14 research outputs found
A minimal classical sequent calculus free of structural rules
Gentzen's classical sequent calculus LK has explicit structural rules for
contraction and weakening. They can be absorbed (in a right-sided formulation)
by replacing the axiom P,(not P) by Gamma,P,(not P) for any context Gamma, and
replacing the original disjunction rule with Gamma,A,B implies Gamma,(A or B).
This paper presents a classical sequent calculus which is also free of
contraction and weakening, but more symmetrically: both contraction and
weakening are absorbed into conjunction, leaving the axiom rule intact. It uses
a blended conjunction rule, combining the standard context-sharing and
context-splitting rules: Gamma,Delta,A and Gamma,Sigma,B implies
Gamma,Delta,Sigma,(A and B). We refer to this system M as minimal sequent
calculus.
We prove a minimality theorem for the propositional fragment Mp: any
propositional sequent calculus S (within a standard class of right-sided
calculi) is complete if and only if S contains Mp (that is, each rule of Mp is
derivable in S). Thus one can view M as a minimal complete core of Gentzen's
LK.Comment: To appear in Annals of Pure and Applied Logic. 15 page
Computational interpretation of classical logic with explicit structural rules
We present a calculus providing a Curry-Howard correspondence to classical logic represented in the sequent calculus with explicit structural rules, namely weakening and contraction. These structural rules introduce explicit erasure and duplication of terms, respectively. We present a type system for which we prove the type-preservation under reduction. A mutual relation with classical calculus featuring implicit structural rules has been studied in detail. From this analysis we derive strong normalisation property
From X to Pi; Representing the Classical Sequent Calculus in the Pi-calculus
We study the Pi-calculus, enriched with pairing and non-blocking input, and
define a notion of type assignment that uses the type constructor "arrow". We
encode the circuits of the calculus X into this variant of Pi, and show that
all reduction (cut-elimination) and assignable types are preserved. Since X
enjoys the Curry-Howard isomorphism for Gentzen's calculus LK, this implies
that all proofs in LK have a representation in Pi.Comment: International Workshop on Classical Logic and Computation (CL&C'08),
Reykjavik, Iceland, July 200
Three Dimensional Proofnets for Classical Logic
Classical logic and more precisely classical sequent calculi are currently the subject of several studies that aim at providing them with an algorithmic meaning. They are however ruled by an annoying syntactic bureaucracy which is a cause of pathologic non-confluence. An interesting patch consists in representing proofs using proofnets. This leads (at least in the propositional case) to cut-elimination procedures that remain confluent and strongly normalising without using any restricting reduction strategy. In this paper we describe a presentation of sequents in a two-dimensional space as well as a presentation of proofnets and sequent calculus derivations in a three-dimensional space. These renderings admit interesting geometrical properties: sequent occurrences appear as parallel segments in the case of three-dimensional sequent calculus derivations and the De Morgan duality is expressed by the fact that negation stands for a ninety degree rotation in the case of two-dimensional sequents and three-dimensional proofnets
Principles of Superdeduction
International audienceIn predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, Deduction modulo allows us to make use of the computational part of the theory Th for true computations modulo which deductions are performed. Focussing on the sequent calculus, this paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. We call such a new deduction system "superdeduction''. We introduce a proof-term language and a cut-elimination procedure both based on Christian Urban's work on classical sequent calculus. Strong normalisation is proven under appropriate and natural hypothesis, therefore ensuring the consistency of the embedded theory and of the deduction system. The proofs obtained in such a new system are much closer to the human intuition and practice. We consequently show how superdeduction along with deduction modulo can be used to ground the formal foundations of new extendible proof assistants. We finally present lemuridae, our current implementation of superdeduction modulo
Continuation-Passing Style and Strong Normalisation for Intuitionistic Sequent Calculi
The intuitionistic fragment of the call-by-name version of Curien and
Herbelin's \lambda\_mu\_{\~mu}-calculus is isolated and proved strongly
normalising by means of an embedding into the simply-typed lambda-calculus. Our
embedding is a continuation-and-garbage-passing style translation, the
inspiring idea coming from Ikeda and Nakazawa's translation of Parigot's
\lambda\_mu-calculus. The embedding strictly simulates reductions while usual
continuation-passing-style transformations erase permutative reduction steps.
For our intuitionistic sequent calculus, we even only need "units of garbage"
to be passed. We apply the same method to other calculi, namely successive
extensions of the simply-typed λ-calculus leading to our intuitionistic
system, and already for the simplest extension we consider (λ-calculus
with generalised application), this yields the first proof of strong
normalisation through a reduction-preserving embedding. The results obtained
extend to second and higher-order calculi