Computational Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] Computational Semantics: Lambda Calculus Semantic Analysis Problems One Solution: λ-Calculus Scott Farrar λ-calculus and FOL λ-calculus and CLMA, University of Washington compositionality The semantics of [email protected] words based on syntactic category February 24, 2010 1/37 Computational Today's lecture Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] 1 Semantic Analysis Semantic Analysis Problems Problems One Solution: λ-Calculus λ-calculus and FOL λ-calculus and 2 One Solution: λ-Calculus compositionality The semantics of λ-calculus and FOL words based on syntactic category λ-calculus and compositionality 3 The semantics of words based on syntactic category 2/37 Computational Semantic analysis Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- Definition [email protected] Semantic analysis is the derivation of a semantic Semantic Analysis Problems representation from a string of words (perhaps marked up One Solution: with syntactic structure). In other words, map sentences of λ-Calculus λ-calculus and FOL λ-calculus and NL onto logical formulas. compositionality The semantics of words based on Map Jim loves Betty to love(JIM; BETTY ) syntactic category There are several competing approaches for doing this, as there are several competing standards for the right semantic representation (use of event vs. relations). 3/37 Computational Compositionality Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- Definition [email protected] Recall the principle of compositionality: the meaning of a Semantic Analysis Problems complex expression is a function of the meaning of its parts. One Solution: λ-Calculus λ-calculus and FOL λ-calculus and The assumption is that we should be able to assign each compositionality The semantics of \part" a meaning, then build larger structures, guided by the words based on syntax of the language. syntactic category The syntax of NL and the syntax of predicate logic are similar, but ultimately not one-to-one compatible: translation between the two is a non-trivial task. 4/37 9e 9b [SailBoat(b) ^ HeelingEvent(e) ^ actor(e; b)] My sailboat is on the bottom. 9e [SpatialLocating(e) ^ theme(e; MYSB) ^ loc(e; SEAFLOOR)] Computational Event structure Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] Semantic Analysis A sailboat heels. Problems One Solution: λ-Calculus λ-calculus and FOL λ-calculus and compositionality The semantics of words based on syntactic category 5/37 My sailboat is on the bottom. 9e [SpatialLocating(e) ^ theme(e; MYSB) ^ loc(e; SEAFLOOR)] Computational Event structure Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] Semantic Analysis A sailboat heels. Problems One Solution: λ-Calculus 9e 9b [SailBoat(b) ^ HeelingEvent(e) ^ actor(e; b)] λ-calculus and FOL λ-calculus and compositionality The semantics of words based on syntactic category 5/37 9e [SpatialLocating(e) ^ theme(e; MYSB) ^ loc(e; SEAFLOOR)] Computational Event structure Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] Semantic Analysis A sailboat heels. Problems One Solution: λ-Calculus 9e 9b [SailBoat(b) ^ HeelingEvent(e) ^ actor(e; b)] λ-calculus and FOL λ-calculus and compositionality The semantics of My sailboat is on the bottom. words based on syntactic category 5/37 Computational Event structure Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] Semantic Analysis A sailboat heels. Problems One Solution: λ-Calculus 9e 9b [SailBoat(b) ^ HeelingEvent(e) ^ actor(e; b)] λ-calculus and FOL λ-calculus and compositionality The semantics of My sailboat is on the bottom. words based on syntactic category 9e [SpatialLocating(e) ^ theme(e; MYSB) ^ loc(e; SEAFLOOR)] 5/37 Let's assume strict compositionality and say that the meaning of each syntactic constituent contributes to the meaning of the parent constituent. We could come up with something like XP.sem to stand for the semantics of some constituent XP. Definition Semantic attachment refers to the adornment of phrase structure rules with such semantic information. Computational Semantic attachments Semantics: Lambda Calculus Consider the problem of two-place predicates in a Scott Farrar CLMA, University non-event-style semantics: we need to map Jim loves Betty of Washington far- [email protected] to something like: Semantic Analysis love(JIM; BETTY ) Problems One Solution: λ-Calculus λ-calculus and FOL . λ-calculus and compositionality The semantics of words based on syntactic category 6/37 Definition Semantic attachment refers to the adornment of phrase structure rules with such semantic information. Computational