5.1.) 72 395.67 T
(Composition: syntax) 108 395.67 T
1 F
2.26 (At the syntactic level, recursive composition of structures may be modelled in an ICMAUS) 72 371.67 P
0.09 (framework in much the same manner as can be done with rewrite rules. The rules, above, may be) 72 355.67 P
(translated as:) 72 339.67 T
4 F
(\050 f af \051) 108 313.67 T
(\050 f not \050 f \051 \051) 108 298.67 T
(\050 f \050 f \051 implies \050 f \051 \051) 108 283.67 T
(\050 f \050 f \051 and \050 f \051 \051) 108 268.67 T
(\050 f \050 f \051 or \050 f \051 \051) 108 253.67 T
(\050 f \050 f \051 equiv \050 f \051 \051) 108 238.67 T
1 F
0.76 (and each instance of the trio of contiguous symbols, \324) 72 214.67 P
4 F
0.84 (\050 f) 335.46 214.67 P
1 F
0.76 (\051\325, may be expanded by alignment and) 351.15 214.67 P
-0.55 (unification with any of the patterns which start with \324) 72 198.67 P
4 F
-0.61 (\050 f) 322.7 198.67 P
1 F
-0.55 (\325 and ends with \324\051\325 \050cf. Section 4.6.2, above.) 332.76 198.67 P
(See also Wolff \050submitted, b\051\051. Fig. 7 shows how a composite formula) 72 182.67 T
4 F
(\050\050not \050af implies af\051\051 and af\051) 108 156.67 T
1 F
(\050in which \324) 72 132.67 T
4 F
(af) 124.66 132.67 T
1 F
(\325 stands for \324atomic formula\325\051 may be aligned with the above patterns.) 134.66 132.67 T
(-------------------------------------------------------------------------) 108 106.67 T
(Please insert Fig. 7 about here) 144 91.67 T
(-------------------------------------------------------------------------) 108 76.67 T
0.48 (Fig. 7. An alignment between a composite formula and some of the patterns defining the) 108 52.67 P
(structure of formulae.) 108 38.67 T
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(- 21 -) 293 757 T
72 18 540 720 R
7 X
V
0 F
0 X
(5.2.) 72 712 T
(Composition: semantics) 108 712 T
1 F
-0.27 (When the syntactic forms in a composite structure have explicit semantic values, there is a need to) 72 688 P
1.36 (ensure that values can be \324passed\325 from one structure to another so that the overall value of a) 72 672 P
-0.24 (composite structure may be found. To see how this may be achieved with ICMAUS, consider first) 72 656 P
(how it may be done with Prolog.) 72 640 T
-0.18 (For the sake of clarity and because alignments can otherwise become very large, a very simple LS) 72 614 P
2.68 (example will be considered, the negation of an atomic formula: \324) 72 598 P
3 F
2.68 (\330) 409.77 598 P
1 F
2.68 (canfly\050) 418.33 598 P
2 F
2.68 (x) 452.31 598 P
1 F
2.68 (\051\325. Although the) 457.64 598 P
-0.19 (example is simple, it captures all the essentials of how values may be passed from one structure to) 72 582 P
0.59 (another and it should be clear from this example how the ideas will generalise to more complex) 72 566 P
(cases.) 72 550 T
(The LS version of \324) 72 524 T
3 F
(\330) 165.98 524 T
1 F
(canfly\050) 174.54 524 T
2 F
(x) 208.52 524 T
1 F
(\051\325 may be expressed in Prolog with a set of clauses like these:) 213.85 524 T
(not_canfly\050X, V\051 :- canfly\050X, T\051, not\050T, V\051.) 108 498 T
(canfly\050robin, 1\051.) 108 468 T
(canfly\050penguin, 0\051.) 108 453 T
(not\0501, 0\051.) 108 423 T
(not\0500, 1\051.) 108 408 T
(If a query is applied like:) 72 384 T
(?- not_canfly\050penguin, V\051.) 108 358 T
0.4 (the variable \324X\325 in the first line is instantiated as \324penguin\325, the variable \324T\325 is instantiated as \3240\325) 72 334 P
(and the variable \324V\325 is instantiated as \3241\325 - which is the correct result.) 72 318 T
-0.27 (For present purposes, the key point is that, in accordance with the fundamentals of Prolog, the two) 72 292 P
1.99 (appearances of \324T\325 in the first line represent a single variable with a single value so that its) 72 276 P
(instantiation as \3240\325 has the effect of \324passing\325 that value from \324canfly\050X, T\051\325, to \324not\050T, V\051\325.) 72 260 T
0.63 (To achieve this kind of effect with ICMAUS, the Prolog clauses may be translated into patterns) 72 234 P
(like these:) 72 218 T
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72 18 540 720 R
7 X
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4 F
0 X
(not_canfly X ; V ; canfly X ; T ; % not T ; V ; % $) 108 712 T
(canfly robin 1 %) 108 682 T
(canfly penguin 0 %) 108 667 T
(not 1 0 %) 108 637 T
(not 0 1 %) 108 622 T
(X robin ;) 108 592 T
(X penguin ;) 108 577 T
(T 1 ;) 108 562 T
(T 0 ;) 108 547 T
(V 1 ;) 108 532 T
(V 0 ;) 108 517 T
1 F
-0.29 (In this translation, the last six patterns may be regarded as type definitions of the \324variables\325, \324) 72 493 P
4 F
-0.32 (X ;) 518.65 493 P
1 F
-0.29 (\325,) 533 493 P
(\324) 72 477 T
4 F
(T ;) 76 477 T
1 F
(\325 and \324) 90 477 T
4 F
(V ;) 121.32 477 T
1 F
(\325 \050as described in Section 4.4\051.) 136 477 T
(The query \324?- not_canfly\050penguin, V\051.\325 may be translated as:) 72 451 T
4 F
(not_canfly penguin V ; $) 108 425 T
1 F
-0.5 (To simplify the alignment for the sake of clarity, the first pattern in the translated \324program\325 above,) 72 401 P
(will be reduced to \324) 72 385 T
4 F
(not_canfly canfly X ; T ; % not T ; V ; %) 165.31 385 T
1 F
( $\325.) 373.42 385 T
0.07 (Fig. 8 shows what appears to be the best alignment between the \324query\325 pattern and the \050reduced\051) 72 359 P
0.33 (patterns defining the negation of the \324canfly\325 relation. Notice that the pattern \324) 72 343 P
4 F
0.37 (not_canfly canfly) 450.25 343 P
0.27 (X ; T ; % not T ; V ; %) 72 327 P
1 F
0.24 ( $\325 appears twice in the alignment, in accordance with the generalisation) 190.08 327 P
0.38 (of the alignment problem described in Section 3.3. Notice also that although each symbol in this) 72 311 P
-0.2 (pattern appears twice in the figure - which suggests the possibility of matching and alignment - no) 72 295 P
(such symbol can be matched and aligned with itself.) 72 279 T
(-------------------------------------------------------------------------) 108 253 T
(Please insert Fig. 8 about here) 144 238 T
(-------------------------------------------------------------------------) 108 223 T
1.48 (Fig. 8. An alignment of patterns showing how a \324value\325 may be \324passed\325 between two) 108 199 P
(instances of a \324variable\325.) 108 185 T
2 (In Fig. 8, the symbol \324penguin\325 in the \324query\325 has the effect of selecting the pattern \324) 72 161 P
4 F
2.23 (canfly) 508.66 161 P
-0.29 (penguin 0 %) 72 145 P
1 F
-0.26 (\325 so that, of the two main contenders for alignment with \324) 138.13 145 P
4 F
-0.29 (T ;) 409.91 145 P
1 F
-0.26 (\325, \324) 423.63 145 P
4 F
-0.29 (T 0 ;) 437.36 145 P
1 F
-0.26 (\325 is selected and,) 460.8 145 P
0.75 (in effect, \324) 72 129 P
4 F
0.84 (T ;) 123.15 129 P
1 F
0.75 (\325 receives \324) 137.99 129 P
4 F
0.84 (0) 192.79 129 P
1 F
0.75 (\325 as its \324value\325. Since the two instances of \324) 199.47 129 P
4 F
0.84 (T ;) 412.86 129 P
1 F
0.75 (\325 are aligned with each) 427.7 129 P
-0.34 (other, that \324value\325 is, in effect, \324passed\325 from \324) 72 113 P
4 F
-0.38 (canfly X ; T ; %) 290.85 113 P
1 F
-0.34 (\325 to \324) 369.63 113 P
4 F
-0.38 (not T ; V ; %) 392.27 113 P
1 F
-0.34 (\325. Then that value) 456.39 113 P
0.07 (of \324) 72 97 P
4 F
0.08 (T ;) 89.07 97 P
1 F
0.07 (\325 selects \324) 103.15 97 P
4 F
0.08 (not 0 1 %) 149.28 97 P
1 F
0.07 (\325 so that, in effect, \324) 200.22 97 P
4 F
0.08 (V ;) 294.89 97 P
1 F
0.07 (\325 receives \324) 309.65 97 P
4 F
0.08 (1) 363.1 97 P
1 F
0.07 (\325 as its value. This, of course, is the) 369.77 97 P
(\324correct\325 result.) 72 81 T
1.15 (The foregoing analysis may seem to create undue complexity out of the apparent simplicity of) 72 55 P
0.43 (\324) 72 39 P
3 F
0.43 (\330) 76 39 P
2 F
0.43 (p) 84.55 39 P
1 F
0.43 (\325. But, as noted in Section 2.2, any such impression is probably a matter more of appearance) 90.55 39 P
2.54 (than reality. Any other attempt to analyse the composition of LS structures, including their) 72 23 P
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(- 23 -) 293 757 T
72 18 540 720 R
7 X
