Mathematical Synthesis of Equational Deduction Systems. Marcelo Fiore. Computer Laboratory University of Cambridge

Transcription

1 Mathematical Synthesis of Equational Deduction Systems Marcelo Fiore Computer Laboratory University of Cambridge TLCA VII.2009

2 Context concrete theories meta-theories

3 Context concrete theories meta-theories Calculi features: For: higher-order, formalisation, linearity, reasoning, sharing, computation, graphics, translation, type dependency,......

4 concrete theories Context meta-theories Calculi features: For: higher-order, formalisation, linearity, reasoning, sharing, computation, graphics, translation, type dependency,......

5 concrete theories Context meta-theories Calculi features: For: higher-order, formalisation, linearity, reasoning, sharing, computation, graphics, translation, type dependency, [1] mathematical models Programme

6 concrete theories Context meta-theories Calculi features: For: higher-order, formalisation, linearity, reasoning, sharing, computation, graphics, translation, type dependency, Programme [1] mathematical models [2] meta-theories

7 This Talk Part I Mathematical algebraic framework and methodology for the synthesis of deduction systems for equational reasoning and computation by rewriting.

8 This Talk Part I Mathematical algebraic framework and methodology for the synthesis of deduction systems for equational reasoning and computation by rewriting. Semantics Algebraic model theory Syntax Initial-algebra semantics ( compositionality) structural recursion induction principle theory of translations

9 This Talk Part II TLCA Typed Lambda Calculi: [simply-typed] λ-calculus, Martin Löf type theory,...? What are Typed Lambda Calculi?

10 This Talk Part II TLCA Typed Lambda Calculi: [simply-typed] λ-calculus, Martin Löf type theory,...? What are Typed Lambda Calculi?! Simple Type Theories are Second-Order Equational Presentations!

11 Part I Equational Meta-Logic

12 Universal Algebra Syntax Semantics

13 Universal Algebra Syntax Semantics signatures: Σ = {Σ n Set } n N

14 Universal Algebra Syntax signatures: Σ = {Σ n Set } n N Semantics algebras: A = {Σ n A n A} n N

15 Universal Algebra Syntax signatures: Σ = {Σ n Set } n N Semantics algebras: A = {Σ n A n A} n N free constructions: V ρ T(V) A!ρ #

16 Universal Algebra Syntax signatures: Σ = {Σ n Set } n N terms, variables Semantics algebras: A = {Σ n A n A} n N free constructions: V ρ T(V) A!ρ #

17 Universal Algebra Syntax signatures: Σ = {Σ n Set } n N terms, variables, and substitution: T(V) A V A t, ρ t[ρ] Semantics algebras: A = {Σ n A n A} n N free constructions: V ρ T(V) A!ρ #

18 Universal Algebra Syntax signatures: Σ = {Σ n Set } n N terms, variables, and substitution: T(V) A V A t, ρ t[ρ] equations: V t t Semantics algebras: A = {Σ n A n A} n N free constructions: V ρ T(V) A!ρ #

19 Universal Algebra Syntax signatures: Σ = {Σ n Set } n N terms, variables, and substitution: T(V) A V A t, ρ t[ρ] equations: V t t Semantics algebras: A = {Σ n A n A} n N free constructions: V ρ T(V) A!ρ # validity: A = t t iff ρ A V.t[ρ] = t [ρ]

20 Birkhoff s Deduction Problem Devise a deduction system such that t t is derivable from a set of equations E soundness completeness for all A = E, A = t t

21 Birkhoff s Deduction Problem Devise a deduction system such that t t is derivable from a set of equations E soundness completeness for all A = E, A = t t Birkhoff s Equational Logic of Universal Algebra (t t ) E t t t i t i (i = 1,...,n) f(t 1,...,t n ) f(t 1,...,t n) (f : n) t t t[ρ] t [ρ] (ρ a substitution)

22 Analysis of Universal Algebra Syntax signatures: Σ = {Σ n Set } n N terms, variables, and substitution: T(V) A V A t, ρ t[ρ] equations: V t t Semantics algebras: A = {Σ n A n A} n N free constructions: V ρ T(V) A!ρ # validity: A = t t iff ρ A V.t[ρ] = t [ρ]

23 Monadic Algebra Syntax Semantics T a strong monad on a smc category algebras: A = (TA A)

24 Monadic Algebra Generalised Syntax generalised terms: t : I TV (Kleisli maps) Semantics T a strong monad on a smc category algebras: A = (TA A)

