Grammatical resources: logic, structure and control

Transcription

1 Grammatical resources: logic, structure and control Michael Moortgat & Dick Oehrle

2 1 Grammatical composition Grammar logic: the vocabulary Base logic: residuation Residuation Kripke structures Base logic: completeness Canonical model Truth lemma: structural modalities Incompleteness of NL wrt tree models Structural postulates: frame constraints Structural postulates: Proof theory: axiomatic presentations Alternative deductive presentations The base logic: residuation laws The base logic: adjointness, monotonicity Equivalence: Lambek Došen Equivalence: Došen Lambek Unary connectives

3 2.7 Structural options: Structural options: Proof terms: categorical combinators Combinators: adjointness, monotonicity Combinators: structural postulates Combinators: examples Proof theory: Gentzen presentation Gentzen calculus: the base logic Gentzen calculus: unary connectives Eliminability of Cut Equivalence of deductive and Gentzen presentations Equivalence (cont d) Structural rules Implicit structural rules ( sugaring ) Unary connectives: structural options Sugared presentation of KT 4 modalities Cut Elimination Cut elimination: case analysis Principal cut on A /A Principal cut on A A

4 3.14 Permutation case Semantics: the Curry-Howard interpretation The meaning of proofs Semantic domains Typed lambda terms Term assigment: LP Term assignment: unary connectives LP terms for sublinear systems Term assignment: top down Natural deduction Binary connectives Natural deduction: control features Combinator derivation of the N.D. rules Mapping from N.D. to Gentzen derivations Mapping from N.D. to Gentzen derivations (cont d) Interpretation: term assignment to proofs Term assignment: control features

5 1. Grammatical composition 1.1. Grammar logic: the vocabulary φ, ψ p atom φ diamond φ box φ ψ product, fusion φ\ψ left division φ/ψ right division models: M = W, R 2, R 3, V W : linguistic resources, signs R 3 : grammatical composition R 2 : structural control

6 1.2. Base logic: residuation Fusion operators as existential modalities: V ( A) = {x y(r 2 xy & y V (A)} V ( A) = {x y(r 2 yx y V (A)} V (A B) = {z x y[r 3 zxy & x V (A) & y V (B)]} V (C/B) = {x y z[(r 3 zxy & y V (B)) z V (C)]} V (A\C) = {y x z[(r 3 zxy & x V (A)) z V (C)]} Residuation laws: A B A B A C/B A B C B A\C

7 1.3. Residuation Let A = (A, A ) and B = (B, B ) be partially ordered sets. Consider a pair of functions f : A B and g : B A The pair (f, g) is called residuated if the inequalities of ( ) hold. ( ) fx B y iff x A gy Alternatively, a pair of functions (f, g) is characterized as residuated by requiring f and g to be isotone ( ), and by having the compositions fg and gf satisfy the inequalities of ( ). ( ) if x A y (x B y) then fx B fy (gx A gy) ( ) fgx B x, x A gfx

8 1.4. Kripke structures C/B A y A\C B z R 3 xyz A y B x A B C A x B A C/B iff A B C A B iff A B A B C iff B A\C R 2 xy

9 1.5. Base logic: completeness soundness, completeness M = W, R 2, R 3, v A B is provable iff v(a) v(b) for every valuation on every frame

10 1.6. Canonical model Define the canonical model for mixed (2,3) frames as M = W, R 2, R 3, where W is the set of formulae F(/,, \,, ) R 3 (A, B, C) iff A B C, R 2 (A, B) iff A B A v(p) iff A p. truth lemma. For any formula φ, M, A = φ iff A φ. The canonical model is a countermodel for every non-theorem of the base logic. Suppose v(a) v(b) but A B. If A B with the canonical valuation on the canonical frame, A v(a) but A v(b) so v(a) v(b). Contradiction.

11 1.7. Truth lemma: structural modalities Below the direction that requires a little thinking. ( ) Assume A V ( B). We have to show A B. A V ( B) implies A such that R 2 AA and A v(b). By inductive hypothesis, A B. By Isotonicity for this implies A B. We have A A by (Def R 2 ) in the canonical frame. By Transitivity, A B. ( ) Assume A V ( B). We have to show A B. A V ( B) implies that A such that R 2 A A we have A V (B). Let A be A. R 2 A A holds in the canonical frame since A A. By inductive hypothesis we have A B, i.e. A B. By Residuation this gives A B.

12 1.8. Incompleteness of NL wrt tree models A tree frame is a groupoid frame satisfying (i) acyclicity, and (ii) unique splittability: (Rxyz Rxy z ) (x = x y = y ). B A y NL A (A\(B C)) A C C A\(B C) z x A (A\(B C)) B C A C (but valid in tree models!)