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0 X
(semantics, in terms of primitive operations is likely to yield similar complexity.) 72 712 T
0.12 (It seems that the analysis should generalise to any combination of relations, functions and logical) 72 686 P
(operators and the passing of values amongst such structures.) 72 670 T
0 F
(6.) 72 634 T
(DEDUCTION) 108 634 T
1 F
0.27 (This section considers both LS and putative LR forms of deduction. The latter provides a natural) 72 605 P
1.56 (introduction to the area of nonmonotonic and uncertain reasoning considered in the following) 72 589 P
(section.) 72 573 T
0 F
(6.1.) 72 547 T
(Modelling \324standard\325 forms of deduction) 108 547 T
3 F
(") 108 523 T
2 F
(x) 116.56 523 T
1 F
(: bird\050) 121.88 523 T
2 F
(x) 151.55 523 T
1 F
(\051) 156.88 523 T
3 F
(\311) 163.87 523 T
1 F
( canfly\050) 172.43 523 T
2 F
(x) 209.41 523 T
1 F
(\051) 214.74 523 T
(bird\050Tweety\051) 108 508 T
3 F
(\134) 108 493 T
1 F
( canfly\050Tweety\051.) 118.36 493 T
1.64 (Rather inelegantly, LS versions of the two premises of this) 72 469 P
2 F
1.64 (modus ponens) 373.42 469 P
1 F
1.64 ( form of syllogistic) 443.4 469 P
(reasoning may be modelled in Prolog like this:) 72 453 T
(bird_implication\050X, V1, V2\051 :-) 108 427 T
(bird\050X, V1\051,) 144 412 T
(canfly\050X, V2\051,) 144 397 T
(implies\050V1, V2, 1\051.) 144 382 T
(bird\050tweety, 1\051.) 108 352 T
(canfly\050_, 1\051.) 108 322 T
(implies\0501, 1, 1\051.) 108 292 T
(implies\0501, 0, 0\051.) 108 277 T
(implies\0500, 1, 1\051.) 108 262 T
(implies\0500, 0, 1\051.) 108 247 T
1.72 (Given the query \324?- bird_implication\050X, V1, V2\051.\325 \050which would be true\051, the variable \324X\325 is) 72 223 P
-0.69 (instantiated as \324tweety\325, the variable \324V2\325 is instantiated as \3241\325, thus confirming \324canfly\050tweety, 1\051\325,) 72 207 P
(i.e., that Tweety can fly.) 72 191 T
2.69 (Given the discussion in Section 5.2, above, it appears that this treatment of) 72 165 P
2 F
2.69 (modus ponens) 468.98 165 P
1 F
2.95 (reasoning, with explicit values for TRUE and FALSE, could, in principle, be modelled as) 72 149 P
-0.73 (ICMAUS. No attempt will be made here to show the corresponding alignments because they would) 72 133 P
(take too much space.) 72 117 T
2.95 (Since any of the other basic forms of LS reasoning \050) 72 91 P
2 F
2.95 (modus tollens) 352.15 91 P
1 F
2.95 (, constructive dilemma,) 421.44 91 P
-0.54 (disjunctive syllogism etc.\051 may be modelled in Prolog in a similar way, it appears that, in principle,) 72 75 P
(these may also be modelled as ICMAUS.) 72 59 T
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7 X
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0 F
0 X
(6.2.) 72 712 T
(Alternatives to standard forms of deduction) 108 712 T
1 F
1.24 (Of course, Prolog, as it is normally used, takes advantage of \324negation as failure\325 to achieve a) 72 688 P
2.89 (significant simplification of) 72 672 P
2 F
2.89 (modus ponens) 217.01 672 P
1 F
2.89 ( and other forms of reasoning. For example, the) 288.23 672 P
-0.17 (implication that if something is a bird then that something can fly would normally be expressed in) 72 656 P
(Prolog, in reversed form, like this:) 72 640 T
(canfly\050X\051 :- bird\050X\051.) 108 614 T
0.65 (This version of the implication illustrates some of the ideas described in the Introduction and in) 72 590 P
-0.35 (Section 4.6. The clause \324canfly\050X\051 :- bird\050X\051.\325 may be seen simply as a \324pattern\325 linking the ability) 72 574 P
2.55 (to fly with the concept of a bird. In this pattern, the symbol \324:-\325 is largely redundant. The) 72 558 P
0.2 (implication may be seen as partial pattern matching: if \324bird\050X\051\325 is successfully matched then we) 72 542 P
-0.25 (may infer \324canfly\050X\051\325. If the implication is viewed in this way, then, as suggested in Section 4.6.3,) 72 526 P
-0.72 (the distinction between IMPLIES and AND, in their \324naturalistic\325 interpretations, becomes blurred.) 72 510 P
0.25 (These observations might suggest that the way forward in any attempt to develop LR versions of) 72 484 P
(logic would be to translate Prolog clauses like \324canfly\050X\051 :- bird\050X\051.\325 into patterns like this:) 72 468 T
4 F
(canfly X ; bird X ;) 108 442 T
1 F
0.11 (and to model the workings of Prolog using ICMAUS. The discussion in Section 5.2 suggests that) 72 418 P
0.34 (this is possible and that, for example, ICMAUS could model the \324passing\325 of values between the) 72 402 P
(two instances of \324X\325 in clauses like \324canfly\050X\051 :- bird\050X\051.\325.) 72 386 T
0.1 (But the flexibility of the ICMAUS framework suggest that there may be better opportunities than) 72 360 P
1.07 (this. Knowledge that \050most\051 birds can fly is simply one item of information amongst the many) 72 344 P
-0.31 (things that we know about birds: that they have wings, feathers, beak, crop, that they lay eggs, that) 72 328 P
-0.42 (some birds have an individual name, and so on. It seems unnecessarily cumbersome to record each) 72 312 P
-0.37 (of these pieces of information as a separate clause like \324canfly\050X\051 :- bird\050X\051.\325. It seems much more) 72 296 P
(natural to record them all together in a single pattern like this:) 72 280 T
4 F
(bird name ; canfly ; wings ; feathers ; beak ; crop ; lays_eggs ; ... $) 108 254 T
1 F
1.08 (In accordance with the remarks in Section 2.2 about the relationship between sensory data and) 72 230 P
0.01 (stored knowledge, this kind of representation seems to reflect empirical input much more directly) 72 214 P
0.05 (than a set of separate Prolog clauses. With a representation of knowledge like this, there seems to) 72 198 P
0.25 (be a better chance of integrating logic with other aspects of cognition such as pattern recognition) 72 182 P
(or learning.) 72 166 T
2.14 (With this kind of description of a typical bird, we may suppose that, for each \324field\325 in the) 72 140 P
0.98 (description \050an alphabetic symbol followed by \324) 72 124 P
4 F
1.09 (;) 307.87 124 P
1 F
0.98 (\325\051, there are zero or more other patterns which) 311.21 124 P
1.65 (expand the information for that field, in the same way that symbols in a rewrite rule may be) 72 108 P
-0.1 (expanded by other rewrite rules. For example, typical \050pet\051 names for birds may be specified such) 72 92 P
(as \324) 72 76 T
4 F
(name Tweety ;) 88.99 76 T
1 F
(\325, \324) 167.69 76 T
4 F
(name Chirpy ;) 181.68 76 T
1 F
(\325 and so on.) 256.37 76 T
-0.64 (Given patterns like these, the existence of a particular bird, Tweety, may be specified with a pattern) 72 50 P
(like this:) 72 34 T
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72 18 540 720 R
7 X
V
4 F
0 X
(bird Tweety $) 108 712 T
1 F
0.54 (And the inference that Tweety can fly may be made from what appears to be the best alignment) 72 688 P
(and unification amongst these patterns, shown in Fig. 9.) 72 672 T
(-------------------------------------------------------------------------) 108 646 T
(Please insert Fig. 9 about here) 144 631 T
(-------------------------------------------------------------------------) 108 616 T
0.08 (Fig. 9. Alignment and unification of patterns related to birds. The ellipsis \050\324...\325\051 represents) 108 592 P
(other characteristics of birds which cannot be listed in the space available.) 108 578 T
(There are \050at least\051 three points of interest here:) 72 554 T
(\245) 72 528 T
0.04 (With this version of the relationship between being a bird and the ability to fly, there is no) 108 528 P
0.01 (need to repeat the \324variable\325, \324) 108 512 P
4 F
0.01 (name ;) 252.64 512 P
1 F
0.01 (\325, as is done with the corresponding variable, \324X\325, in) 289.33 512 P
0.32 (the Prolog version. The possibility that a bird has a name is just one of the many features) 108 496 P
(of any bird which may be stored once in a pattern alongside those other features.) 108 480 T
(\245) 72 454 T
0.67 (The probability that Tweety can fly is not the only inference that may be made from the) 108 454 P