25 Monadic Algebra Generalised Syntax generalised terms: t : I TV (Kleisli maps) variables: V TV Semantics T a strong monad on a smc category algebras: A = (TA A)

26 Monadic Algebra Generalised Syntax generalised terms: t : I TV (Kleisli maps) variables: V TV substitution: σ : TV [V,A] A Semantics T a strong monad on a smc category algebras: A = (TA A)

27 Monadic Algebra Generalised Syntax generalised terms: t : I TV (Kleisli maps) variables: V TV substitution: σ : TV [V,A] A Semantics T a strong monad on a smc category algebras: A = (TA A) interpretation: I [V,A] t id TV [V,A] [t] A σ

28 Monadic Algebra Generalised Syntax generalised terms: t : I TV (Kleisli maps) variables: V TV substitution: σ : TV [V,A] A generalised equations: t t : I TV Semantics T a strong monad on a smc category algebras: A = (TA A) interpretation: I [V,A] t id TV [V,A] [t] A σ

29 Monadic Algebra Generalised Syntax generalised terms: t : I TV (Kleisli maps) variables: V TV substitution: σ : TV [V,A] A generalised equations: t t : I TV Semantics T a strong monad on a smc category algebras: A = (TA A) interpretation: I [V,A] t id TV [V,A] [t] A σ validity: A = t t iff [[t]] = [[t ]]

30 Deduction Problem Devise a deduction system such that t t : I TV is derivable from a set of generalised equations E soundness completeness for all A = E, A = t t

31 Equational Meta-Logic Rules

32 Equational Meta-Logic Rules (Subst) t 1 t 1 : U TV t 2 t 2 : V TW t 1 [t 2 ] t 1 [t 2 ] : U TW

33 Equational Meta-Logic Rules (Subst) t 1 t 1 : U TV t 2 t 2 : V TW t 1 [t 2 ] t 1 [t 2 ] : U TW {e i : U i U} i I a cover (LocChar) te i t e i : U i TV (i I) t t : U TV

34 Equational Meta-Logic Rules (Subst) t 1 t 1 : U TV t 2 t 2 : V TW t 1 [t 2 ] t 1 [t 2 ] : U TW {e i : U i U} i I a cover (LocChar) te i t e i : U i TV (i I) t t : U TV (Ext) t t : U TV I t I t : I U T(I V)

35 Soundness If t t : I TV is derivable from E then A = t t, for all A = E.

36 Soundness If t t : I TV is derivable from E then A = t t, for all A = E. Internal Soundness and Completeness For TV the free algebra satisfying E and q : TV TV the associated quotient map, the following are equivalent: 1. A = t t : U TV, for all A = E 2. TV = t t : U TV 3. qt = qt : U TV

37 Reconstruction of Birkhoff s Equational Logic Set syntactic structure = signature: Σ = {Σ n Set } n N

38 Reconstruction of Birkhoff s Equational Logic Σ-Alg Set T Σ interpretation T Σ (X) = X + n N Σ n (T Σ X) n syntactic structure = signature: Σ = {Σ n Set } n N

39 Reconstruction of Birkhoff s Equational Logic interpretation Σ-Alg Set T Σ T Σ (X) = X + n N Σ n (T Σ X) n syntactic structure = signature: Σ = {Σ n Set } n N concretion Equational Logic V t t

40 Substitution Rule (Subst) V s s (LocChar) W t x t x (x V) W {t x } x V {t x} x V W s[t x /x] x V s [t x/x] x V

41 Reconstruction of Goguen and Meseguer s Multi-Sorted Equational Logic Set S syntactic structure = signature: Σ = {Σ σ Set S } σ S General method for the extension from the mono-sorted to the multi-sorted case.

42 Reconstruction of Goguen and Meseguer s Multi-Sorted Equational Logic Σ-Alg Set S T Σ (T Σ X) s = X s + σ=(s 1...s n ) S Σ σ,s n i=1 (T ΣX) si interpretation syntactic structure = signature: Σ = {Σ σ Set S } σ S General method for the extension from the mono-sorted to the multi-sorted case.

43 Reconstruction of Goguen and Meseguer s Multi-Sorted Equational Logic Σ-Alg Set S T Σ (T Σ X) s = X s + σ=(s 1...s n ) S Σ σ,s n i=1 (T ΣX) si interpretation syntactic structure = signature: Σ = {Σ σ Set S } σ S concretion Equational Logic x : σ t t : s General method for the extension from the mono-sorted to the multi-sorted case.

