GF + MMT = GLF From Language to Semantics through LF Michael - PowerPoint PPT Presentation

GF + MMT = GLF From Language to Semantics through LF Michael Kohlhase Jan Frederik Schaefer Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg LFMTP Vancouver, June 22, 2019 1 / 35

Natural Language Semantics run ′ ( mary ′ ) ∧ happy ′ ( john ′ ) “ Mary runs and John is happy. ” ∀ x . love ′ ( x , mary ′ ) “ Everyone loves Mary. ” “ He loves her. ” ∃ X M , Y F . love’ ( X M , Y F ) ¬ ◇ run ′ ( john ′ ) “ John isn’t allowed to run. ” 2 / 35

Natural Language Semantics Definition NL semantics studies the meaning of NL utterances How could we do this? Look at a fragment of English and define a suitable logic [Mon70] ↝ we could cheat a little: run ′ ( mary ′ ) ∧ happy ′ ( mary ′ ) “ Mary runs. She is happy. ” ↝ describe the translation as well 3 / 35

Natural Language Semantics induces M = ⟨D , I⟩ ⊧ ⊆ FL × FL ⊧ ≡ ⊢ C ? I ϕ formulae ⊢ C ⊆ FL × FL FL Analysis ⊧ N L ≡ ⊢ C ? induces ⊧ N L ⊆ NL × NL NL Comp Ling 4 / 35

Natural Language Understanding (NLU) Systems Semantics Semantic Construction Analysis Syntax Logic Logic (compositional) (inferential) Tree Expression Expression parsing NL Utterance 5 / 35

The Grammatical Logical Framework (GLF) parsing Semantics NL Utterance Syntax Logic Construction Tree Expression GF MMT (Grammatical Framework) Transition trivial: compatible logical frameworks GF = grammar development framework + MMT = logic development framework GLF = semantics development framework 6 / 35

The Example ∀ x . run ′ ( x ) “ Everyone runs. ” ∃ x . happy ′ ( x ) “ Someone is happy. ” happy ′ ( john ′ ) ∧ happy ′ ( mary ′ ) “ John and Mary are happy. ” Fragment of English Target logic: FOL 7 / 35

The Grammatical Framework (GF) [Ran11] ● GF is a programming language for multilingual grammar applications ● Abstract syntax : describes parse trees ● Concrete syntaxes : language-specific linearization rules makeStmt abstract syntax john be_happy ⋯ ⋯ “ Johann ist ⋯ “ John is happy ” concrete syntaxes gl¨ ucklich ” 8 / 35

Describing the Fragment in GF – Abstract Syntax abstract Gossip = { cat Actor; Action; Stmt; fun everyone : Actor; someone : Actor; makeStmt : Actor -> Action -> Stmt; twoOf : Actor -> Actor -> Actor; } abstract GossipLex = Gossip ** { fun makeStmt john, mary : Actor; run : Action; twoOf be_happy be_happy : Action; } mary john makeStmt (twoOf john mary) be_happy 9 / 35

Describing the Fragment in GF – Concrete Syntax concrete GossipEng of Gossip = { lincat Actor = Str; Action = Str; Stmt = Str; lin everyone = "everyone"; someone = "someone"; makeStmt actor action = actor ++ action; twoOf a b = a ++ "and" ++ b; } concrete GossipLexEng of GossipLex = GossipEng ** { lin john = "John"; mary = "Mary"; run = "runs"; be_happy = "is happy"; } 10 / 35

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Describing the Fragment in GF – Concrete Syntax Problem “ John is happy ” vs “ John and Mary are happy ” Solutions ● More sophisticated grammar rules ● Use the resource grammar library 12 / 35

