Strong Normalisation of Cut-Elimination in Classical Logic

Cut-free proofs in Herbelin’s sequent calculus are in 1-1 correspondence with normal natural deduction proofs. For this reason Herbelin’s sequent calculus has been considered a privileged middle-point between L-systems and natural deduction. However, this bijection does not extend to proofs containing cuts and Herbelin observed that his cutelimination procedure is not isomorphic to β-reduction.In this paper we equip Herbelin’s system with rewrite rules which, at the same time: (1) complete in a sense the cut elimination procedure firstly proposed by Herbelin; and (2) perform the intuitionistic “fragment” of the tq-protocol - a cut-elimination procedure for classical logic defined by Danos, Joinet and Schellinx. Moreover we identify the subcalculus of our system which is isomorphic to natural deduction, the isomorphism being with respect not only to proofs but also to normalisation.Our results show, for the implicational fragment of intuitionistic logic, how to embed natural deduction in the much wider world of sequent calculus and what a particular cut-elimination procedure normalisation is.KeywordsClassical LogicIntuitionistic LogicReduction RuleNatural DeductionSequent CalculusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A mechanical proof of the Knuth---Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet's well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth---Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.

System I is a simply-typed lambda calculus with pairs, extended with an equational theory obtained from considering the type isomorphisms as equalities. In this work we propose an extension of System I to polymorphic types, adding the corresponding isomorphisms. We provide non-standard proofs of subject reduction and strong normalisation, extending those of System I.

This paper presents simple, syntactic strong normalization proofs for the simply-typed i¾?-calculus and the polymorphic i¾?-calculus (system F ) with the full set of logical connectives, and all the permutative reductions. The normalization proofs use translations of terms and types of to terms and types of and from to .

A proof of ∀t∃nSN(t, n) (term t performs at most n reduction steps) is given, based on strong computability predicates. Using modified realizability, a bound on reduction lengths is extracted from it. This upper bound is compared with the one Gandy defines, using strictly monotonic functionals. This reveals a remarkable connection between his proof and Tait's. We show the details for simply typed λ-calculus and Gödel's T. For the latter system, program extraction yields considerably sharper upper bounds.KeywordsFormal ProofFinite TypePredicate SymbolReduction SequenceExistential QuantifierThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

When looking at systems of typed A calculus from a logical point of view, there are some interesting questions that arise. One of them is whether the formulasas-types embedding from the logic into the typed A calculus is complete, that is, whether the types that are inhabited in the typed A calculus are provable (as formulas) in the logic. It is well-known that this is not a vacuous question: the 's tandard' formulas-as-types embedding from higher order predicate logic into the Calculus of Constructions is not complete. (See [Berardi 1989], [Geuvers 1989] or [Geuvers 1993].) Another interesting issue is whether the typed )~ calculus approach can help to solve questions about the logics or vice versa. An example of such a fruitful interaction is the proof of (strong) normalization for the Calculus of Constructions, which has as corollary in higher order predicate logic that cut elimination terminates. In this paper we want to treat questions of conservativity between systems of typed A calculi (and hence between the logical systems that correspond with them according to the formulas-as-types embedding). On the one hand this is an issue of interest for the typed A calculi themselves. (Can new type forming operators create inhabitants of previously empty types?) On the other hand, however, this is a nice example of how the formulas-as-types embedding can help to solve questions about logics by making use of typed A calculi and vice versa. If one sees a typed A calculus as a logical system, one takes one specific universe ( 'sort ' in the terminology of Pure Type Systems) to be interpreted as the universe of all formulas. Let's call this universe Prop. Now suppose that $1 is a system of typed A calculus containing the universe Prop, and suppose that $2 is a system that extends $1.