44 Methodology Model abstract syntax T S universe of discourse

45 Methodology Model abstract syntax T S universe of discourse Theory Deductive System axioms A f f : C TA co-arities arities sound for a canonical algebraic model theory + framework for completeness

46 Methodology Model abstract syntax T S universe of discourse Theory Deductive System axioms A f f : C TA co-arities arities sound for a canonical algebraic model theory + framework for completeness concretion interpretation syntactic structure Equational Logical Framework A Γ t t

47 Part II The Universal Algebra of Simple Type Theory

48 Simply Typed Theories types algebraic terms algebraic with binding

49 Simply Typed Theories algebraic simply typed theories dependently typed types unstructured algebraic terms algebraic algebraic with binding algebraic with binding algebraic with binding

50 The paradigmatic simple type theory: λ-calculus (β) (λx.m)n = M[N/x] ( ) (η) λx.mx = M x FV(M) Simply-typed λ-calculus, computational λ-calculus,...

51 The paradigmatic simple type theory: λ-calculus (β) (λx.m)n = M[N/x] ( ) (η) λx.mx = M x FV(M) Simply-typed λ-calculus, computational λ-calculus,... The syntactic theory should account for: variables and meta-variables variable binding and α-equivalence capture-avoiding and meta substitution mono and multi sorting

52 Algebraic Model syntactic structure = arities: an operator of arity n = (n 1...n k ) takes k arguments, respectively binding n i variables.

53 Algebraic Model finite sets (contexts) and functions (renamings) Set F X Set F is a functor XΓ (Γ F) F(Γ, ) Set(XΓ,X ) E.g. the object of variables is VΓ = Γ syntactic structure = arities: an operator of arity n = (n 1...n k ) takes k arguments, respectively binding n i variables.

54 Algebraic Model finite sets (contexts) and functions (renamings) Set F X Set F is a functor XΓ (Γ F) F(Γ, ) Set(XΓ,X ) E.g. the object of variables is VΓ = Γ syntactic structure = arities: an operator of arity n = (n 1...n k ) takes k arguments, respectively binding n i variables. signature: Σ = {Σ n Set F } n N

55 Algebraic Model Σ finite sets (contexts) and functions (renamings) Set F Σ(X) = n=(n 1...n k ) N Σ n k interpretation X Set F is a functor XΓ (Γ F) i=1 XVn i F(Γ, ) Set(XΓ,X ) E.g. the object of variables is VΓ = Γ syntactic structure = arities: an operator of arity n = (n 1...n k ) takes k arguments, respectively binding n i variables. signature: Σ = {Σ n Set F } n N

56 Algebraic Model Σ finite sets (contexts) and functions (renamings) Set F Σ(X) = n=(n 1...n k ) N Σ n k interpretation X Set F is a functor XΓ (Γ F) i=1 XVn i F(Γ, ) Set(XΓ,X ) E.g. the object of variables is VΓ = Γ syntactic structure = arities: an operator of arity n = (n 1...n k ) takes k arguments, respectively binding n i variables. signature: Σ = {Σ n Set F } n N + substitution

57 Algebras with Substitution (Σ-monoids) algebra structure: ΣX ξ X substitution structure: ( monoid V e X X m X Γ XΓ X (XΓ) subject to the laws of substitution ) subject to the compatibility condition: Σ(X) X Σ(X X) Σm ΣX ξ X X X m X ξ

58 Signature. λ Pre-Models with Substitution Σ λ = 0@0 (application) λ(1) (abstraction) Algebras. app : L 2 L abs : L V L var : V L L L : subst L 2 L = (L L) 2 subst 2 L 2 app L L L subst L app L V L (L L) V subst V L V abs L L L subst L abs

59 Monadic Model Σ-Mon M Σ Set F Thm: 1. M Σ (X) = V + X M Σ (X) + Σ(M Σ X)

60 Monadic Model Σ-Mon M Σ Set F Thm: 1. M Σ (X) = V + X M Σ (X) + Σ(M Σ X) 2. For Σ induced by a binding signature, concretion M Σ is a strong monad. Syntactic theory for variables, meta-variables, variable binding, α-equivalence, capture-avoiding substitution, meta-substitution.