Describing the Fragment in GF – Concrete Syntax concrete GossipEng of Gossip = { param Plurality = Sg | Pl; lincat Actor = {s : Str; p : Plurality }; Action = Plurality => Str; Stmt = Str; lin everyone = {s = "everyone"; p = Sg }; someone = {s = "someone"; p = Sg }; makeStmt actor action = actor.s ++ action ! actor.p ; twoOf a b = {s = a.s ++ "and" ++ b.s; p = Pl }; } 13 / 35

Describing the Fragment in GF – Concrete Syntax Resource Grammar Library : grammar rules for 36 languages concrete GossipEng of Gossip = open SyntaxEng, DictEng in { lincat Actor = NP; Action = VP; Stmt = S; lin everyone = everyone_NP; someone = someone_NP; makeStmt actor action = mkS (mkCl actor action); twoOf a b = mkNP and_Conj a b; } 14 / 35

Where are we? makeStmt “ John and Mary ↦ twoOf be_happy are happy ” mary john parsing Semantics NL Utterance Syntax Logic Construction Tree Expression GF MMT (Grammatical Framework) 15 / 35

MMT – “anything you can do we can do meta” [RK13] ● You may remember “Rapid Prototyping Formal Systems in MMT: 5 Case Studies” [MR19] ● Meta meta theories/meta meta tool set ● Little theories ● Bring your own logic ● Logic development environment ● Foundation-independent 16 / 35

From GF to MMT Abstract syntax (GF) Language theory (MMT) abstract Gossip = { theory Gossip : ur:?LF = cat : type ∥ Actor; Actor Action : type ∥ Action; : type ∥ Stmt; Stmt fun everyone : Actor ∥ everyone : Actor; : Actor ∥ someone : Actor; someone makeStmt : makeStmt : Actor → Action → Stmt ∥ Actor->Action->Stmt; twoOf:Actor → Actor → Actor ∥ twoOf:Actor->Actor->Actor; } � makeStmt ↦ makeStmt (twoOf john mary) be happy twoOf be_happy mary john 17 / 35

Where are we? makeStmt (twoOf john mary) be happy “ John and Mary are happy ” makeStmt (twoOf john mary) be happy parsing Semantics NL Utterance Syntax Logic Construction Tree Expression GF MMT (Grammatical Framework) 18 / 35

Target Logic and Domain Theory in MMT theory FOL : ur:?LF = ∥ prop : type | # o : o → o → o | # 1 ∧ 2 ∥ and : o → o | # ¬ 1 ∥ neg : o → o → o | # 1 ∨ 2 | or = [x,y] ¬ ( ( ¬ x) ∧ ( ¬ y) ) ∥ ∥ ind : type | # ι forall : ( ι → o) → o | # ∀ 1 ∥ exists : ( ι → o) → o | # ∃ 1 | = [p] ¬ ( ∀ [x] ( ¬ p x)) ∥ � theory DomainTheory : ?FOL = ∥ mary : ι | # mary’ ∥ john : ι | # john’ : ι → o | # run’ 1 ∥ run happy : ι → o | # happy’ 1 ∥ � 19 / 35

Where are we? makeStmt (twoOf john mary) be happy “ John and Mary (happy’ john’) ∧ (happy’ mary’) are happy ” makeStmt (twoOf john mary) be happy parsing Semantics NL Utterance Syntax Logic Construction Tree Expression GF MMT (Grammatical Framework) 20 / 35

Semantics Construction in MMT Gossip FOL Actor , Action , Stmt o ,ι ∧ , ¬ , ∨ everyone , someone , ∀ , ∃ makeStmt , twoOf GossipLex DomainTheory john ′ , mary ′ , john , mary , run ′ , happy ′ run , be happy 21 / 35

Naive Approach view GossipSem : ?Gossip -> ?FOL = = o ∥ Stmt = ι → o ∥ Action = ι ∥ Actor everyone = ??? ∥ = ??? ∥ someone makeStmt = [a, ϕ ] ϕ a ∥ = ??? ∥ twoOf � view GossipLexSem : ?GossipLex -> ?DomainTheory = include ?GossipSem ∥ = john’ ∥ john = mary’ ∥ mary = [x] run’ x ∥ run be_happy = [x] happy’ x ∥ � 22 / 35

