Glue Semantics for Proof Theorists
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This document provides an overview of glue semantics, a theory of the syntax-semantics interface of natural language that uses linear logic for meaning composition. Glue semantics distinguishes between a meaning logic for semantic representations and a glue logic for specifying how chunks of meaning are assembled. It discusses how linear logic is well-suited for modeling linguistic resources and applications of glue semantics, including examples using lexical functional grammar. The document also covers identity criteria for proofs in glue semantics through lambda equivalence and the Curry-Howard isomorphism.
![Introduction
Glue Semantics
Glue in Action
Identity Criteria for Proofs
Two proofs of A, A → B B:
[A]1 A→B
→E
A→B A B
→E →, 1
B A→B A
→E
B
These are not really distinct proofs:
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![Introduction
Glue Semantics
Glue in Action
Lambda-Equivalence of Proof Terms
Include proof terms in previous derivations:
[x : A]1 f :A→B
→E
f :A→B a:A f (x ) : B
→E → I, 1
f (a) : B λx .f (x ) : A → B a:A
(λx .f (x ))(a) : B
Note: f (a) = (λx .f (x ))(a)
λ-equivalence of proof terms: semantic identity of derivations.
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![Introduction
Glue Semantics
Glue in Action
Quantifier Scope: Everyone saw something
everyone: (g −◦ f ) −◦ f
Premises saw: h −◦ (g −◦ f )
something: (h −◦ f ) −◦ f
Derivations:
∃∀ ∀∃
h −◦ (g −◦ f ) [h]
h −◦ (g −◦ f ) [h] g −◦ f [g]
g −◦ f (g −◦ f ) −◦ f f
f h −◦ f (h −◦ f ) −◦ f
h −◦ f (h −◦ f ) −◦ f f
f g −◦ f (g −◦ f ) −◦ f
f
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![Introduction
Glue Semantics
Glue in Action
With Meaning Terms
saw : h −◦ (g −◦ f ) [y : h]
saw (y ) : g −◦ f [x : g]
saw (y )(x ) : f
λy .saw (y )(x ) : h −◦ f everyone : (h −◦ f ) −◦ f
everyone(λy .saw (y )(x )) : f
λx .everyone(λy .saw (y )(x )) : g −◦ f something : (g −◦
something(λx .everyone(λy .saw (y )(x ))) : f
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Glue Semantics for Proof Theorists
- 1. Glue Semantics for Proof Theorists Valeria de Paiva Nuance Communications, CA, USA Abstract Proof Theory – April, 2013
- 2. Introduction Glue Semantics Glue in Action Introduction This talk is about the application of proof theoretic methods to the semantics of natural languages like English. Proof Theory had its beginnings as the poor cousin of Model Theory in Mathematical Logic. But it got a big boost from its use in Computer Science. Proof theory has applications in the design and specification of programming languages (type theories, compilers), in the foundations of security and as well as being essential to Artificial Intelligence and Automated Deduction. Proof Theory also has extensive applications in Computational Semantics of natural language. Here we concentrate on one application to the syntax-semantics interface: Glue Semantics 2 / 23
- 3. Introduction Glue Semantics Glue in Action Glue semantics? Glue semantics is a theory of the syntax-semantics interface of natural language that uses linear logic for meaning composition. Distinguish two separate logics in semantic interpretation 1. Meaning logic: target logical representation 2. Glue logic: logical specification of how chunks of meaning are assembled In principle, Glue uses any of several alternative grammar formalisms and any of the mainstream semantics. In practice, Glue started for LFG, with a vanilla Montague-style logic for meanings. Glue analyses have been proposed within HPSG, Context-free grammar, Categorial grammar, and TAG. Meaning languages in glue analyses include Discourse Representation Theory, First-order logic, and Natural Semantic Metalanguage(NSM). 3 / 23
