| { | |
| "paper_id": "T75-2035", | |
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| "date_generated": "2023-01-19T07:43:09.146236Z" | |
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| "title": "Formal Reasoning ~nd Language Understanding Systems", | |
| "authors": [ | |
| { | |
| "first": "Raymond", | |
| "middle": [], | |
| "last": "Reiter", | |
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| "laboratory": "", | |
| "institution": "University of British Columbia", | |
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| "email": "" | |
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| "abstract": [], | |
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| "text": "Finally, there is, or should be [5, 18] a specification of the semantics of these formal languages. There seem to be three dominant proposals for semantic representations:", | |
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| "text": "[5,", | |
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| "text": "(I) Procedural semantics [16, 17] where the underlying representation consists of procedures in some executable language.", | |
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| "text": "(2) Network structures [11, 13, 14 ] which represent knowledge by appropriate graphical data structures.", | |
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| "text": "(3) Logical representation [3, 7, 12] which express world knowledge by formulae in some formal calculus. In this connection it is worth observing that, contrary to some prevailing opinions, formal reasoning does not preclude fuzzy or imprecise reasoning. There are no a priori reasons why notions like \"probably\", \"possibly\", etc . cannot be formalized within a logical calculus and new imprecise knowledge deduced from old by means of prefectly definite and precise rules of inference. In the remainder of this paper I discuss two paradigms for formal reasoning with which I have worked -resolution and natural deduction -and argue in favour of the latter approach.", | |
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| "text": ". cannot be formalized within a logical calculus and new imprecise knowledge deduced from old by means of prefectly definite and precise rules of inference. In the remainder of this paper I discuss two paradigms for formal reasoning with which I have worked -resolution and natural", | |
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| "text": "I also indicate how other semantic representations -procedures and networks -might fit into this paradigm. Finally, I discuss some problems deriving from computational linguistics which have not been seriously considered by researchers in formal inference but which I think might fruitfully be explored within a logical framework.", | |
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| "text": "A. Resolution [10] The resolution principle is based on five key concepts, two of which (the elimination of quanifiers through the introduction of Skolem functions, unification) are of particular relevance to problems in the representation of linguistic deep structures. C) etc. The formula (2) has two clauses in its canonical form: ", | |
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| "text": "[10]", | |
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| "start": 271, | |
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| "text": "C)", | |
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| "section": "Paradigms for Formal Reasoning", | |
| "sec_num": "2." | |
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| { | |
| "text": "EQUATION", | |
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| "text": "EQUATION", | |
| "ref_id": "EQREF", | |
| "raw_str": "AN'IMAL(x) ~ HAS-AS-PART(x, nose(x)) ~x~ V NOSE(nose(x))", | |
| "eq_num": "(3)" | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "~T(F) V HAS-AS-PART(y,nose(F)) CAT(y) NOSE(nose(y)) (4) i.e.", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "cats have noses. If in addition it is known that CAT(fritz), then by unifying this on CAT(y) in (4), we can deduce the two clauses HAS-AS-PART(fritz,nose(fritz))", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "(5.1) NOSE(nose(fritz)) (5.2) (v) Completeness", | |
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| "text": "Resolution is a refutation loJ~ic i.e. if T is some statement to be proved, the clausal form of its negation is added to the clauses representing the knowledge base, and an attempt is made to derive a contradiction by means of the single resolution inference rule.", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "For exmple, to prove that Fritz has a nose i.e. (~z)[NOSE(x) A HAS-AS-PART(fritz,z)] first negate, yielding__ (z)[N--6-~E(z) ~ HAS-AS-PART(fritz,z)], then remove the universal quantifier which i~ds the clause NOSE(z)v HAS-AS-PART(fritz,z).", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "Resolving with (5.1) yields NOSE(nose(fritz)) which contradicts (5.2). ", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "The general fact that cats are animals has no representation in the logical component, but is represented in the net by appropriately linked CAT and ANIMAL nodes. Now the question \"Does Fritz have a nose?\" translates to an attempt to prove HAS-AS-PART(fritz{CAT}, y{NOSE}). If we could unify this with (6) the question would be answered.", | |
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| "text": "However, a term (in this case x) cannot unify with another term (fritz) unless their types are compatible. To determine compatibilitythe unifier calls on the semantic net processor to check whether a path of superset links connects node CAT to node ANIMAL.", | |
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| "text": "In this case there is such a path, so the unificaton succeeds.", | |
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| "text": "Notice how each component benefits from the presence of the other. The logic benefits by processing fewer, and considerably more compact formulae than would otherwise be necessary.", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "(Compare (6) with", | |
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| "text": "(2)).", | |
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| "text": "In particular, compactification eliminates many logical connectives, which has the effect of reducing the number of applications of rules of inference in deriving a result. This is so because these rules are \"connective driven\".", | |
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| "section": "I) The elimination of quantifiers", | |
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| "text": "Since It is possible, for example, to define \"most-of\" in some set theoretic formalism which effectively says \"more than 80%\", but I find this approach unsatisfying. A differenct approach, borrowing on the successful treatment of \"there exists\" in logic, might define \"most-of\" as a Skolem function with certain properties peculiar to our understanding of the meaning of \"most of\".", | |
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| "text": "Thus, one property of the \"Skolem function\" most-of is that it unifies with any term of the same type as the argument to most-of; the unifier returns the atom \"probably\".", | |
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| "text": "Thus, \"Most dogs bark\" becomes something like BARK(most-of(x{DOG})), and \"Does Fido bark?\" translates to BARK(fido{DOG}).", | |
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| "text": "Unification succeeds and we conclude something like PROBABLY(BARK(fido{DOG})).", | |
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| "text": "Clearly there are plenty of problems here not least what we mean by \"probably\", but the example gives the flavour of a possible logical approach, as well as an indication how certain kinds of \"fuzzy\" reasoning might be modeled in an extended logic.", | |
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| "text": "(ii) Different levels of memory -contexts for wanting, needing etc. Consider representing \"x wants P\" in some logical formalism, where P is an arbitrary proposition.", | |
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| "text": "In specifying the properties of \"WANT\" we shall need (among other things) some kind of schema of the form WANTS(x,P) A Q WANTS(x, anything derivable from P and Q) Notice that within a want-context there is no commitment to the truth value of a formula -x may want a unicorn.", | |
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| "text": "The role of the schema (7) is assumed by the logic which knows which intercontextual inferences are legal. Consequently, it is important to better understand this trade-off between computation and deduction (or the particular and the general) and we can hope that in the future researchers in formal reasoning will clarify some of the issues. In this connection it is worth remarking that the distinction between computation and deduction is by no means clear [4] .", | |
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