0.17 (alignment and unification shown in Fig. 9. We may also conclude that Tweety has wings,) 108 438 P
0.7 (feathers and all the other characteristics of birds. Given the information that Tweety is a) 108 422 P
-0.56 (bird, it seems much more natural to make all those inferences in one operation than to make) 108 406 P
0.06 (each inference separately as seems to be required with some LS approaches to this kind of) 108 390 P
-0.59 (problem. Notice that this observation is independent of whether or not the operation is done) 108 374 P
0.41 (in series or in parallel. This is because completion of the unification \050by serial or parallel) 108 358 P
1.41 (processes\051 leads at once to the inference that Tweety has all the attributes listed in the) 108 342 P
(pattern describing the concept of a bird.) 108 326 T
(\245) 72 300 T
0.08 (The given inference is an example of) 108 300 P
2 F
0.08 (inheritance) 288.41 300 P
1 F
0.08 ( of the attributes of a class by an instance) 343.07 300 P
0.28 (of the class. \050In other cases, of course, the attributes of a class may be inherited by a sub-) 108 284 P
3.39 (class of the given class\051. This form of reasoning has been incorporated in several) 108 268 P
-0.6 (programming languages \050originally Simula \050Birtwistle) 108 252 P
2 F
-0.6 (et al) 369.33 252 P
1 F
-0.6 (., 1973\051, followed by Smalltalk,) 389.73 252 P
0.73 (C++ and others\051 and has been the subject of other research more closely related to logic) 108 236 P
(\050see, for example, \050Horty, 1994\051\051.) 108 220 T
0.81 (This last point will be picked up again in Section 7 where ICMAUS proposals like the one just) 72 194 P
(outlined are developed in the context of nonmonotonic and uncertain reasoning.) 72 178 T
0 F
(6.3.) 72 152 T
(Chains of reasoning) 108 152 T
1 F
1.16 (Before leaving this section, it is appropriate to mention the idea which is discussed more fully) 72 128 P
3.25 (elsewhere \050Wolff, 1996, submitted, a\051, that ICMAUS may accommodate the \324chaining\325 of) 72 112 P
1.53 (reasoning which is a familiar feature of everyday reasoning and, indeed, of LS reasoning and) 72 96 P
(Prolog.) 72 80 T
0.44 (If A implies B and B implies C then we may conclude that A implies C - and likewise when the) 72 54 P
-0.2 (chains are longer. In keeping with the discussion in Section 6.2, above, each of the implications in) 72 38 P
0.37 (any chain like this may be represented as a simple association of symbols in a pattern: \324) 72 22 P
4 F
0.41 (A B) 497.88 22 P
1 F
0.37 (\325, \324) 517.63 22 P
4 F
0.41 (B) 531.99 22 P
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1.18 (C) 72 712 P
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1.06 (\325, \324) 80.66 712 P
4 F
1.18 (smoke fire) 95.71 712 P
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1.06 (\325, \324) 152.24 712 P
4 F
1.18 (fire burn) 167.29 712 P
1 F
1.06 (\325, and so on. It is not hard to see that such patterns may be linked) 212.48 712 P
(together by ICMAUS and chains of reasoning may thereby be modelled.) 72 696 T
0.72 (By contrast with the relatively rigid chaining embodied in Prolog, the ICMAUS framework can) 72 670 P
0.95 (accommodate more flexible linking of patterns which is, perhaps, more in keeping with human) 72 654 P
(thinking. Relevant examples are presented in \050Wolff, 1996, submitted, a\051.) 72 638 T
0 F
(7.) 72 602 T
(NONMONOTONIC AND UNCERTAIN REASONING) 108 602 T
1 F
2.4 (Although concepts associated with propositional logic and first-order logic provide a tightly) 72 573 P
-0.71 (organised and internally consistent view of logic, they seem to lack the flexibility needed to capture) 72 557 P
0.15 (the relaxed style of everyday reasoning where uncertainty is the rule and where something which) 72 541 P
0.48 (is true in one context may easily be overridden in another. This section describes in outline how) 72 525 P
-0.18 (the ICMAUS concepts may be applied to three ideas which seem to be closer in spirit to this more) 72 509 P
0.1 (flexible kind of reasoning: inheritance of attributes, abductive reasoning and inductive reasoning.) 72 493 P
0 (Each of these are large subjects so, in the space available, only a sketch of the possibilities can be) 72 477 P
(given.) 72 461 T
0 F
(7.1.) 72 435 T
(Inheritance and default reasoning) 108 435 T
1 F
3.98 (This section considers how the ICMAUS framework may accommodate three aspects of) 72 411 P
2.81 (inheritance: the inheritance of attributes through two or more levels, cross classification or) 72 395 P
0.89 (\324multiple inheritance\325, and the way in which default attributes may be negated or overridden in) 72 379 P
(specific contexts.) 72 363 T
2 F
(7.1.1.) 72 337 T
(Inheritance Through Two or More Levels) 108 337 T
1 F
-0.71 (In the example given in Section 6.2, Tweety inherits attributes directly from the single class \324birds\325.) 72 313 P
0.61 (But it is widely recognised that classes may form hierarchies which may be arbitrarily deep and) 72 297 P
0.01 (that a given entity may inherit attributes from all levels in such a hierarchy. In Fig. 10, the pattern) 72 281 P
0.61 (at the top which describes Tweety as a swallow is aligned with other patterns so that, as before,) 72 265 P
-0.24 (Tweety may be seen to inherit the general characteristics of birds and, in addition, Tweety inherits) 72 249 P
0.21 (characteristics which are specific to swallows: that they can fly long distances and that they have) 72 233 P
(pointed wings.) 72 217 T
(-------------------------------------------------------------------------) 108 191 T
(Please insert Fig. 10 about here) 144 176 T
(-------------------------------------------------------------------------) 108 161 T
0.34 (Fig. 10. Alignment and unification of patterns modelling the inheritance of attributes in a) 108 137 P
(two-level hierarchy.) 108 123 T
2.7 (It should be clear from this example how ICMAUS could accommodate the inheritance of) 72 99 P
(attributes through class hierarchies which are arbitrarily deep.) 72 83 T
2 F
(7.1.2.) 72 57 T
(Cross-Classification or \324Multiple Inheritance\325) 108 57 T
1 F
1.63 (A given entity may belong in two or more classes which are not sub-classes, one of another.) 72 33 P
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1.08 (Amongst living things, the classes \324male\325 and \324female\325 cut across classes such as \324bird\325, \324fish\325,) 72 712 P
3.17 (\324mammal\325 and so on. The way in which ICMAUS may accommodate this kind of cross-) 72 696 P
-0.13 (classification or \324multiple inheritance\325 is suggested by the example in Fig. 11 where a living thing) 72 680 P
0.29 (called \324Chirpy\325 \050let\325s give Tweety a rest!\051 inherits attributes from a pattern representing the class) 72 664 P
(\324male\325 and, at the same time, inherits attributes from the class \324bird\325.) 72 648 T
(-------------------------------------------------------------------------) 108 622 T
(Please insert Fig. 11 about here) 144 607 T
(-------------------------------------------------------------------------) 108 592 T
(Fig. 11. Alignment and unification of patterns modelling multiple inheritance.) 108 568 T
1.19 (This example suggests that gender characteristics are totally independent of classifications like) 72 544 P
-0.31 (\324bird\325, \324fish\325 and so on, but of course we know that they combine in complex and subtle ways. The) 72 528 P
-0.71 (example in Fig. 10 suggests how attributes may combine in an ICMAUS framework but more work) 72 512 P
(is needed in this area.) 72 496 T
2 F
(7.1.3.) 72 470 T
(Negation of a Default Attribute) 108 470 T
1 F
1.5 (The chief role of the overworked \324Tweety\325 in discussions like these is to be a bird which, by) 72 446 P
0.37 (default, has the ability to fly but loses this ability if we learn that Tweety is a penguin. Although) 72 430 P
-0.23 (this kind of adjustment of our expectations is a familiar part of everyday reasoning, it is a problem) 72 414 P
1.63 (for LS which has the property of) 72 398 P
2 F
1.63 (monotonicity) 243.03 398 P
1 F