Type Raising [Mon74] Problem Actor = ι everyone : Actor = ? Solution Actor = ( ι → o) → o john = [ ϕ ] ϕ john’ everyone = [ ϕ ] ∀ [x] ( ϕ x) = [ ϕ ] ∀ ϕ Example “ everyone runs ” ↦ ↝ β ([ ϕ ] ∀ [x] ( ϕ x)) run’ ∀ [x] (run’ x) 23 / 35

Better Approach view GossipSem : ?Gossip -> ?FOL = = o ∥ Stmt = ι → o ∥ Action = ( ι → o) → o ∥ Actor everyone = [ ϕ ] ∀ ϕ ∥ = [ ϕ ] ∃ ϕ ∥ someone makeStmt = [a, ϕ ] a ϕ ∥ = [a1,a2] [ ϕ ] (a1 ϕ ) ∧ (a2 ϕ ) ∥ twoOf � view GossipLexSem : ?GossipLex -> ?DomainTheory = include ?GossipSem ∥ = [ ϕ ] ϕ john’ ∥ john = [ ϕ ] ϕ mary’ ∥ mary = [x] run’ x ∥ run be_happy = [x] happy’ x ∥ � 24 / 35

Better Approach view GossipSem : ?Gossip -> ?FOL = = o ∥ Stmt = ι → o ∥ Action = ( ι → o) → o ∥ Actor everyone = [ ϕ ] ∀ ϕ ∥ = [ ϕ ] ∃ ϕ ∥ someone makeStmt = [a, ϕ ] a ϕ ∥ = [a1,a2] [ ϕ ] (a1 ϕ ) ∧ (a2 ϕ ) ∥ twoOf � These views are described in NL semantics papers like [Mon74]: view GossipLexSem : ?GossipLex -> ?DomainTheory = include ?GossipSem ∥ = [ ϕ ] ϕ john’ ∥ john = [ ϕ ] ϕ mary’ ∥ mary = [x] run’ x ∥ run be_happy = [x] happy’ x ∥ � 24 / 35

Example “ John and Mary are happy ” ↓ parse makeStmt (twoOf john mary) be happy ↓ semantics construction ([a, ϕ ] a ϕ ) (([a1,a2] [ ϕ ] (a1 ϕ ) ∧ (a2 ϕ )) ([ ϕ ] ϕ john’) ([ ϕ ] ϕ mary’)) ([x] happy’ x) ↓ simplify ([a1,a2, ϕ ] (a1 ϕ ) ∧ (a2 ϕ )) ([ ϕ ] ϕ john’) ([ ϕ ] ϕ mary’) happy’ ↓ simplify ([ ϕ ] ( ϕ john’) ∧ ( ϕ mary’)) happy’ ↓ simplify (happy’ john’) ∧ (happy’ mary’) 25 / 35

Other Examples (from paper) Adding Transitive Verbs ( ↝ more type raising) “ John and Mary love everyone ” ↓ ∀ [x: ι ](love’ john’ x) ∧ (love’ mary’ x) 26 / 35

Other Examples (from paper) (Multi) Modal Logic Modalities: ● deontic – something is obligatory ( � d � ) or permissible ( ⟪ d ⟫ ) ● epistemic – someone believes something is true ( � e john’ � ) or possible ( ⟪ e john’ ⟫ ). “ John doesn’t believe that Mary has to run ” ↓ ¬ � (e john’) �� d � (run’ mary’) . 27 / 35

GLF Script Please enter a sentence: John isn’t allowed to run I got the following interpretations: ¬⟪ d ⟫ (run’ john’) Please enter a sentence: Mary believes that John doesn’t have to run I got the following interpretations: � (e mary’) � ¬ � d � (run’ john’) Please enter a sentence: 28 / 35