LF P A Logical Framework with External Predicates

Transcription

1 LF P A Logical Framework with External Predicates Petar Maksimović in collaboration with Furio Honsell, Marina Lenisa, Luigi Liquori, and Ivan Scagnetto Mathematical Institute of the Serbian Academy of Sciences and Arts, Serbia Faculty of Technical Sciences, University of Novi Sad, Serbia INRIA Sophia Antipolis Méditerranée, France Università di Udine, Italy Fourth Workshop on Formal and Automated Theorem Proving and Applications, February 2-3, 2012, Belgrade, Serbia

2 Briefly about LF The Harper-Honsell-Plotkin Logical Framework LF a logical framework based on the λπ-calculus Dependent types - types depending on terms Based on the Curry-Howard isomorphism Basis for the proof assistant Twelf

3 The ideas behind LF P The main ideas Develop a way of easily and smoothly encoding logics with arbitrary structural side-conditions in LF, Separate derivation from verification/computation, Increase modularity, and Optimize performance.

4 The pseudo-syntax of LF P Σ S Σ ::= Σ, a:k Σ, c:σ Signatures Γ C Γ ::= Γ, x:σ Contexts K K K ::= Type Πx:σ.K Kinds σ, τ, ρ F σ ::= a Πx:σ.τ σ N L P N,σ [ρ] Families M, N O M ::= c x λx:σ.m M N L P N,σ [M] U N,σ P [M] Objects Figure: The pseudo-syntax of LF P

5 So, what is new? Predicates on derivable typing judgements P(Γ Σ N : σ) Truth verified via an external call to a logical system, Can inspect the signature, context, term, and the type.

6 So, what is new? Predicates on derivable typing judgements P(Γ Σ N : σ) Truth verified via an external call to a logical system, Can inspect the signature, context, term, and the type. Locked types (L P N,σ [ρ]), locked objects (LP N,σ [M]), and unlocked objects (UN,σ P [M]),

7 So, what is new? Predicates on derivable typing judgements P(Γ Σ N : σ) Truth verified via an external call to a logical system, Can inspect the signature, context, term, and the type. Locked types (L P N,σ [ρ]), locked objects (LP N,σ [M]), and unlocked objects (UN,σ P [M]), Introduction rules: Γ Σ ρ : Type Γ Σ N : σ Γ Σ M : ρ Γ Σ N : σ Γ Σ L P N,σ [ρ] : Type Γ Σ L P N,σ [M] : LP N,σ [ρ]

8 So, what is new? Predicates on derivable typing judgements P(Γ Σ N : σ) Truth verified via an external call to a logical system, Can inspect the signature, context, term, and the type. Locked types (L P N,σ [ρ]), locked objects (LP N,σ [M]), and unlocked objects (UN,σ P [M]), Introduction rules: Γ Σ ρ : Type Γ Σ N : σ Γ Σ M : ρ Γ Σ N : σ Γ Σ L P N,σ [ρ] : Type Γ Σ L P N,σ [M] : LP N,σ [ρ] Elimination rule: Γ Σ M : L P N,σ [ρ] Γ Σ N : σ P(Γ Σ N : σ) Γ Σ U P N,σ [M] : ρ

9 So, what is new? Predicates on derivable typing judgements P(Γ Σ N : σ) Truth verified via an external call to a logical system, Can inspect the signature, context, term, and the type. Locked types (L P N,σ [ρ]), locked objects (LP N,σ [M]), and unlocked objects (UN,σ P [M]), Introduction rules: Γ Σ ρ : Type Γ Σ N : σ Γ Σ M : ρ Γ Σ N : σ Γ Σ L P N,σ [ρ] : Type Γ Σ L P N,σ [M] : LP N,σ [ρ] Elimination rule: Γ Σ M : L P N,σ [ρ] Γ Σ N : σ P(Γ Σ N : σ) Γ Σ U P N,σ [M] : ρ L-reduction: U P N,σ [LP N,σ [M]] L M.

10 Properties of LF P The main properties Confluence, Strong Normalization - Yes, immediately.

11 Properties of LF P The main properties Confluence, Strong Normalization - Yes, immediately. Subject Reduction - Yes, with certain conditions imposed on predicates (closure under signature and context weakening and permutation, substitution, and βl-reduction.

12 Properties of LF P The main properties Confluence, Strong Normalization - Yes, immediately. Subject Reduction - Yes, with certain conditions imposed on predicates (closure under signature and context weakening and permutation, substitution, and βl-reduction. Decidability - Yes, if predicates used are decidable.

13 Properties of LF P The main properties Confluence, Strong Normalization - Yes, immediately. Subject Reduction - Yes, with certain conditions imposed on predicates (closure under signature and context weakening and permutation, substitution, and βl-reduction. Decidability - Yes, if predicates used are decidable. A canonical version of the system (LF C P ) was also developed, dealing only with βη-long normal forms, for easy formulation and proofs of adequacy of the encodings.

14 Encoded examples The untyped λ-calculus (using HOAS)

15 Encoded examples The untyped λ-calculus (using HOAS) The untyped λ-calculus with call-by-value reduction strategy β-reduction in M N occurs only if N is a value

16 Encoded examples The untyped λ-calculus (using HOAS) The untyped λ-calculus with call-by-value reduction strategy β-reduction in M N occurs only if N is a value Modal logics S 4 and S 5 in Hilbert and Natural deduction style Rule applicable if formula does not depend on assumptions

17 Encoded examples The untyped λ-calculus (using HOAS) The untyped λ-calculus with call-by-value reduction strategy β-reduction in M N occurs only if N is a value Modal logics S 4 and S 5 in Hilbert and Natural deduction style Rule applicable if formula does not depend on assumptions Non-commutative linear logic Conditions on occurence and order of assumptions

18 Encoded examples The untyped λ-calculus (using HOAS) The untyped λ-calculus with call-by-value reduction strategy β-reduction in M N occurs only if N is a value Modal logics S 4 and S 5 in Hilbert and Natural deduction style Rule applicable if formula does not depend on assumptions Non-commutative linear logic Conditions on occurence and order of assumptions A simple imperative language with Hoare-like logic Pre- and post-conditions

19 The end of the presentation Thank you for your attention! Any questions?