1.63 (: any conclusion which is valid in one state of) 305.7 398 P
(knowledge remains valid whatever new knowledge is acquired \050Ginsberg, 1994\051.) 72 382 T
1.77 (It seems that ICMAUS may provide a framework in which the nonmonotonicity of everyday) 72 356 P
1.74 (reasoning may be accommodated. In Fig. 12, the second pattern in the alignment describes a) 72 340 P
1.05 (penguin as a bird in which the attribute \324canfly\325 has been qualified with \324no\325 \050and the attribute) 72 324 P
0.14 (\324wings\325 has been qualified with \324stubby\325\051. Given the information that Tweety is a penguin \050in the) 72 308 P
0.08 (top pattern\051, it seems that alignment and unification of patterns achieves the effect of negating the) 72 292 P
0.67 (expectation that Tweety would have the ability to fly \050and also leads us to expect that Tweety\325s) 72 276 P
(wings would be \324stubby\325\051.) 72 260 T
(-------------------------------------------------------------------------) 108 234 T
(Please insert Fig. 12 about here) 144 219 T
(-------------------------------------------------------------------------) 108 204 T
(Fig. 12. Alignment and unification of patterns modelling the negation of an attribute.) 108 180 T
0.41 (Assuming that the pattern describing birds remains in the store of knowledge, then knowing that) 72 156 P
-0.36 (penguins cannot fly should not disturb the expectations, derived by alignment and unification with) 72 140 P
(the pattern for birds, that other birds - swallows, guillemots, parrots and so on - can fly.) 72 124 T
0.21 (The similarity between Fig. 12 and Fig. 10 suggests that negation of an attribute does not require) 72 98 P
0.26 (special status and may be treated in exactly the same way as any other qualifier of an attribute in) 72 82 P
(an inheritance hierarchy.) 72 66 T
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(7.2.) 72 712 T
(Abductive reasoning) 108 712 T
1 F
0.12 (If we know that something can fly, it is not unreasonable to think that it might be a bird. It could,) 72 688 P
0.49 (of course, be an aeroplane, a bat, a flying squirrel and so on but, given the world as we know it,) 72 672 P
(there is a good chance that it would be a bird.) 72 656 T
0.5 (Given the LS implication that all birds can fly, this is back-to-front \324abductive\325 reasoning which) 72 630 P
0.08 (does not fit easily into an LS framework. The suggestion here is that ICMAUS may offer a better) 72 614 P
-0.6 (home for this kind of reasoning which, like nonmonotonic reasoning, is a prominent feature of how) 72 598 P
(we ordinarily think.) 72 582 T
-0.26 (To avoid an) 72 556 P
2 F
-0.26 (ad hoc) 131.54 556 P
1 F
-0.26 ( treatment for inductive reasoning within the ICMAUS framework, it would be) 163.6 556 P
-0.48 (natural to express the association between being a bird and being able to fly in a simple pattern like) 72 540 P
-0.16 (the one shown in Fig. 9 which includes all the other attributes normally associated with birds. The) 72 524 P
-0.23 (alignment and unification in Fig. 13 suggests how, if we know only that \324Tweety\325 can fly, we may) 72 508 P
-0.3 (infer that he, she or it is a bird - and we may also infer that this entity has wings, feathers and other) 72 492 P
0.31 (attributes of birds. Of course, with this example, we must discount any other knowledge we may) 72 476 P
(happen to have about the kinds of names that birds may be given.) 72 460 T
(-------------------------------------------------------------------------) 108 434 T
(Please insert Fig. 13 about here) 144 419 T
(-------------------------------------------------------------------------) 108 404 T
(Fig. 13. An abductive inference as alignment and unification of patterns.) 108 380 T
0.5 (The sceptical reader may object that the abductive inference modelled in Fig. 13 says more than) 72 356 P
0.44 (we are entitled to extract from the original implication that \322All birds can fly\323. Contrary to what) 72 340 P
(seems reasonable, the right-to-left inference suggests that anything which can fly is a bird!) 72 324 T
-0.28 (The suggested answer to this objection is a reminder that all inferences in an ICMAUS framework) 72 298 P
3.31 (are probabilistic and the observation that the probability of any inference depends on the) 72 282 P
(knowledge which is stored.) 72 266 T
0.38 (If the system only knows one pattern and that pattern shows an association between being a bird) 72 240 P
-0.17 (and having the ability to fly then it would indeed infer with a high probability that anything which) 72 224 P
0.4 (can fly is a bird. A little reflection suggests that if our knowledge was equally restricted then we) 72 208 P
(would probably reach the same conclusion.) 72 192 T
-0.45 (But if a more realistic range of patterns is stored, including the facts that aeroplanes, bats and some) 72 166 P
1.39 (kinds of squirrels can fly, then the knowledge that some entity can fly should lead to a set of) 72 150 P
2 (alternative alignments highlighting the things which are known to be able to fly but without) 72 134 P
(ascribing a high probability to any one of these inferences.) 72 118 T
1.07 (In summary, the suggestion is that the asymmetry in the normal LS formulation of implication) 72 92 P
0.09 (may, with some profit, be replaced by a system of inference in which knowledge is stored simply) 72 76 P
2.08 (as \324patterns\325 and that asymmetries associated with inferences, if they exist, emerge from the) 72 60 P
-0.48 (probabilities associated with each alignment which, in turn, depend on the constellation of patterns) 72 44 P
(which is stored at any one time.) 72 28 T
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2.85 (The potential profit from adopting this standpoint is a uniform treatment of deduction and) 72 712 P
0.34 (abduction within a framework which apparently has potential to assimilate other aspects of logic) 72 696 P
(and computing.) 72 680 T
1.54 (The foregoing is, of course, only a sketch of how abductive reasoning may be modelled with) 72 654 P
(ICMAUS. Future work will seek to give quantitative precision to the ideas outlined here.) 72 638 T
0 F
(7.3.) 72 612 T
(Inductive reasoning) 108 612 T
1 F
0.14 (In Wolff \0501996\051, it is suggested that there is a closer connection between inductive reasoning and) 72 588 P
2.37 (deductive reasoning than has traditionally been thought. Information may be compressed by) 72 572 P
-0.2 (pattern matching, unification and search and, as a by-product, recurrent patterns may be identified) 72 556 P
2.72 (\050Wolff, 1995b\051. Then by partial pattern recognition \050which is also IC by pattern matching,) 72 540 P
1.3 (unification and search\051, the observed parts of a recurrent pattern may lead to the prediction or) 72 524 P
(inference of the unseen parts.) 72 508 T
1.17 (It should be clear from the foregoing discussion that both these processes of pattern matching,) 72 482 P
-0.15 (unification and search may be modelled as ICMAUS. The identification of recurrent patterns may) 72 466 P
0.26 (be seen as unsupervised inductive learning. Examples are described in Wolff \0501991, 1995b\051. The) 72 450 P
0.14 (partial matching and merging of patterns, with a corresponding inference of the unmatched parts,) 72 434 P
(is illustrated by the examples described in Sections 4.1 and 6.2, above \050see also Wolff, 1996\051.) 72 418 T
0 F
(8.) 72 382 T
(CONCLUSION) 108 382 T
1 F
0.62 (This article has described in outline how IC by) 72 353 P
2 F
0.62 (multiple alignment) 305.25 353 P
1 F
0.62 ( \050with unification and search\051) 396.21 353 P
0.76 (may accommodate varied aspects of logic in its \324standard\325 form and at the same time offers the) 72 337 P
0.98 (potential for simplification and rationalisation in some areas of logic. The proposed framework) 72 321 P
0.8 (seems to be particularly well suited to the relatively flexible kind of reasoning which we use in) 72 305 P
2.46 (everyday thinking which is characteristically nonmonotonic and which rarely has the all-or-) 72 289 P
(nothing certainty of standard logic.) 72 273 T