- 4. Introduction Glue Semantics Glue in Action Linear Implication and (Multiplicative) Conjunction To assemble meanings we use (intuitionistic) multiplicative Linear Logic. Traditional implication: A, A → B B A, A → B A∧B Re-use A Linear implication: A, A −◦ B B A, A −◦ B A⊗B Cannot re-use A Traditional conjunction: A ∧ B A Discard B Linear conjunction: A⊗B A Cannot discard B 4 / 23
- 5. Introduction Glue Semantics Glue in Action The Linguistic Appeals of Linear Logic Resource usage: appealing idea for thinking about linguistic issues. 1. How a string of words provides a sequence of resources that can be consumed to construct a syntactic analysis of a sentence. Lambek Calculus++ 2. How word meanings provide a collection of resources that can be used to construct the meaning of a sentence. (example) 3. How linguistic context can make certain resources available, such as possible pronoun antecedents, that can be used to flesh out the interpretations of he, she or it. Only dealing with 2 above. To begin with it looks like the proof semantics we’re used to. 5 / 23
- 6. Introduction Glue Semantics Glue in Action Example: 6 / 23
- 7. Introduction Glue Semantics Glue in Action Linguistic applications of linear logic Categorial and type-logical grammar (Moortgat, vanBenthem): including parsing categorial grammars (Morrill, Hepple) and compositional semantics of categorial grammars (Morrill, Carpenter) Resource-based reformulations of other grammatical theories Minimalism (Retore,Stabler) Lexical Functional Grammar (Saraswat,Muskens) Tree Adjoining Grammar (Abrusci) AI issues such as the frame problem (White) or planning (Dixon) with linguistic relevance ‘Glue semantics’ (a version of categorial semantics without an associated categorial grammar?) (Dalrymple, Lamping & Gupta)) 7 / 23
- 8. Introduction Glue Semantics Glue in Action Identity Criteria for Proofs Two proofs of A, A → B B: [A]1 A→B →E A→B A B →E →, 1 B A→B A →E B These are not really distinct proofs: 8 / 23
- 9. Introduction Glue Semantics Glue in Action Lambda-Equivalence of Proof Terms Include proof terms in previous derivations: [x : A]1 f :A→B →E f :A→B a:A f (x ) : B →E → I, 1 f (a) : B λx .f (x ) : A → B a:A (λx .f (x ))(a) : B Note: f (a) = (λx .f (x ))(a) λ-equivalence of proof terms: semantic identity of derivations. 9 / 23
- 10. Introduction Glue Semantics Glue in Action Curry-Howard Isomorphism (CHI) CHI = Pairing of proof rules with operations on proof terms But doesn’t work for all logics, or proof systems Defines interesting identity criteria for proofs Syntactically distinct derivations corresponding to same proof Intimate relation between logic and type-theory. Varied applications, e.g. — Proofs as programs — Semantic construction for natural language 10 / 23
- 11. Introduction Glue Semantics Glue in Action Example: Using LFG Grammar 11 / 23
- 12. Introduction Glue Semantics Glue in Action Cutting and Pasting 1... 12 / 23
- 13. Introduction Glue Semantics Glue in Action Example: Input to Semantic Interpretation Lexicon Word Meaning Glue John john ↑ where ↑= g Fred fred ↑ where ↑= h saw λy .λx . see(x , y ) ↑ .OBJ −◦ (↑ .SUBJ −◦ ↑) where ↑= f , f .OBJ = h, f .SUBJ = g Constituents g, h, f : semantic resources, consuming & producing meanings 13 / 23
- 14. Introduction Glue Semantics Glue in Action Lexical Premises: Their nature saw λy .λx . see(x , y ) : h −◦ (g −◦ f ) Meaning Term Glue Formula (Propositional LL) Atomic propositions (f , g, h): • Correspond to syntactic constituents found in parsing • Denote resources used in semantic interpretation (Match production & consumption of constituent meanings) Meaning terms: • Expressions in some chosen meaning language • Language must support abstraction and application • . . . but otherwise relatively free choice 14 / 23
