february 2009introduction to semantics1 logic, representation and inference introduction to...

February 2009 Introduction to Semantics 1

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? – Provide a workable definition of meaning.– Explain semantic relations between

expressions.


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


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


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.


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


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.


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?


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.


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.


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)


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.


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: ), (, “,”


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: ), (, “,”


Anatomy of FOL

With these symbols we can make expressions of different types– Expressions for referring to things

• 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:

xy teaches(lecturer(x),x) & studies(y,subject(x))xy likes(x,subjectOf(y)) studies(x,y)


word POS Logic Representation

csai proper noun individual constant

csai

student common noun 1 place predicate

student(x)

easy adjective 1 place predicate

easy(x)

easy interesting course

adj/noun 1 place predicate

easy(x) & interesting(x) & course(x)

snores intrans verb 1 place predicate

snore (x)

studies trans. verb 2 place predicate

study(x,,y)

gives ditrans verb 3 place pred give(x,y,z)

Logical Form of Phrases


Logical Forms of Sentences

• John kicks Fido:

kick(john, fido)

• Every student wrote a program

xy( stud(x) prog(y) & write(x,y))

yx(stud(x) prog(y) & write(x,y))

• Semantic ambiguity related to quantifier scope


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


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


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.


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.


Swrite(bertrand,principia)

NPbertrand

VPy.write(y,principia)

Vx.y.write(y,x)

NPprincipia

bertrand

writes principia

Parse Tree with Logical Forms

february 2009introduction to semantics1 logic, representation and inference introduction to...

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