13 1.9. Structural postulates: frame constraints Correspondence theory: structural postulates introduce restrictions on the interpretation of the composition relation R. Frame constraints for Associativity and Commutativity ( x, y, z, u W ): (ass) (A B) C A (B C) t.rtxy & Rutz v.rvyz & Ruxv (comm) A B B A Rxyz Rxzy L, NLP, LP A B iff v(a) v(b) for every valuation v on every ternary frame satisfying (ass), (comm), (ass)+(comm), respectively.

14 1.10. Structural postulates: Unary and interaction postulates,. 4 : A A T : A A K1 : (A B) A B K2 : (A B) A B K : (A B) A B Below the correponding frame conditions ( x, y, z, w W ). 4 : (Rxy & Ryz) Rxz T : Rxx K : (Rwx & Rxyz) y z (Ry y & Rz z & Rwy z )

15 2. Proof theory: axiomatic presentations 2.1. Alternative deductive presentations The base logic Residuation laws (a) Adjointness laws, monotonicity (b) Equivalence of (a) and (b) Options for structural resource management Composition : Associativity, Commutativity Control : Reflexivity, Transitivity, Distributivity Characteristic theorems Proof terms: categorical combinators Residuation, adjointness, monotonicity Structural combinators Identifying proofs: categorical equations

16 2.2. The base logic: residuation laws - Derivability: reflexivity, transitivity A A A B B C A C - Residuation laws for, / and, \: A B C A C/B A B C B A\C A C/B A B C B A\C A B C

17 2.3. The base logic: adjointness, monotonicity - Derivability: reflexivity, transitivity A A A B B C A C - Adjointness laws: A/B B A B B\A A A (A B)/B A B\(B A) - Monotonicity laws: A B C D A C B D A B C D A/D B/C A B C D D\A C\B

18 2.4. Equivalence: Lambek Došen A/B A/B (A/B) B A A B A B A (A B)/B A/C A/C (A/C) C A A B (A/C) C B A/C B/C C/B C/B (C/B) B C A B B (C/B)\C A (C/B)\C (C/B) A C C/B C/A A B B (B C)/C C D D B\(B D) A (B C)/C A C B C C B\(B D) B C B D A C B D

19 2.5. Equivalence: Došen Lambek A B C A (A B)/B (A B)/B C/B A C/B A C/B B B A B (C/B) B (C/B) B C A B C

20 2.6. Unary connectives Residuation laws: A B A B A B A B Alternatively: Adjointness laws, Isotonicity. A A A A A B A B A B A B

21 2.7. Structural options: - Structural postulates: associativity, commutativity - Characteristic theorems: associativity A (B C) (A B) C A B B A A/B (A/C)/(B/C), B\A (C\B)\(C\A) B/C (A/B)\(A/C), C\B (C\A)/(B\A) A/B B/C A/C, C\B B\A C\A (A\B)/C A\(B/C) A/(B C) (A/C)/B, (A B)\C B\(A\C) - Characteristic theorems: commutativity if A B\C then B A\C A/B B\A A B/(B/A), A (A\B)\B A/B C\B C\A, B/C B\A A/C

22 2.8. Structural options: - Transitivity (4), Reflexivity (T ), Distributivity (K). 4 : A A T : A A K : (A B) A B - Characteristic theorems ( versions of K, T, 4): 4 : T : K / : K \ : A A A A (A/B) A/ B (B\A) B\ A

23 2.9. Proof terms: categorical combinators Deductions of the form f : A B, where f is a process for deducing B from A. 1 A : A A f : A B µ(f) : A B f : A B C β(f) : A C/B f : A B C γ(f) : B A\C f : A B g : B C g f : A C g : A B µ 1 (g) : A B g : A C/B β 1 (g) : A B C g : B A\C γ 1 (g) : A B C

24 2.10. Combinators: adjointness, monotonicity 1 A : A A f : A B g : B C g f : A C unit : A A co-unit : A A unit / : A/B B A co-unit / : A (A B)/B unit \ : B B\A A co-unit \ : A B\(B A) f : A B (f) : A B f : A B (f) : A B f : A B g : C D f g : A C B D f : A B g : C D f/g : A/D B/C f : A B g : C D g\f : D\A C\B

25 2.11. Combinators: structural postulates α A,B,C : A (B C) (A B) C : α 1 A,B,C π A,B : A B B A ρ A : A A τ A : A A κ A,B : (A B) A B