The Polymorphic Rewriting-calculus: [Type Checking vs. Type Inference

Functional Reactive Programming (FRP) models reactive systems with events and signals, which have previously been observed to correspond to the "eventually" and "always" modalities of linear temporal logic (LTL). In this paper, we define a constructive variant of LTL with least fixed point and greatest fixed point operators in the spirit of the modal mu-calculus, and give it a proofs-as-programs interpretation as a foundational calculus for reactive programs. Previous work emphasized the propositions-as-types part of the correspondence between LTL and FRP; here we emphasize the proofs-as-programs part by employing structural proof theory. We show that the type system is expressive enough to enforce liveness properties such as the fairness of schedulers and the eventual delivery of results. We illustrate programming in this calculus using (co)iteration operators. We prove type preservation of our operational semantics, which guarantees that our programs are causal. We give also a proof of strong normalization which provides justification that our programs are productive and that they satisfy liveness properties derived from their types.

The fact that classical mathematical proofs of simply existential statements\ncan be read as programs was established by Goedel and Kreisel half a century\nago. But the possibility of extracting useful computational content from\nclassical proofs was taken seriously only from the 1990s on when it was\ndiscovered that proof interpretations based on Goedel's and Kreisel's ideas can\nprovide new nontrivial algorithms and numerical results, and the Curry-Howard\ncorrespondence can be extended to classical logic via programming concepts such\nas continuations and control operators.\n The workshop series "Classical Logic and Computation" aims to support a\nfruitful exchange of ideas between the various lines of research on\ncomputational aspects of classical logic. This volume contains the abstracts of\nthe invited lectures and the accepted contributed papers of the third CL&C\nworkshop which was held jointly with the workshop "Program Extraction and\nConstructive Mathematics" at the University of Brno in August 21-22, 2010, as a\nsatellite of CSL and MFCS. The workshops were held in honour of Helmut\nSchwichtenberg who became "professor emeritus" in September 2010.\n The topics of the papers include the foundations, optimizations and\napplications of proof interpretations such as Hilbert's epsilon substitution\nmethod, Goedel's functional interpretation, learning based realizability and\nnegative translations as well as special calculi and theories capturing\ncomputational and complexity-theoretic aspects of classical logic such as the\nlambda-mu-calculus, applicative theories, sequent-calculi, resolution and\ncut-elimination\n

We propose a new direct presentation of full (propositional) classical logic by means of proof nets, for which strong normalization and confluence hold true. Our proposal is based on the breaking of symmetry between propositional connectives, combined with a principle of focusing/defocusing.

Call-by-value, call-by-name, and strong normalization for the classical sequent calculus

On the Computational Representation of Classical Logical Connectives

The stack calculus is a functional language in which is in a Curry-Howard correspondence with classical logic. It enjoys confluence but, as well as Parigot's lambda-mu, does not admit the Bohm Theorem, typical of the lambda-calculus. We present a simple extension of stack calculus which is for the stack calculus what Saurin's Lambda-mu is for lambda-mu.

This paper introduces a cut-elimination procedure of the intuitionistic sequent calculus and shows that it is isomorphic to the proof reduction of the intuitionistic natural deduction with general elimination and explicit substitution. It also proves strong normalization and Church-Rosser property of the cut-elimination procedure by projecting the sequent calculus to the natural deduction with general elimination without explicit substitution.KeywordsClassical LogicReduction RuleNatural DeductionSequent CalculusGeneral EliminationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The symmetric λμ -calculus is the λμ -calculus introduced by Parigot in which the reduction rule μ ′, which is the symmetric of μ, is added. We give arithmetical proofs of some strong normalization results for this calculus. We show (this is a new result) that the μμ ′-reduction is strongly normalizing for the un-typed calculus. We also show the strong normalization of the βμ μ ′-reduction for the typed calculus: this was already known but the previous proofs use candidates of reducibility where the interpretation of a type was defined as the fix point of some increasing operator and thus, were highly non arithmetical.KeywordsInduction HypothesisClassical LogicReduction RuleNatural DeductionAnnual IEEE SymposiumThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.