Semantic attachments Semantics: Lambda Calculus Consider the problem of two-place predicates in a Scott Farrar CLMA, University non-event-style semantics: we need to map Jim loves Betty of Washington far- [email protected] to something like: Semantic Analysis love(JIM; BETTY ) Problems One Solution: λ-Calculus λ-calculus and FOL . λ-calculus and compositionality The semantics of Let's assume strict compositionality and say that the words based on syntactic category meaning of each syntactic constituent contributes to the meaning of the parent constituent. We could come up with something like XP.sem to stand for the semantics of some constituent XP. 6/37 Computational Semantic attachments Semantics: Lambda Calculus Consider the problem of two-place predicates in a Scott Farrar CLMA, University non-event-style semantics: we need to map Jim loves Betty of Washington far- [email protected] to something like: Semantic Analysis love(JIM; BETTY ) Problems One Solution: λ-Calculus λ-calculus and FOL . λ-calculus and compositionality The semantics of Let's assume strict compositionality and say that the words based on syntactic category meaning of each syntactic constituent contributes to the meaning of the parent constituent. We could come up with something like XP.sem to stand for the semantics of some constituent XP. Definition Semantic attachment refers to the adornment of phrase structure rules with such semantic information. 6/37 Computational Semantic attachments Semantics: Lambda Calculus Scott Farrar CLMA, University of Washington far- [email protected] Semantic Analysis Problems One Solution: λ-Calculus λ-calculus and FOL λ-calculus and compositionality The semantics of words based on syntactic category 7/37 S.sem = NP.sem + VP.sem VP.sem = V.sem + NP.sem V.sem = love.sem love.sem = love(x; y) NP.sem = NNP.sem NNP.sem = Betty.sem or Jim.sem Betty.sem = BETTY Jim.sem = JIM Computational Example: Semantic attachments Semantics: Lambda Calculus Scott Farrar Assume that the + symbol stands for the compositionality CLMA, University of Washington far- operator: [email protected] Semantic Analysis Problems One Solution: λ-Calculus λ-calculus and FOL λ-calculus and compositionality The semantics of words based on syntactic category loves(JIM; BETTY ) 8/37 VP.sem = V.sem + NP.sem V.sem = love.sem love.sem = love(x; y) NP.sem = NNP.sem NNP.sem = Betty.sem or Jim.sem Betty.sem = BETTY Jim.sem = JIM Computational Example: Semantic attachments Semantics: Lambda Calculus Scott Farrar Assume that the + symbol stands for the compositionality CLMA, University of Washington far- operator: [email protected] Semantic Analysis S.sem = NP.sem + VP.sem Problems One Solution: λ-Calculus λ-calculus and FOL λ-calculus and compositionality The semantics of words based on syntactic category loves(JIM; BETTY ) 8/37 V.sem = love.sem love.sem = love(x; y) NP.sem = NNP.sem NNP.sem = Betty.sem or Jim.sem Betty.sem = BETTY Jim.sem = JIM Computational Example: Semantic attachments Semantics: Lambda Calculus Scott Farrar Assume that the + symbol stands for the compositionality CLMA, University of Washington far- operator: [email protected] Semantic Analysis S.sem = NP.sem + VP.sem Problems One Solution: VP.sem = V.sem + NP.sem λ-Calculus λ-calculus and FOL λ-calculus and compositionality The semantics of words based on syntactic category loves(JIM; BETTY ) 8/37 love.sem = love(x; y) NP.sem = NNP.sem NNP.sem = Betty.sem or Jim.sem Betty.sem = BETTY Jim.sem = JIM Computational Example: Semantic attachments Semantics: Lambda Calculus Scott Farrar Assume that the + symbol stands for the compositionality CLMA, University of Washington far- operator: [email protected] Semantic Analysis S.sem = NP.sem + VP.sem Problems One Solution: VP.sem = V.sem + NP.sem λ-Calculus λ-calculus and FOL λ-calculus and V.sem = love.sem compositionality The semantics of words based on syntactic category loves(JIM; BETTY ) 8/37 NP.sem = NNP.sem NNP.sem = Betty.sem or Jim.sem Betty.sem = BETTY Jim.sem = JIM Computational Example: Semantic attachments Semantics: Lambda Calculus Scott Farrar Assume that the + symbol stands for the compositionality CLMA, University of Washington far- operator: [email protected] Semantic Analysis S.sem = NP.sem + VP.sem Problems One Solution: VP.sem = V.sem + NP.sem λ-Calculus λ-calculus and FOL λ-calculus and V.sem = love.sem compositionality The semantics of love.sem = love(x; y) words based on syntactic category loves(JIM; BETTY ) 8/37 NNP.sem = Betty.sem or Jim.sem Betty.sem = BETTY Jim.sem = JIM Computational Example: Semantic attachments Semantics: Lambda Calculus