26 2.12. Combinators: examples - Lifting: 1 A/B : A/B A/B β 1 (1 A/B ) : (A/B) B A γ(β 1 (1 A/B )) : B (A/B)\A - Monotonicity: if A B than A/C B/C 1 A/C : A/C A/C β 1 (1 A/C ) : (A/C) C A f : A B f β 1 (1 A/C ) : (A/C) C B β(f β 1 (1 A/C )) : A/C B/C

27 3. Proof theory: Gentzen presentation 3.1. Gentzen calculus: the base logic - Sequents Γ A with Γ S, A F. - Structures S ::= F S S. - Γ[ ]: term Γ containing a distinguished occurrence of subterm. [Ax] A A A Γ[A] C Γ[ ] C [Cut] [/R] Γ B A Γ A/B [\R] B Γ A Γ B\A B Γ[A] C Γ[A/B ] C B Γ[A] C Γ[ B\A] C [/L] [\L] [ L] Γ[A B] C Γ[A B] C Γ A B Γ A B [ R]

28 3.2. Gentzen calculus: unary connectives Formulas: F ::= A F/F F F F\F F F Structures: S ::= F S S S Logical rules: unary connectives Γ A Γ A R Γ[ A ] B Γ[ A] B L Γ A Γ A R Γ[A] B Γ[ A ] B L

29 3.3. Eliminability of Cut Let be the formula translation of a structured database : ( 1 2 ) = 1 2, A = A, for A F theorem (Lambek 58) For every arrow f : A B there is a Gentzen proof of A B, and for every proof of a sequent Γ B there is an arrow f : Γ B. cut elimination theorem (Lambek 58) The Cut rule is admissible: every theorem has a cut-free proof. corollaria. Decidability, subformula property.

30 3.4. Equivalence of deductive and Gentzen presentations For every arrow f : A B there is a Gentzen proof of A B, and for every proof of a sequent Γ B there is an arrow f : Γ B. ( ). 1 A becomes [Ax], trans is a special case of [Cut]. Derivation of the residuation laws: A A B B A B A B R A (A B)/B /R B B A B C (A B)/B B C /L Cut A, B C A C/B /R B B C C A C/B B B A B (C/B) B R C/B B C /L (C/B) B C L Cut A B C A B C L

31 3.5. Equivalence (cont d) ( ). Replace structural terms Γ by the corresponding product formula Γ. Derivation of [Ax], [/R], [\R], [ L] is immediate. [ R] is the monotonicity law for. Derivation of [/L]: Context empty: f : A C g : B f/g : A/B C/ β 1 (f/g) : (A/B) C f : Γ[A] C. π(f) : A C Γ g : B β 1 (π(f)/g) : (A/B) C Γ Context non-empty:. π 1 (β 1 (π(f)/g)) : Γ[(A/B) ] C

32 3.6. Structural rules Γ[ 2 1 ] A Γ[ 1 2 ] A [P] Γ[( 1 2 ) 3 ] A Γ[ 1 ( 2 3 )] A [A]

33 3.7. Implicit structural rules ( sugaring ) Sequents S F where S ::= F F, S. L: implicit Associativity, interpreting S as a sequence. LP: implicit Associativity+Permutation, interpreting S as a multiset. (The context variables Γ, Γ can be empty.) [Ax] A A A Γ, A, Γ C Γ,, Γ C [Cut] [/R], B A A/B [\R] B, A B\A B Γ, A, Γ C Γ, A/B,, Γ C B Γ, A, Γ C Γ,, B\A, Γ C [/L] [\L] [ L] Γ, A, B, Γ C Γ, A B, Γ C A B, A B [ R]

34 3.8. Unary connectives: structural options Transitivity (4), Reflexivity (T ); interaction, : Distributivity (K). 4 : A A T : A A K1 : (A B) A B K2 : (A B) A B K : (A B) A B Gentzen rules: Γ[ ] A Γ[ ] A 4 Γ[ 1 2 ] A Γ[ 1 2 ] A KΓ[ ] A Γ[ ] A T

35 3.9. Sugared presentation of KT 4 modalities Compiling out the structural punctuation. We write Γ, Γ, Γ for a term Γ of which the (pre)terminal subterms are all of the form A, A, A, respectively. The 4(Cut) step is a series of replacements (read bottom-up) of terminal A by A via Cuts depending on 4. Γ[A] B Γ[ A ] B L Γ[ A] B T Γ[A] B Γ[ A] B L(S4) Γ A Γ A L Γ A 4(Cut) Γ A K Γ A R Γ A Γ A R(S4)

36 3.10. Cut Elimination Proof: a constructive algorithm for stepwise transformation of a derivation involving Cut inferences into a Cut-free derivation. Strategy: induction on the complexity d of Cut inferences, measured in the number of connective occurrences. A Γ, A, Γ B Cut rule: Γ,, Γ Cut B Complexity: d(cut) = d( ) + d(γ) + d(γ ) + d(a) + d(b) Cut formula: A Targets: instances of Cut which have themselves been derived without using the Cut rule. Show that in the derivation in question such a Cut inference can be replaced by one or two Cuts of lower degree. Repeat the process until all Cuts have been removed.