61 Second-Order Syntactic Theory (I) Syntax: For X an object of meta-variables, t M Σ (X) Γ ::= x (x Γ) ( M X(l) M[t 1,...,t l ] t i (M Σ X) Γ f ( ( x 1 )t 1,...,( x k )t k ) ) ( f Σ ( x1... x k ) t i (M Σ X) Γ, xi ) )

62 Second-Order Syntactic Theory (I) Syntax: For X an object of meta-variables, t M Σ (X) Γ ::= x (x Γ) ( M X(l) M[t 1,...,t l ] t i (M Σ X) Γ f ( ( x 1 )t 1,...,( x k )t k ) ) ( f Σ ( x1... x k ) t i (M Σ X) Γ, xi Capture-avoiding substitution: M Σ (X) M Σ (X) M Σ (X) ( M Σ (X) ( ) ) M Σ (X) Γ M Σ (X) Γ ) )

63 Second-Order Syntactic Theory (I) Syntax: For X an object of meta-variables, t M Σ (X) Γ ::= x (x Γ) ( M X(l) M[t 1,...,t l ] t i (M Σ X) Γ f ( ( x 1 )t 1,...,( x k )t k ) ) ( f Σ ( x1... x k ) t i (M Σ X) Γ, xi Capture-avoiding substitution: M Σ (X) M Σ (X) M Σ (X) ( M Σ (X) ( ) ) M Σ (X) Γ M Σ (X) Γ Meta-substitution: M Σ (X) ( M Σ (Y) ) X MΣ (Y) ( M Σ (X) Γ l N X(l) ( (M Σ Y) Vl )Γ M Σ(Y) Γ ) ) )

64 Second-Order Syntactic Theory (II) Canonical specification and derived correct definition of variable renaming, capture-avoiding simultaneous substitution, meta-variable renaming, meta-substitution. Canonical algebraic model theory.

65 Second-Order Equational Presentations x 1,...,x n ; M 1 [m 1 ],...,M k [m k ] t t Second-Order Algebraic Models [[t]] = [[t ]] : V n A Vm 1... A Vm k A

66 Second-Order Equational Presentations x 1,...,x n ; M 1 [m 1 ],...,M k [m k ] t t λ-calculus (β) M[1], N[0] λ ( (x)m[ x ] N[ ] M [ N[ ] ] (η) M[ ] λ ( (x) M[ ]@ x ) M[ ] Second-Order Algebraic Models [[t]] = [[t ]] : V n A Vm 1... A Vm k A

67 Second-Order Equational Presentations x 1,...,x n ; M 1 [m 1 ],...,M k [m k ] t t λ-calculus (β) M[1], N[0] λ ( (x)m[ x ] N[ ] M [ N[ ] ] (η) M[ ] λ ( (x) M[ ]@ x ) M[ ] Second-Order Algebraic Models [[t]] = [[t ]] : V n A Vm 1... A Vm k A λ-models with substitution L V L abs id L L (β) app L L L subst (V L) V (var L) V L id L (η) abs (L L) V app V L V

68 Second-Order Equational Logic (I) (Subst) t 1 t 1 : U TV t 2 t 2 : V TW t 1 [t 2 ] t 1 [t 2 ] : U TW

69 Second-Order Equational Logic (I) (Subst) t 1 t 1 : U TV t 2 t 2 : V TW t 1 [t 2 ] t 1 [t 2 ] : U TW Substitution Rule x; M 1 [m 1 ],...,M k [m k ] t u y (i) 1,...,y(i) m i ; N 1 [n 1 ],...,N l [n l ] t i u i x; N 1 [n 1 ],...,N l [n l ] t{m i := (y (i) 1,...,y(i) m i )t i } u{m i := (y (i) 1,...,y(i) m i )u i } (i = 1,...,k)

70 Second-Order Equational Logic (II) (LocChar) {e i : U i U} i I a cover te i t e i : U i TV (i I) t t : U TV Local Character Rule ı : {x 1,...,x k } {y 1,...,y l } y 1,...,y l ;...,M i [m i ],... t[ı] u[ı] x 1,...,x k ;...,M i [m i ],... t u

71 Second-Order Equational Logic (III) (Ext) t t : U TV I t I t : I U T(I V) Extension Rule x;...,m i [m i ],... t 1 t 2 x,y 1,...,y l ;...,M i [m i + l],... t 1 # t 2 # t # = t { M i := (z (i) 1,...,z(i) m i ) M i [z (i) 1,...,z(i) m i, y 1,...,y l ] }

72 Algebraic Simple Type Theory Algebraic model theory & Second-order equational logic for second-order equational presentations + Theory of translations