0.92 (These proposals originate in a programme of research which seeks to integrate a wide range of) 72 247 P
-0.11 (concepts in computing and cognition within a framework of information compression by multiple) 72 231 P
(alignment, unification and search.) 72 215 T
0.86 (There are many questions still to be answered about these proposals. I hope that what has been) 72 189 P
0.04 (presented has shown sufficient promise to encourage other researchers to develop these ideas and) 72 173 P
(explore their potential applications.) 72 157 T
0 F
(ACKNOWLEDGEMENTS) 72 126 T
1 F
0.46 (I am grateful to John Hornsby of the School of Electronic Engineering and Computing Systems,) 72 102 P
1.9 (Bangor, for advice on mathematical formulae and to anonymous referees for comments on a) 72 86 P
(previous version of the paper.) 72 70 T
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1 F
-0.69 (Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. \050eds.\051 \0501992\051) 72 688 P
2 F
-0.69 (Handbook of Logic in Computer) 386.74 688 P
(Science) 108 672 T
1 F
(, Vols. 1-4. Clarendon Press, Oxford.) 144.65 672 T
1.6 (Allison, L. and Yee, C. N. \0501990\051 Minimum message length encoding and the comparison of) 72 646 P
(macromolecules.) 108 630 T
2 F
(Bulletin of Mathematical Biology) 192.65 630 T
1 F
(, 52 \0503\051, 431-453.) 352.98 630 T
2.89 (Allison, L., Wallace, C. S. and Yee, C. N. \0501992\051 Finite-state models in the alignment of) 72 604 P
(macromolecules.) 108 588 T
2 F
(Journal of Molecular Evolution) 192.65 588 T
1 F
(, 35, 77-89.) 344.98 588 T
0.31 (Allison, L. and Wallace, C. S. \0501994\051 The posterior probability distribution of alignments and its) 72 562 P
0.7 (application to parameter estimation of evolutionary trees and to optimization of multiple) 108 546 P
(alignments.) 108 530 T
2 F
(Journal of Molecular Biology) 166.67 530 T
1 F
(, 39, 418-430.) 309.66 530 T
-0.33 (Attneave F \0501954\051 Informational aspects of visual perception.) 72 504 P
2 F
-0.33 (Psychological Review) 368.33 504 P
1 F
-0.33 (, 61, 183-193.) 473.66 504 P
0.93 (Barlow, H. B. \0501969\051 Trigger features, adaptation and economy of impulses. In K. N. Leibovic) 72 478 P
(\050ed\051,) 108 462 T
2 F
(Information Processing in the Nervous System) 133.32 462 T
1 F
(. Springer, New York, pp. 209-230.) 356.3 462 T
0.17 (Barton, G. J. \0501990\051 Protein multiple sequence alignment and flexible pattern matching.) 72 436 P
2 F
0.17 (Methods) 498.67 436 P
(in Enzymology) 108 420 T
1 F
(, 183, 403-428.) 178.99 420 T
0 (Birtwistle, G. M., Dahl, O-J., Myhrhaug, B. and Nygaard, K. \0501973\051) 72 394 P
2 F
0 (Simula Begin) 402.67 394 P
1 F
0 (. Van Nostrand) 467.01 394 P
(Reinhold, New York.) 108 378 T
-0.33 (Chan, S. C., Wong, A. K. C. and Chiu, D. K. Y. \0501992\051 A survey of multiple sequence comparison) 72 352 P
(methods.) 108 336 T
2 F
(Bulletin of Mathematical Biology) 154.67 336 T
1 F
(, 54 \0504\051, 563-598.) 315 336 T
-0.42 (Cook, C. M. and Rosenfeld, A. \0501976\051 Some experiments in grammatical inference. In J. C. Simon) 72 310 P
(\050ed.\051,) 108 294 T
2 F
(Computer Oriented Learning Processes) 136.32 294 T
1 F
(. Noordhoff, Leyden, pp 157-174.) 327.97 294 T
(Copi, I. M. \0501986) 72 268 T
2 F
(\051 Introduction to Logic) 156 268 T
1 F
(. Seventh Edition. MacMillan, New York.) 265.67 268 T
-0.71 (Davis, M. \0501993\051 First order logic. In Gabbay, Hogger and Robinson \050eds.\051 \0501993-4\051, Vol. 1, 31-65.) 72 242 P
2.84 (Day, W. H. E. and McMorris, F. R. \0501992\051 Critical comparison of consensus methods for) 72 216 P
(molecular sequences.) 108 200 T
2 F
(Nucleic Acids Research) 213.64 200 T
1 F
(, 20 \050) 327.61 200 T
2 F
(5) 352.61 200 T
1 F
(\051, 1093-1099.) 358.61 200 T
-0.4 (Eisinger, N. and Ohlbach, H. J. \0501993\051 Deduction systems based on resolution. In Gabbay, Hogger) 72 174 P
(and Robinson \050eds.\051 \0501993-4\051, Vol. 1, 183-271.) 108 158 T
0.02 (Gabbay, D. M., Hogger, C. J. and Robinson, J. A. \050eds.\051 \0501993-4) 72 132 P
2 F
0.02 (\051 Handbook of Logic in Artificial) 381.75 132 P
(Intelligence and Logic Programming) 108 116 T
1 F
(. Vols. 1-3. Clarendon Press, Oxford.) 286.32 116 T
1.37 (Gammerman, A. J. \0501991\051. The representation and manipulation of the algorithmic probability) 72 90 P
0.91 (measure for problem solving.) 108 74 P
2 F
0.91 (Annals of Mathematics and Artificial Intelligence) 255.96 74 P
1 F
0.91 (, 4, 281-) 498.18 74 P
(300.) 108 58 T
0.03 (Ginsberg, M. L. \0501994\051 AI and nonmonotonic reasoning. In Gabbay, Hogger and Robinson \050eds.\051) 72 32 P
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7 X
V
0 X
(\0501993-4\051, Vol. 3, 1-33.) 108 712 T
1.29 (Horty, J. F. \0501994\051 Some direct theories of nonmonotonic inheritance. In Gabbay, Hogger and) 72 686 P
(Robinson \050eds.\051 \0501993-4\051, Vol. 3, 111-187.) 108 670 T
4.89 (Huffman, D. A. \0501952\051 A method for the construction of minimum-redundancy codes.) 72 644 P
2 F
(Proceedings of the IRE) 108 628 T
1 F
(, 40, 1098-1101.) 219.65 628 T
0.62 (Klir, G. and Yuan, B. \0501995\051) 72 602 P
2 F
0.62 (Fuzzy sets and fuzzy logic: theory and applications) 217.05 602 P
1 F
0.62 (. Prentice-Hall,) 466.39 602 P
(Upper Saddle River, NJ.) 108 586 T
0.42 (Klop, J. W. \0501992\051 Term rewriting systems. In Abramsky., Gabbay and Maibaum. \050eds.\051 \0501992\051,) 72 560 P
(Vol. 2, 1-116.) 108 544 T
(Kosko, B. \0501994\051) 72 518 T
2 F
(Fuzzy thinking: the new science of fuzzy logic) 158.33 518 T
1 F
(. Flamingo, London.) 377.3 518 T
1.23 (Li, M. and Vit\207nyi, P. \0501993\051 An introduction to Kolmogorov complexity and its applications.) 72 492 P
(Springer-Verlag, Heidelberg.) 108 476 T
0.26 (Rissanen, J. \0501987\051 Stochastic complexity.) 72 450 P
2 F
0.26 (Journal of the Royal Statistical Society B,) 280.62 450 P
1 F
0.26 (49\0503) 485.75 450 P
0 F
0.26 (\051) 507.75 450 P
1 F
0.26 (, 223-) 511.74 450 P
(239.) 108 434 T
0.65 (Solomonoff, R. J \0501964\051 A formal theory of inductive inference, Parts I and II.) 72 408 P
2 F
0.65 (Information and) 461.02 408 P
(Control) 108 392 T
1 F
(, 7, 1-22 and 224-254.) 145.34 392 T
0.11 (Storer, J. A. \0501988\051) 72 366 P
2 F
0.11 (Data Compression Methods and Theory.) 168.11 366 P
1 F
0.11 (Computer Science Press, Rockville,) 367.34 366 P
(Maryland.) 108 350 T
1.48 (Taylor, W. R. \0501988\051 Pattern matching methods in protein sequence comparison and structure) 72 324 P
(prediction.) 108 308 T
2 F
(Protein Engineering) 162.66 308 T
1 F
(, 2 \0502\051, 77-86.) 260.99 308 T
(Von B\216k\216sy G \0501967\051) 72 282 T
2 F
(Sensory Inhibition) 180.65 282 T
1 F
(, Princeton University Press, Princeton, NJ.) 268.98 282 T
1.08 (Wallace, C. S. and Boulton, D. M. \0501968\051 An information measure of classification.) 72 256 P
2 F
1.08 (Computer) 492 256 P
(Journal,) 108 240 T
1 F
(11, 185-195.) 151.33 240 T
-0.26 (Wallace, C. S. and Freeman, P. R. \0501987\051 Estimation and inference by compact coding.) 72 214 P
2 F
-0.26 (Journal of) 490.59 214 P
(the Royal Statistical Socieity B,) 108 198 T
1 F
(49\0503\051, 240-252.) 262 198 T
3.04 (Watanabe, S. \0501972\051 Pattern recognition as information compression. In S. Watanabe \050ed.\051) 72 172 P
2 F
(Frontiers of Pattern Recognition) 108 156 T
1 F
(. Academic Press, New York.) 265.67 156 T
(Watanabe, S. \0501985\051) 72 130 T
2 F
(Pattern Recognition: Human and Mechanical) 173.64 130 T
1 F
(. Wiley, New York.) 393.61 130 T
(Wolff, J. G. \050submitted, a\051 Multiple alignment and computing.) 72 104 T
-0.14 (Wolff, J. G. \050submitted, b\051 Parsing as information compression by multiple alignment, unification) 72 78 P
(and search.) 108 62 T
0.95 (Wolff, J. G. \0501996\051 Learning and reasoning as information compression by multiple alignment,) 72 36 P
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-0.73 (unification and search. In A. Gammerman \050ed.\051,) 108 712 P
2 F
-0.73 (Computational Learning and Probabilistic) 336.5 712 P
-0.14 (Reasoning) 108 696 P
1 F
-0.14 (, Wiley, New York, 67-85. \050This is a revised version of the paper \324Learning and) 158.66 696 P