37 3.11. Cut elimination: case analysis Case 1 The base case of the recursion: one of the Cut premises is an Axiom. The other premise is identical to the conclusion, and the application of Cut can be pruned. Case 2 Permutation conversions. The active formula in the left or right premise of Cut is not the Cut formula. One shows that the logical rule introducing the main connective of the active formula and the Cut rule can be permuted, pushing the Cut inference upwards, with a decrease in degree because a connective is now introduced lower in the proof. Case 3 Principal Cuts. The active formula in the left and right premise of Cut make up the Cut formula A. One reduces the degree by splitting the Cut formula up into its two immediate subformulae, and applying Cuts on these.

38 3.12. Principal cut on A /A, A A A /A /R A Γ, A, Γ B Γ, A /A,, Γ B /L Γ,,, Γ Cut B is transformed into, A A Γ, A, Γ B A Γ,, A, Γ B Γ,,, Γ Cut B Cut

39 3.13. Principal cut on A A A A, A A R Γ, A, A, Γ B Γ, A A, Γ B L Γ,,, Γ Cut B is transformed into A Γ, A, A, Γ B A Γ, A,, Γ B Γ,,, Γ Cut B Cut

40 3.14. Permutation case A, A, A, A /A,, A /L Γ, A, Γ B Γ,, A /A,,, Γ B Cut is transformed into, A, A Γ, A, Γ B A Γ,, A,, Γ B Γ,, A /A,,, Γ /L B Cut

41 4. Semantics: the Curry-Howard interpretation 4.1. The meaning of proofs The Curry-Howard-de Bruyn morphism. Slogan: formulas-as-types. Declarative unit: labelled formulae x : A with A: formula, seen as a type x: label in semantic representation language (typed λ term, meaning recipe) Sequents: x 1 : A 1,..., x n : A n t : B Proof: computation of denotation recipe t of type B out of parameters x i of type A i. Rules of inference: operations on semantic labels Proof transformations: term equations

42 4.2. Semantic domains Language of semantic type formulae: A ::= e (individuals, entities) t (truth values) F ::= A F F F F Interpretation: frames F = {D A } A F based on some non-empty set E, the domain of discourse. D e = E (domain of discourse) D t = {0, 1} (set of truth values) D A B = D A D B (Cartesian products) D A B = D D A B (function spaces)

43 4.3. Typed lambda terms Let V A be the set of variables of type A. The set Λ of typed λ terms is {T A } A F, where for all A, B F: T A ::= V A (variables) (T B A T B ) (application) (T A B ) 0 (left projection) (T B A ) 1 (right projection) T A B ::= λv A T B (abstraction) T A B ::= T A, T B (pairing)

44 4.4. Term assigment: LP Notation: x, y, z for variables, t, u, v for arbitrary terms; u[t/x] for the substitution of term t for variable x in term u. In sequents x 1 : A 1,..., x n : A n t : B, the antecedent x i are distinct. x : A x : A (Ax) Γ t : A x : A, u : B Γ, u[t/x] : B (Cut) Γ, x : A, y : B, t : C Γ, y : B, x : A, t : C (P ) t : A Γ, x : B u : C Γ,, y : A B u[(y t)/x] : C ( L) Γ, x : A t : B Γ λx.t : A B ( R) Γ t : A u : B Γ, t, u : A B ( R) Γ, x : A, y : B t : C Γ, z : A B t[(z) 0 /x, (z) 1 /y] : C ( L)

45 4.5. Term assignment: unary connectives Syntax of typed lambda terms: clauses for,. Destructors and, corresponding to rules of use for and. Constructors and, for rules of proof. M A ::=... (M A ) (M A ) M A ::= (M A ) M A ::= (M A ) Term decorated Gentzen rules. The, cases. Γ t : A Γ t : A R Γ t : A Γ t : A R Γ[ y : A ] t : B Γ[x : A] t[ x/y] : B L Γ[y : A] t : B Γ[ x : A ] t[ x/y] : B L

46 4.6. LP terms for sublinear systems Mapping t : F F from syntactic to semantic types, which interprets complex types modulo directionality. t(a/b) = t(b\a) = t(b) t(a), t(a B) = t(a) t(b) Term assignment for sublinear calculi NL, L, NLP using Λ(LP) as the language of semantic composition. Structural rules, if any, are neutral with respect to term assignment: they manipulate formulae with their associated term labels.