- 15. Introduction Glue Semantics Glue in Action The Form of Glue Derivations Γ M:f where • Γ is set of lexical premises (instantiated by parse) • f is (LL atom corresponding to) sentential constituent • M is meaning term produced by derivation (Semantic) Ambiguity Often (many) alternative derivations Γ Mi : f each producing a different meaning term Mi for f Need to find all alternative derivations (efficiently!) 15 / 23
- 16. Introduction Glue Semantics Glue in Action Alternative Derivations: Modifier Scope Consider phrase “alleged criminal from London” λx . criminal(x ) : f λP. alleged(P) : f −◦ f λPλx . from(lon, x ) ∧ P(x ) : f −◦ f There are two normal derivations, resulting in: 1. λx . from(lon, x ) ∧ alleged(criminal)(x ) : f 2. alleged(λx . from(lon, x ) ∧ criminal(x )) : f 16 / 23
- 17. Introduction Glue Semantics Glue in Action Two normal derivations 17 / 23
- 18. Introduction Glue Semantics Glue in Action Skeleton-Modifier Derivations Modifier: any formula equivalent to φ −◦ φ Initial derivation separating modifiers from skeleton g g −◦ h −◦ f h h −◦ f skeleton f g −◦ h −◦ f a −◦ (f −◦ f ) ⇒ a a −◦ (f −◦ f ) b −◦ (h −◦ f ) −◦ (h −◦ f ) modifier g, h, a, b f −◦ f b b −◦ ((h −◦ f ) −◦ (h −◦ f )) modifier (h −◦ f ) −◦ (h −◦ f ) Final derivation inserts modifiers — All scope ambiguities due to modifier insertion 18 / 23
- 19. Introduction Glue Semantics Glue in Action Quantifier Scope: Everyone saw something everyone: (g −◦ f ) −◦ f Premises saw: h −◦ (g −◦ f ) something: (h −◦ f ) −◦ f Derivations: ∃∀ ∀∃ h −◦ (g −◦ f ) [h] h −◦ (g −◦ f ) [h] g −◦ f [g] g −◦ f (g −◦ f ) −◦ f f f h −◦ f (h −◦ f ) −◦ f h −◦ f (h −◦ f ) −◦ f f f g −◦ f (g −◦ f ) −◦ f f 19 / 23
- 20. Introduction Glue Semantics Glue in Action With Meaning Terms saw : h −◦ (g −◦ f ) [y : h] saw (y ) : g −◦ f [x : g] saw (y )(x ) : f λy .saw (y )(x ) : h −◦ f everyone : (h −◦ f ) −◦ f everyone(λy .saw (y )(x )) : f λx .everyone(λy .saw (y )(x )) : g −◦ f something : (g −◦ something(λx .everyone(λy .saw (y )(x ))) : f 20 / 23
- 21. Introduction Glue Semantics Glue in Action Glue Sales Pitch Linguistically powerful & flexible approach Interesting analyses of scope, control (Asudeh), event-based semantics (Fry), intensional verbs (Dalrymple), context dependence, coordination. But many other phenomena still to do Grammar & semantics engineering Applicable to grammars besides LFG based ones Steep learning curve for writing lexical entries But turns out to allow plentiful re-use of “lingware” Can be implemented efficiently: Lev, also in NLTK open source github 21 / 23
- 22. Introduction Glue Semantics Glue in Action Conclusions For linguists: lots of language engineering to do, on a principled basis. For proof theorists: for this application cuts-with-axioms are not a negligible cut, they are the most important cuts ever. Counting how many there are and which derivations/proofs they give rise to, is solving the ambiguity of language problem! but you need a good grammar module.. also the application sits "in-between" the proof-search and the proof-normalization paradigms... 22 / 23
- 23. Introduction Glue Semantics Glue in Action References PhD thesis of Asudeh and Lev (Stanford) and Kokkonidis (Oxford) Crouch and van Genabith (Linear Logic for Linguists) Online Bibliography http://users.ox.ac.uk/ lina1301/GlueBibliography.htm plus Google code http://nltk.googlecode.com/svn/trunk/doc/contrib/sem/gluesem.pd https://github.com/nltk/nltk/blob/master/examples/grammars/ 23 / 23