1.5 (reasoning as compression by multiple alignment and unification,\325 presented at Applied) 108 680 P
0.46 (Decision Technologies \32595 \050ADT \32595\051, Brunel University, London, April 1995, Stream 1) 108 664 P
(\050Computational Learning and Probabilistic Reasoning\051, 1995, 223-236.\051) 108 648 T
0.47 (Wolff, J. G. \0501995a\051 Computing as compression: an overview of the SP theory and system.) 72 622 P
2 F
0.47 (New) 518.66 622 P
(Generation Computing) 108 606 T
1 F
(, 13, 187-214.) 219 606 T
-0.33 (Wolff, J. G. \0501995b\051 Computing as compression: SP20) 72 580 P
2 F
-0.33 (. New Generation Computing) 333 580 P
1 F
-0.33 (, 13: 215-241.) 473.33 580 P
0.09 (Wolff, J. G. \0501994\051 A scaleable technique for best-match retrieval of sequential information using) 72 554 P
(metrics-guided search.) 108 538 T
2 F
(Journal of Information Science,) 219.64 538 T
1 F
(20 \0501\051, 16-28.) 375.29 538 T
0.94 (Wolff, J. G. \0501993\051 Computing, cognition and information compression.) 72 512 P
2 F
0.94 (AI Communications,) 430.45 512 P
1 F
0.94 ( 6) 530.06 512 P
(\0502\051, 107-127.) 108 496 T
(Wolff, J. G. \0501991\051) 72 470 T
2 F
(Towards a Theory of Cognition and Computing) 166.98 470 T
1 F
(. Ellis Horwood, Chichester.) 395.68 470 T
0.21 (Wolff, J. G. \0501990\051 Simplicity and power: some unifying ideas in computing.) 72 444 P
2 F
0.21 (Computer Journal,) 448.46 444 P
1 F
(33 \0506\051, 518-534. \050Reproduced in Wolff \0501991\051, Chapter 4\051.) 108 428 T
3.36 (Zipf, G. K. \0501949\051) 72 402 P
2 F
3.36 (Human Behaviour and the Principle of Least Effort) 176.44 402 P
1 F
3.36 (. Addison-Wesley,) 446.98 402 P
(Cambridge, Mass.) 108 386 T
FMENDPAGE
%%EndPage: "32" 33
%%Page: "33" 33
612 792 0 FMBEGINPAGE
72 747 540 765 R
7 X
0 K
V
1 12 Q
0 X
(- 33 -) 293 757 T
72 18 540 720 R
7 X
V
0 F
0 X
(FIGURE CAPTIONS) 72 712 T
1 F
(Fig. 1) 72 688 T
(. A scene in which some objects are partially obscured by other objects.) 100.01 688 T
(Fig. 2. Examples of truth table definitions of logical operators.) 72 662 T
2.13 (Fig. 3. Alignment and unification of an \324input\325 pattern with part of the LS definition of the) 72 636 P
(IMPLIES operation. Symbols which are the result of unification are shown in bold type.) 72 620 T
0.04 (Fig. 4. Alignment and unification of a \324variable\325 with a \324value\325. The ellipses \050\324) 72 594 P
4 F
0.04 (...) 447.44 594 P
1 F
0.04 (\325\051 represent other) 457.44 594 P
(symbols which may lie to the left and right of \324) 72 578 T
4 F
(x ;) 297.98 578 T
1 F
(\325.) 310.66 578 T
0.94 (Fig. 5. An alignment amongst patterns comprising an alphabetic string \050\324) 72 552 P
4 F
1.04 (a b c x y z d e f) 430.69 552 P
1 F
0.94 (\325\051, a) 519.74 552 P
-0.41 (pattern containing a variable \050\324) 72 536 P
4 F
-0.46 (1 a b c) 218.31 536 P
1 F
-0.41 (\244) 255.83 536 P
4 F
-0.46 ( ; d e f) 261.83 536 P
1 F
-0.41 ( #\325\051 and patterns which constitute a \324type definition\325) 293.35 536 P
(of the variable.) 72 520 T
1.54 (Fig. 6. Alignment and unification of patterns for the assignment of an \324actual parameter\325 to a) 72 494 P
(\324formal parameter\325.) 72 478 T
-0.15 (Fig. 7. An alignment between a composite formula and some of the patterns defining the structure) 72 452 P
(of formulae.) 72 436 T
-0.2 (Fig. 8. An alignment of patterns showing how a \324value\325 may be \324passed\325 between two instances of) 72 410 P
(a \324variable\325.) 72 394 T
0.67 (Fig. 9. Alignment and unification of patterns related to birds. The ellipsis \050\324...\325\051 represents other) 72 368 P
(characteristics of birds which cannot be listed in the space available.) 72 352 T
-0.57 (Fig. 10. Alignment and unification of patterns modelling the inheritance of attributes in a two-level) 72 326 P
(hierarchy.) 72 310 T
(Fig. 11. Alignment and unification of patterns modelling multiple inheritance.) 72 284 T
(Fig. 12. Alignment and unification of patterns modelling the negation of an attribute.) 72 258 T
(Fig. 13. An abductive inference as alignment and unification of patterns.) 72 232 T
FMENDPAGE
%%EndPage: "33" 34
%%Page: "34" 34
612 792 1 FMBEGINPAGE
200.73 496.48 146.89 456.05 146.89 361.69 265.92 344.36 265.92 427.16 5 Y
5 X
0 K
V
3 H
2 Z
0 X
N
204.2 496.6 485.19 548.33 556.27 480.68 265.57 420.99 4 L
9 X
V
0 X
N
556.27 479.71 556.27 410.69 269.07 345.51 3 L
N
243.47 487.58 243.47 514.27 265.99 514.27 265.99 470.43 4 Y
1 X
V
0 X
N
255.67 513.38 M
262.29 530.38 206.89 517.03 210.07 529.33 D
214.3 545.71 150.61 527.53 160.59 543.29 D
N
326.93 359.61 326.93 410.56 361.02 418.2 361.02 367.25 4 Y
3 X
V
4 H
0 X
N
421.85 439.49 421.85 408.88 528.47 434.08 528.47 462.89 4 Y
4 X
V
0 X
N
363.83 307.74 615.11 383.76 2 L
6 X
V
3 H
0 X
N
563.66 393.63 553.49 366.21 2 L
6 X
V
0 X
N
363.83 308.21 374.68 323.32 327.38 343.62 317.3 328.04 4 Y
6 X
V
0 X
N
375.46 324.27 449.13 360.15 390.97 378.56 327.38 343.15 4 L
6 X
V
0 X
N
450.49 361.1 464.78 376.93 406.05 395.73 390.97 379.89 4 L
9 X
V
0 X
N
563.9 405.95 563.9 392.26 2 L
6 X
V
0 X
N
448.35 360.15 563.9 392.26 2 L
6 X
V
0 X
N
395.62 317.59 M
403.04 306.39 427.67 308.77 437.12 313.02 D
448.43 318.08 451.45 325.91 449.19 332.77 D
2 X
V
0 X
N
553.82 364.93 M
561.23 353.13 585.87 355.64 595.31 360.12 D
606.63 365.45 609.65 373.7 607.38 380.92 D
2 X
V
0 X
N
464.2 377.28 564.23 405.58 499.09 419.99 404.88 395.43 4 L
6 X
V
0 X
N
571.57 406.81 517.33 419.47 502.07 419.99 563.66 405.76 4 Y
6 X
V
0 X
N
571.01 406.28 615.11 383.15 554.06 365.68 563.66 391.89 563.66 404.23 5 Y
3 X
V
0 X
N
449.6 361.13 563.66 392.05 552.98 365.32 364.2 309.25 375.43 324.97 5 Y
3 X
V
0 X
N
328.6 343.53 374.37 323.49 2 L
6 X
V
0 X
N
451.22 363.04 464.78 377.81 563.1 404.7 562.53 392.05 4 Y
2 X
V
0 X
N
464.64 377.15 463.08 338.9 2 L
6 X
V
0 X
N
515.82 391.79 516.59 354.01 2 L
6 X
V
0 X
N
497.88 581.71 M
478.14 587.75 428.98 569.46 461.48 560.13 D
484.45 553.54 462.7 547.74 457.07 547.61 D
432.36 547.01 405.28 520.53 436.12 504.17 D
441.35 501.39 470.87 507.41 459.28 498.62 D
449.21 490.99 429.91 476.09 450.46 464.87 D
462.1 458.5 483.55 461.08 498.98 465.95 D
4 X
V
0 X
N
575.79 550.13 M
593.76 545.95 642.47 539.5 620.34 517.55 D
611.31 508.59 644.65 504.41 646 492.49 D
647.49 479.36 632.35 473.23 619.37 472.49 D
610.16 471.97 601.15 469.6 593.02 465.76 D
4 X
V
0 X
N
484.89 566.83 M
466.67 567.71 450.29 593.37 465.86 610.13 D
473.02 617.83 479.19 598.57 482.71 614.45 D
485.98 629.16 495.78 653.07 499.66 646.19 D
507.69 631.97 517.66 652.62 520.83 653.58 D
535.4 658 544.63 631.96 554.65 639.27 D
566.97 648.25 590.79 633.13 592.29 616.12 D
592.77 610.68 603.14 615.66 606.77 606.09 D
613.43 588.56 576.47 587.37 592 580.88 D
603.67 575.99 609.12 554.57 592.7 553.47 D
5 X
V
0 X
N
527.17 454.9 M
520.37 441.84 499.07 428.79 488.91 444.04 D
486.16 448.17 494.3 463.4 482.07 458.86 D
468.88 453.97 445.22 474.45 460.22 486.13 D
474.28 497.08 454.98 490.92 453.17 492.77 D
437.44 508.9 439.9 532.5 450.51 545.11 D
461.41 558.07 481.61 536.28 483.48 550.13 D
486.7 574.05 513.13 600.6 531.4 593.92 D
532.8 593.41 536.92 600.15 542.67 604.78 D
556.7 616.08 577.73 602.05 581.16 582.71 D
582.23 576.64 566.89 571.54 580.72 569.16 D
593.2 567.01 604.29 543.25 589.18 533.6 D
589.22 533.63 563.16 530.09 581.43 531.2 D
595.85 532.08 638.42 522.32 626.32 494.16 D
621.04 481.87 584.2 486.66 604.68 477.14 D
618.16 470.88 614.03 442.07 599.05 443.16 D
589.29 443.86 584.97 437.4 577.2 432.74 D
558.78 421.7 550.64 470.16 541.97 458.24 D
6 X
V
0 X
N
541.97 458.24 M
532.65 451.58 515.38 453.83 516.28 435.68 D
518.32 394.52 520.24 346.72 487.7 315.48 D
486.87 314.68 470.9 303.31 480.66 300.38 D
489.79 297.63 501.87 327.97 506.64 314.56 D
509.37 306.88 490.83 281 499.68 284.2 D
513.41 289.13 514.37 319.27 527.17 311.02 D
532.32 307.7 526.01 289.13 536.33 290.84 D
549.93 293.08 533.32 320.23 545.49 313.31 D
551.82 309.71 576.09 286.82 570.15 302.86 D
564.08 319.3 547.64 342.82 547.97 366.35 D
548.42 398.19 537.61 430.36 544.08 461.58 D
1 X
V
0 X
N
515.89 526.74 M
501.62 536.6 492.91 522.45 485.59 515.06 D
