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Logic, Representation and Inference

Logic, Representation and Inference. Introduction to Semantics What is semantics for? Role of FOL Montague Approach. Semantics. Semantics is the study of the meaning of NL expressions Expressions include sentences, phrases, and sentences. What is the goal of such study?

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Logic, Representation and Inference

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  1. Logic, Representation and Inference Introduction to Semantics What is semantics for? Role of FOL Montague Approach Introduction to Semantics

  2. Semantics • Semantics is the study of the meaning of NL expressions • Expressions include sentences, phrases, and sentences. • What is the goal of such study? • Provide a workable definition of meaning. • Explain semantic relations between expressions. Introduction to Semantics

  3. Examples of Semantic Relations • Synonymy • John killed Mary • John caused Mary to die • Entailment • John fed his cat • John has a cat • Consistency • John is very sick • John is not feeling well • John is very healthy Introduction to Semantics

  4. Different Kinds of MeaningX means Y • Meaning as definition: • a bachelor means an unmarried man • Meaning as intention: • What did John mean by waving? • Meaning as reference:"Eiffel Tower " means Introduction to Semantics

  5. Workable Definition of Meaning • Restrict the scope of semantics. • Ignore irony, metaphor etc. • Stick to the literal interpretations of expressions rather than metaphorical ones. (My car drinks petrol). • Assume that meaning is understood in terms of something concrete. Introduction to Semantics

  6. Concrete Semantics • Procedural semantics: the meaning of a phrase or sentence is a procedure:“Pick up a big red block”(Winograd 1972) • Object–Oriented Semantics: meaning is an instance of a class. • Truth-Conditional Semantics Introduction to Semantics

  7. Truth Conditional Semantics • Key Claim: the meaning of a sentence is identical to the conditions under which it is true. • Know the meaning of "Ġianni ate fish for tea" = know exactly how to apply it to the real world and decide whether it is true or false. • On this view, one task of semantic theory is to provide a system for identifying the truth conditions of sentences. Introduction to Semantics

  8. TCS and Semantic Relations • TCS provides a precise account of semantic relations between sentences. • Examples: • S1 is synonymous with S2. • S1 entails S2 • S1 is consistent with S2. • S1 is inconsistent with S2. • Just like logic! • Which logic? Introduction to Semantics

  9. NL Semantics: Two Basic Issues • How can we automate the process of associating semantic representations with expressions of natural language? • How can we use semantic representations of NL expressions to automate the process of drawing inferences? • We will focus mainly on first issue. Introduction to Semantics

  10. Associating Semantic Representations Automatically • Design a semantic representation language. • Figure out how to compute the semantic representation of sentences • Link this computation to the grammar and lexicon. Introduction to Semantics

  11. Semantic Representation Language • Logical form (LF) is the name used by logicians (Russell, Carnap etc) to talk about the representation of context-independent meaning. • Semantic representation language has to encode the LF. • One concrete representation for logical form is first order logic (FOL) Introduction to Semantics

  12. Why is FOL a good thing? • Has a precise, model-theoretic semantics. • If we can translate a NL sentence S into a sentence of FOL, then we have a precise grasp on at least part of the meaning of S. • Important inference problems have been studied for FOL. Computational solutions exist for some of them. • Hence the strategy of translating into FOL also gives us a handle on inference. Introduction to Semantics

  13. Anatomy of FOL • Symbols of different types • constant symbols: a,b,c • variable symbols: x, y, z • function symbols: f,g,h • predicate symbols: p,q,r • connectives: &, v,  • quantifiers: ,  • punctuation: ), (, “,” Introduction to Semantics

  14. Anatomy of FOL • Symbols of different types • constant symbols: csa3180, nlp, mike, alan, rachel, csai • variable symbols: x, y, z • function symbols: lecturerOf, subjectOf • predicate symbols: studies, likes • connectives: &, v,  • quantifiers: ,  • punctuation: ), (, “,” Introduction to Semantics

  15. Anatomy of FOL With these symbols we can make expressions of different types • Expressions for referring tothings • constant: alan, nlp • variable: x • term: subject(csa3180) • Expressions for stating facts • atomic formula: study(alan,csa3180) • complex formula: study(alan,csa3180) & teach(mike, csa3180) • quantified expression: xy teaches(lecturer(x),x) & studies(y,subject(x))xy likes(x,subjectOf(y))  studies(x,y) Introduction to Semantics

  16. Logical Form of Phrases Introduction to Semantics

  17. Logical Forms of Sentences • John kicks Fido: kick(john, fido) • Every student wrote a program xy( stud(x)  prog(y) & write(x,y)) yx(stud(x)  prog(y) & write(x,y)) • Semantic ambiguity related to quantifier scope Introduction to Semantics

  18. Building Logical Form Frege’s Principle of Compositionality • The POC states that the LF of a complex phrase can be built out of the LFs of the constituent parts. • An everyday example of compositionality is the way in which the “meaning” of arithmetic expressions is computed(2+3) * (4/2) = (5 * 2) =10 Introduction to Semantics

  19. Compositionality for NL • The LF of the whole sentence can be computed from the LF of the subphrases, i.e. • Given the syntactic rule X  Y Z. • Suppose [Y], [Z] are the LFs of Y, and Z respectively. • Then [X] = ([Y],[Z]) where  is some function for semantic combination Introduction to Semantics

  20. Claims of Richard Montague: • Each syntax rule is associated with a semantic rule that describes how the LF of the LHS category is composed from the LF of its subconstituents • 1:1 correspondence between syntax and semantics (rule-to-rule hypothesis) • Functional composition proposed for combining semantic forms. • Lambda calculus proposed as the mechanism for describing functions for semantic combination. Introduction to Semantics

  21. Sentence Rule • Syntactic Rule:S  NP VP • Semantic Rule:[S] = [VP]([NP])i.e. the LF of S is obtained by "applying" the LF of VP to the LF of NP. • For this to be possible [VP] must be a function, and [NP] the argument to the function. Introduction to Semantics

  22. S write(bertrand,principia) VP y.write(y,principia) NP bertrand bertrand V x.y.write(y,x) NP principia writes principia Parse Tree with Logical Forms Introduction to Semantics

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