47 [Ax] x : A x : A u : A Γ[x : A] t : C Γ[ ] t[u/x] : C [Cut] [/R] Γ x : B t : A Γ λx.t : A/B [\R] x : B Γ t : A Γ λx.t : B\A t : B Γ[x : A] u : C Γ[y : A/B ] u[(y t))/x] : C [/L] t : B Γ[x : A] u : C Γ[ y : B\A] u[(y t)/x] : C [\L] [ L] Γ[x : A y : B] t : C Γ[z : A B] t[(z) 0 /x, (z) 1 /y] : C Γ t : A u : B Γ t, u : A B [ R]

48 4.7. Term assignment: top down Goal sequents x 1 : A 1,..., x n : A n M : A with distinct x i for the assumptions. Notation: lower-case x, y, z (t, u, v) for object-level variables (terms), upper case M, M,... meta-level (search) variables; M := t (one-sided) unification: instantiation of search variable M with lambda term t. [Ax] {M := t} t : A M : A

49 [/R] Γ x : B M : A Γ M : A/B {M := λx.m } N : B Γ[(t N) : A] M : C Γ[t : A/B ] M : C [/L] [ L] Γ[(t) 0 : A (t) 1 : B] M : C Γ[t : A B] M : C Γ M : A M : B Γ M, M : A B [ R]

50 5. Natural deduction 5.1. Binary connectives Notation: Γ A for the deduction of a conclusion A from a configuration of assumptions Γ. Axioms: A A. [/I] Γ B A Γ A/B [\I] B Γ A Γ B\A Γ A/B B Γ A Γ B B\A Γ A [/E] [\E] [ I] Γ A B Γ A B A B Γ[A B] C Γ[ ] C [ E] Elimination and Introduction rules for the logical constants (composition) and /, \ (incompleteness). ( ) as the structural counterpart of ( )

51 5.2. Natural deduction: control features Γ A Γ A ( E) Γ A Γ A ( I) Γ A Γ A ( I) A Γ[ A ] B Γ[ ] B ( E) Elimination and Introduction rules for the logical constants ( key ) and ( lock ) as the structural counterpart of

52 5.3. Combinator derivation of the N.D. rules The Axiom case coincides in the two presentations. (/I) and (\I) become the β and γ half of residuation. ( I) is the Monotonicity rule of inference for. (\E): compose Monotonicity with Application. The (/E) case is similar. f : A g : Γ A\B f g : ( Γ) A A\B app \ (f g) : ( Γ) B app \ : A A\B B ( E): f : (Γ[A B]) C. g : A B π(f) : (A B) C Γ π(f) g : C Γ. π 1 (π(f) g) : (Γ[ ]) C

53 5.4. Mapping from N.D. to Gentzen derivations A A A A D (Γ B) A Γ A/B /I D 1 D 2 Γ A B (Γ ) A B I D (Γ B) A Γ A/B /R D 1 D 2 Γ A B (Γ ) A B R

54 5.5. Mapping from N.D. to Gentzen derivations (cont d) D 1 D 1 Γ A/B B /E (Γ ) A D 1 D 2 A B Γ[(A B)] C E Γ[ ] C D 1 D 2 B A A Γ A/B A/B A /L Cut Γ A D 1 D 2 Γ[A B] C A B Γ[A B] C L Cut Γ[ ] C

55 5.6. Interpretation: term assignment to proofs x : A x : A [/I] Γ x : B t : A Γ λx.t : A/B [\I] x : B Γ t : A Γ λx.t : B\A Γ t : A/B u : B Γ (t u) : A Γ u : B t : B\A Γ (t u) : A [/E] [\E] [ I] Γ t : A u : B Γ t, u : A B u : A B Γ[x : A y : B] t : C Γ[ ] t[(u) 0 /x, (u) 1 /y] : C [ E] /, \ E: function application /, \ I: function abstraction E: projection I: pairing

56 5.7. Term assignment: control features Γ t : A Γ t : A ( E) Γ t : A Γ t : A ( I) Γ t : A Γ t : A ( I) u : A Γ[ x : A ] t : B Γ[ ] t[ u/x] : B ( E)