480.87 510.29 469.07 515.2 471.44 504.18 D
473.98 492.41 487.94 498.67 487 484.13 D
486.58 477.57 477.29 476.86 486.3 473.15 D
493.93 470.01 492.34 484.13 500.39 486.24 D
507.08 487.99 518.62 472.64 519.52 482.46 D
520.16 489.38 505.86 498.95 518.01 501.36 D
532.81 504.29 528.95 523.42 517.3 525.9 D
5 X
V
0 X
N
583.54 508.36 M
570.25 497.5 554.42 530.89 548.36 510.03 D
546.86 504.87 555.88 495.99 545.49 493.52 D
536.21 491.31 538.31 472.46 548.31 475.96 D
560.95 480.38 564.76 452.44 577.2 464.57 D
583.07 470.29 568.87 478.77 578.61 482.04 D
585.68 484.42 601.1 492.41 593.13 502.51 D
589.85 506.67 585.55 510.11 580.72 509.19 D
4 X
V
0 X
N
518.71 573.52 M
515.22 571.74 499.61 563.14 515.89 564.26 D
528.45 565.13 525.65 549.15 532.8 547.22 D
542.41 544.62 533.57 559.19 544.78 561.96 D
547.31 562.59 559.3 552.55 556.26 563.49 D
554.33 570.44 547.98 582.05 540.56 573.39 D
533.99 565.72 529.05 587.51 521.53 576.86 D
3 X
V
0 X
N
465.91 377.81 406.58 395.74 2 L
4 H
N
90 450 2.55 2.03 353.61 388.58 G
90 450 2.55 2.03 353.61 388.58 A
264.91 345.38 272.5 346.9 2 L
N
200.46 496.86 266.83 420.61 2 L
3 H
N
1 18 Q
(Fig. 1) 92.25 693.38 T
92.25 266.62 699.75 705.38 C
92.25 266.62 699.75 687.38 C
92.25 266.62 699.75 705.38 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "34" 35
%%Page: "35" 35
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 2) 72 708 T
72 252 720 720 C
72 254 720 702 C
428 462 703 638 R
7 X
0 K
V
0.5 H
2 Z
0 X
N
428 603 703 603 2 L
N
428 567.33 703 567.33 2 L
N
428 531.67 703 531.67 2 L
N
428 496 703 496 2 L
N
608 638 608 462 2 L
N
518 638 518 462 2 L
N
92 462 367 638 R
7 X
V
0 X
N
1 24 Q
(Input 1) 104.34 612.22 T
(Input 2) 191.34 612.22 T
(Output) 283.33 612.22 T
(1) 161 576.59 T
(1) 248 576.59 T
(1) 338 576.59 T
(1) 162 540.59 T
(0) 249 540.59 T
(0) 339 540.59 T
(0) 161 507.59 T
(1) 248 507.59 T
(0) 338 507.59 T
(0) 161 471.59 T
(0) 248 471.59 T
(0) 338 471.59 T
(AND) 92 649 T
(Input 1) 441.34 612.22 T
(Input 2) 528.34 612.22 T
(Output) 620.33 612.22 T
(1) 498 576.59 T
(1) 585 576.59 T
(1) 675 576.59 T
(1) 499 540.59 T
(0) 586 540.59 T
(0) 676 540.59 T
(0) 498 507.59 T
(1) 585 507.59 T
(1) 675 507.59 T
(0) 498 471.59 T
(0) 585 471.59 T
(1) 675 471.59 T
(IMPLIES) 429 649 T
(Input) 325.09 351.22 T
(Output) 411.08 351.22 T
(1) 362.75 313.59 T
(0) 464.75 313.59 T
(0) 362.75 275.59 T
(1) 464.75 275.59 T
(NOT) 299.75 386 T
92 603 367 603 2 L
N
92 567.33 367 567.33 2 L
N
92 531.67 367 531.67 2 L
N
92 496 367 496 2 L
N
272 638 272 462 2 L
N
182 638 182 462 2 L
N
398.25 375 398.25 266 2 L
N
301.25 265 495.25 376 R
N
301.25 340 495.25 340 2 L
N
301.25 302 495.25 302 2 L
N
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "35" 36
%%Page: "36" 36
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 3) 72 708 T
72 252 720 720 C
72 252 720 702 C
1 24 Q
0 X
0 K
(0) 216 563.32 T
(1) 242.4 563.32 T
(;) 268.8 563.32 T
(;) 316.27 563.32 T
(0) 216 520.15 T
(1) 242.4 520.15 T
(;) 268.8 520.15 T
(;) 316.27 520.15 T
(1) 289.87 520.15 T
319 553.32 319 541.32 2 L
1 H
2 Z
N
272 553.32 272 541.32 2 L
N
247 553.32 247 541.32 2 L
N
221 553.32 221 541.32 2 L
N
2 18 Q
(Alignment:) 180 605 T
(Unification:) 180 466.63 T
0 24 Q
(0) 216 424.95 T
(1) 242.4 424.95 T
(;) 268.14 424.95 T
(;) 315.61 424.95 T
1 F
(1) 289.87 424.95 T
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "36" 37
%%Page: "37" 37
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 4) 72 708 T
72 252 720 720 C
72 252 720 702 C
4 24 Q
0 X
0 K
(x) 250.42 562.2 T
(;) 304.56 562.2 T
(x) 250.42 517.84 T
(;) 304.56 517.84 T
(1) 276.82 517.84 T
307 551.71 307 536.71 2 L
1 H
2 Z
N
256 553.2 256 538.2 2 L
N
2 18 Q
(Alignment:) 180 605 T
(Unification:) 180 464.28 T
4 24 Q
(...) 216 562.2 T
(...) 325.63 562.2 T
5 F
(x) 250.42 421.47 T
(;) 305.9 421.47 T
4 F
(...) 216 421.47 T
(...) 328.3 421.47 T
(1) 278.16 421.47 T
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "37" 38
%%Page: "38" 38
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 5) 72 708 T
72 252 720 720 C
73.5 253 718.5 702 C
1 24 Q
0 X
0 K
(\244) 270.14 580.02 T
4 F
(;) 488.19 580.02 T
1 F
(\244) 270.14 531.03 T
(\244) 332.44 531.03 T
4 F
(x) 301.29 531.03 T
(x) 301.29 629 T
(y) 363.59 629 T
(z) 425.89 629 T
1 F
(\244) 332.44 482.05 T
(\244) 394.74 482.05 T
4 F
(y) 363.59 482.05 T
1 F
(\244) 394.74 433.06 T
(\244) 457.04 433.06 T
4 F
(z) 425.89 433.06 T
(;) 488.19 384.08 T
1 F
(\244) 457.04 384.08 T
4 F
(c) 238.99 580.02 T
(b) 206.49 580.02 T
(a) 174 580.02 T
(c) 238.99 629 T
(b) 206.49 629 T
(a) 174 629 T
(d) 514.01 580.02 T
(e) 546.51 580.02 T
(f) 579 580.02 T
(d) 514.01 629 T
(e) 546.51 629 T
(f) 579 629 T
583 620 583 601 2 L
0.5 H
2 Z
N
553 620 553 601 2 L
N
519 620 519 601 2 L
N
244 620 244 601 2 L
N
214 620 214 601 2 L
N
180 620 180 601 2 L
N
275 572.5 275 553.5 2 L
N
337 523.5 337 504.5 2 L
N
400 474.5 400 455.5 2 L
N
462 425.5 462 406.5 2 L
N
491 570 491 405 2 L
N
431 622 431 452 2 L
N
369 618 369 502 2 L
N
306 623 306 550 2 L
N
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "38" 39
%%Page: "39" 39
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 6) 72 708 T
72 252 720 720 C
72 252 720 702 C
4 24 Q
0 X
0 K
(g) 212.66 588.2 T
(\050) 237.07 588.2 T
(\051) 462.22 588.2 T
(1) 413.4 588.2 T
(g) 212.66 543.84 T
(\050) 237.07 543.84 T
(boolean) 259.46 543.84 T
(x) 359.26 543.84 T
(#) 385.66 543.84 T
(;) 441.14 543.84 T
(\051) 462.22 543.84 T
(boolean) 259.46 499.49 T
(#) 385.66 499.49 T
(1) 413.4 499.49 T
(;) 441.14 499.49 T
219 577 219 564 2 L
1 H
2 Z
N
242 578.2 242 568.2 2 L
N
464 578.2 464 568.2 2 L
N
292 538.2 292 520.2 2 L
N
393 538.2 393 522.2 2 L
N
419 582.2 419 521.2 2 L
N
444 535.2 444 519.2 2 L
N
2 18 Q
(Alignment:) 180 631 T
(Unification:) 180 445.92 T
5 24 Q
(g) 213 403.12 T
(\050) 238.07 403.12 T
(boolean) 257.15 403.12 T
4 F
(x) 360.26 403.12 T
5 F
(#) 386.66 403.12 T
(;) 441.48 403.12 T
(\051) 463.22 403.12 T
(1) 414.4 403.12 T
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "39" 40
%%Page: "40" 40
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 7) 54 708 T
54 252 738 720 C
54 252 738 702 C
4 24 Q
0 X
0 K
(not) 149.73 612 T
(af) 281.58 612 T
(implies) 337.25 612 T
(af) 468.07 612 T
(f) 129.22 523.5 T
(not) 149.73 523.5 T
(f) 218.74 523.5 T
(\051) 545.57 523.5 T
(f) 218.74 479.25 T
(f) 261.07 479.25 T
(\051) 315.42 479.25 T
(implies) 337.25 479.25 T
(f) 447.57 479.25 T
(\051) 501.92 479.25 T
(f) 261.07 435 T
(af) 281.58 435 T
(\051) 315.42 435 T
(f) 447.57 435 T
(af) 468.07 435 T
(\051) 501.92 435 T
(\050) 196.92 612 T
(\051) 523.74 612 T
(and) 567.39 612 T
(af) 663.58 612 T
(\051) 719.26 612 T
(\050) 107.4 612 T
(\051) 545.57 612 T
(\050) 65.07 612 T
(\050) 107.4 523.5 T
(\051) 523.74 523.5 T
(\050) 65.07 567.75 T
(f) 86.9 567.75 T
(\050) 239.25 479.25 T
(\051) 545.57 567.75 T
(and) 567.39 567.75 T
(f) 643.08 567.75 T
(\051) 697.43 567.75 T
(\051) 719.26 567.75 T
(f) 643.08 435 T
(af) 663.58 435 T
(\051) 697.43 435 T
(\051) 523.74 479.25 T
(\050) 239.25 435 T
(\050) 621.26 567.75 T
(\050) 621.26 435 T
(\050) 196.92 523.5 T
(\050) 196.92 479.25 T
(\050) 425.74 479.25 T
(\050) 425.74 435 T
(\050) 107.4 567.75 T
(f) 129.22 567.75 T
721 601 721 591 2 L
0.5 H
2 Z
N
525 513 525 503 2 L
N
547 557 547 547 2 L
N
548 602 548 592 2 L
N
504 468 504 458 2 L
N
432 468 432 458 2 L
N
317 469 317 459 2 L
N
243 469 243 459 2 L
N
202 512 202 502 2 L
N
112 557 112 547 2 L
N
112 601 112 591 2 L
N
70 600 70 590 2 L
N
451 472 451 458 2 L
N
222 516 222 502 2 L
N
133 562 133 548 2 L
N
265 472 265 458 2 L
N
699 557 699 460 2 L
N
626 556 626 459 2 L
N
525 600 525 547 2 L
N
201 599 201 546 2 L
N
674 606 674 457 2 L
N
475 604 475 455 2 L
N
290 604 290 455 2 L
N
587 605 587 587 2 L
N
372 604 372 500 2 L
N
166 605 166 544 2 L
N
54 252 738 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "40" 41
%%Page: "41" 41
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 8) 72 708 T
71.5 253.01 720.5 702 C
4 18 Q
0 X
0 K
(not_canfly) 123.5 593.47 T
(penguin) 282.57 628.88 T
(V) 458.43 628.88 T
(;) 502.05 628.88 T
($) 544.65 628.88 T
(not_canfly) 123.5 628.88 T
(flies) 216.96 593.47 T
(X) 259.77 593.47 T
(;) 357.42 593.47 T
(flies) 216.96 558.41 T
(penguin) 282.57 558.41 T
(X) 259.77 523.34 T
(penguin) 282.57 523.34 T
(;) 357.42 523.34 T
(T) 373.22 593.47 T
(;) 415.83 593.47 T
(%) 431.63 593.47 T
(T) 373.22 488.28 T
(0) 395.02 488.28 T
(;) 415.83 488.28 T
(not) 565.46 593.47 T
(T) 601.28 593.47 T
(;) 623.08 593.47 T
(V) 638.88 593.47 T
(;) 661.69 593.47 T
(not_canfly) 98.5 453.22 T
(flies) 193.33 453.22 T
(X) 236.14 453.22 T
(;) 258.94 453.22 T
(T) 274.75 453.22 T
(;) 296.54 453.22 T
(%) 312.35 453.22 T
(not) 339.15 453.22 T
(T) 373.22 453.22 T
(;) 415.83 453.22 T
(V) 458.43 453.22 T
(;) 502.05 453.22 T
(%) 517.85 453.22 T
($) 544.65 453.22 T
(not) 339.15 418.15 T
(0) 395.02 418.15 T
(0) 395.02 558.41 T
(%) 431.63 558.41 T
(V) 458.43 383.09 T
(0) 481.24 383.09 T
(;) 502.05 383.09 T
(0) 481.24 418.15 T
(%) 517.85 418.15 T
($) 677.49 593.47 T
171 621 171 610 2 L
1 H
2 Z
N
232 588 232 572 2 L
N
266 588 266 542 2 L
N
315 622 315 572 2 L
N
315 551 315 536 2 L
N
358 585 358 537 2 L
N
378 588 378 508 2 L
N
379 483 379 471 2 L
N
399 553 399 506 2 L
N
399 482 399 436 2 L
N
418 480 418 467 2 L
N
418 585 418 503 2 L
N
441 588 441 575 2 L
N
351 448 351 432 2 L
N
464 623 464 472 2 L
N
465 447 465 401 2 L
N
485 413 485 400 2 L
N
504 620 504 467 2 L
N
504 446 504 397 2 L
N
526 448 526 434 2 L
N
549 623 549 470 2 L
N
0 180 792 792 C
FMENDPAGE
%%EndPage: "41" 42
%%Page: "42" 42
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 9) 72 708 T
72 252 720 720 C
72 255 720 702 C
4 18 Q
0 X
0 K
(bird) 180.34 593.19 T
(Tweety) 276.96 593.19 T
($) 625.26 593.19 T
(bird) 180.34 554.53 T
(name) 221.14 554.53 T
(;) 345.77 554.53 T
(name) 221.14 515.87 T
(Tweety) 276.96 515.87 T
(;) 345.77 515.87 T
(canfly) 361.58 554.53 T
(;) 419.39 554.53 T
(wings) 435.2 554.53 T
(;) 492.01 554.53 T
(feathers) 507.81 554.53 T
(;) 583.65 554.53 T
(...) 599.45 554.53 T
($) 625.26 554.53 T
2 14 Q
(Alignment:) 144.34 638 T
(Unification:) 139.51 466.35 T
5 18 Q
(bird) 175.18 421.54 T
(name) 219.98 421.54 T
(;) 349.61 421.54 T
(Tweety) 277.79 421.54 T
4 F
(canfly) 366.41 421.54 T
(;) 424.22 421.54 T
(wings) 440.03 421.54 T
(;) 496.83 421.54 T
(feathers) 512.64 421.54 T
(;) 588.47 421.54 T
(...) 604.28 421.54 T
5 F
($) 630.09 421.54 T
195 587 195 571 2 L
1 H
2 Z
N
243 549 243 529 2 L
N
304 588 304 530 2 L
N
348 546 348 531 2 L
N
628 584 628 573 2 L
N
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "42" 43
%%Page: "43" 43
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 10) 71 708 T
71 252 720 720 C
72 252 719 702 C
4 14 Q
0 X
0 K
(swallow) 209.53 585.41 T
(Tweety) 319.04 585.41 T
2 12 Q
(Alignment:) 106.01 619.69 T
4 14 Q
(swallow) 209.53 557.74 T
(bird) 124.33 557.74 T
(canfly) 382.45 557.74 T
(far) 426.22 557.74 T
(;) 449.76 557.74 T
(wings) 460.85 557.74 T
(pointed) 503.83 557.74 T
(;) 556.95 557.74 T
(name) 276.82 502.39 T
(Tweety) 319.04 502.39 T
(;) 371.36 502.39 T
(bird) 124.33 530.07 T
(name) 276.82 530.07 T
(;) 371.36 530.07 T
(canfly) 382.45 530.07 T
(;) 449.76 530.07 T
(wings) 460.85 530.07 T
(feathers) 568.05 530.07 T
(;) 556.95 530.07 T
($) 655.8 585.41 T
($) 655.8 530.07 T
(;) 625.83 530.07 T
(...) 636.92 530.07 T
(species) 154.87 530.07 T
(species) 154.87 557.74 T
(;) 265.73 557.74 T
(;) 265.73 530.07 T
5 F
(bird) 123.96 420.84 T
(species) 157.6 420.84 T
(swallow) 216.17 420.84 T
(;) 277.06 420.84 T
(name) 288.92 420.84 T
(Tweety) 332.69 420.84 T
(;) 387.35 420.84 T
(canfly) 399.21 420.84 T
4 F
(far) 446.87 420.84 T
5 F
(;) 470.41 420.84 T
(wings) 482.27 420.84 T
4 F
(pointed) 529.15 420.84 T
5 F
(;) 582.27 420.84 T
4 F
(feathers) 594.13 420.84 T
(;) 651.91 420.84 T
(...) 663 420.84 T
5 F
($) 681.88 420.84 T
2 12 Q
(Unification:) 106.01 455.12 T
233 581 233 570 2 L
0.5 H
2 Z
N
267 552 267 542 2 L
N
451 552 451 542 2 L
N
372 523 372 513 2 L
N
558 551 558 541 2 L
N
173 553 173 541 2 L
N
138 553 138 541 2 L
N
294 525 294 512 2 L
N
342 580 342 513 2 L
N
400 555 400 540 2 L
N
478 553 478 540 2 L
N
659 580 659 545 2 L
N
71 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "43" 44
%%Page: "44" 44
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 11) 72 708 T
72 252 720 720 C
72 254 720 702 C
4 18 Q
0 X
0 K
(living_thing) 132 557.33 T
(name) 232.84 557.33 T
(name) 232.84 522.27 T
(Chirpy) 291.66 522.27 T
(;) 357.47 522.27 T
(;) 357.47 557.33 T
(gender) 373.27 557.33 T
(gender) 373.27 487.2 T
(male) 440.11 487.2 T
(...) 489.91 487.2 T
(;) 515.72 487.2 T
(;) 515.72 557.33 T
(class) 531.53 557.33 T
(class) 531.53 452.14 T
(bird) 583.33 452.14 T
(...) 624.14 452.14 T
(;) 649.95 452.14 T
(;) 649.95 557.33 T
($) 665.75 557.33 T
(Chirpy) 291.66 592.39 T
(male) 440.11 592.39 T
(bird) 583.33 592.39 T
($) 665.75 592.39 T
254 553.39 254 534.39 2 L
1 H
2 Z
N
321 586.39 321 537.39 2 L
N
359 550.39 359 535.39 2 L
N
396 552.39 396 501.39 2 L
N
457 587.39 457 500.39 2 L
N
518 549.39 518 502.39 2 L
N
552 552.39 552 465.39 2 L
N
599 587.39 599 466.39 2 L
N
652 548.39 652 467.39 2 L
N
670 586.39 670 574.39 2 L
N
2 14 Q
(Alignment:) 96 630 T
(Unification:) 96 402.62 T
4 18 Q
(living_thing) 132 365.01 T
5 F
(name) 232.84 365.01 T
(Chirpy) 290.65 365.01 T
(;) 358.46 365.01 T
(gender) 375.25 365.01 T
(male) 446.06 365.01 T
4 F
(...) 497.89 365.01 T
5 F
(;) 523.7 365.01 T
(class) 540.49 365.01 T
(bird) 596.33 365.01 T
4 F
(...) 641.13 365.01 T
5 F
(;) 666.94 365.01 T
4 F
($) 683.74 365.01 T
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "44" 45
%%Page: "45" 45
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 12) 72 708 T
72 252 720 720 C
72 252 720 702 C
4 14 Q
0 X
0 K
(penguin) 219.87 597.51 T
(Tweety) 330.19 597.51 T
2 12 Q
(Alignment:) 99.35 631.79 T
4 14 Q
(penguin) 219.87 567.77 T
(bird) 134.67 567.77 T
(canfly) 393.6 567.77 T
(no) 437.37 567.77 T
(;) 467.34 567.77 T
(name) 287.97 511.59 T
(Tweety) 330.19 511.59 T
(;) 382.51 511.59 T
(bird) 134.67 539.26 T
(name) 287.97 539.26 T
(;) 382.51 539.26 T
(canfly) 393.6 539.26 T
(;) 467.34 539.26 T
(wings) 478.43 539.26 T
(feathers) 580.95 539.26 T
(;) 569.86 539.26 T
($) 668.7 597.51 T
($) 668.7 539.26 T
(;) 638.73 539.26 T
(...) 649.82 539.26 T
(species) 165.21 539.26 T
(species) 165.21 567.77 T
(;) 276.88 567.77 T
(;) 276.88 539.26 T
5 F
(bird) 136.29 430.04 T
(species) 169.94 430.04 T
(penguin) 228.51 430.04 T
(;) 290.15 430.04 T
(name) 302.01 430.04 T
(Tweety) 345.78 430.04 T
(;) 400.44 430.04 T
(canfly) 412.3 430.04 T
4 F
(no) 459.96 430.04 T
5 F
(;) 482.73 430.04 T
(wings) 494.59 430.04 T
4 F
(stubby) 541.47 430.04 T
5 F
(;) 589.91 430.04 T
4 F
(feathers) 601.78 430.04 T
(;) 659.56 430.04 T
(...) 670.65 430.04 T
5 F
($) 689.53 430.04 T
2 12 Q
(Unification:) 99.35 464.31 T
4 14 Q
(wings) 478.43 567.77 T
(stubby) 521.41 567.77 T
(;) 569.86 567.77 T
148 564 148 550 2 L
0.5 H
2 Z
N
188 563 188 549 2 L
N
240 594 240 578 2 L
N
278 561 278 550 2 L
N
384 534 384 523 2 L
N
468 561 468 550 2 L
N
571 562 571 551 2 L
N
307 535 307 522 2 L
N
354 593 354 524 2 L
N
409 562 409 551 2 L
N
494 563 494 550 2 L
N
672 593 672 554 2 L
N
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "45" 46
%%Page: "46" 46
612 792 1 FMBEGINPAGE
1 18 Q
0 X
0 K
(Fig. 13) 72 708 T
72 252 720 720 C
72 254 720 702 C
4 18 Q
0 X
0 K
(Tweety) 275.16 565 T
($) 623.46 565 T
(bird) 178.54 526.34 T
(name) 219.34 526.34 T
(;) 343.97 526.34 T
(name) 219.34 487.67 T
(Tweety) 275.16 487.67 T
(;) 343.97 487.67 T
(canfly) 359.78 526.34 T
(;) 417.59 526.34 T
(wings) 433.4 526.34 T
(;) 490.2 526.34 T
(feathers) 506.01 526.34 T
(;) 581.84 526.34 T
(...) 597.65 526.34 T
($) 623.46 526.34 T
2 14 Q
(Alignment:) 142.53 610.41 T
(Unification:) 137.71 438.76 T
4 18 Q
(bird) 175.37 393.95 T
5 F
(name) 216.18 393.95 T
(;) 345.81 393.95 T
(Tweety) 273.99 393.95 T
(canfly) 362.61 393.95 T
4 F
(;) 425.43 393.95 T
(wings) 441.23 393.95 T
(;) 498.04 393.95 T
(feathers) 513.84 393.95 T
(;) 589.68 393.95 T
(...) 605.48 393.95 T
5 F
($) 631.29 393.95 T
4 F
(canfly) 359.78 565 T
240 520 240 501 2 L
1 H
2 Z
N
304 559 304 502 2 L
N
346 517 346 503 2 L
N
381 560 381 540 2 L
N
627 559 627 544 2 L
N
72 252 720 720 C
0 180 792 792 C
FMENDPAGE
%%EndPage: "46" 47
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