2,628 research outputs found
Uniform Proofs of Normalisation and Approximation for Intersection Types
We present intersection type systems in the style of sequent calculus,
modifying the systems that Valentini introduced to prove normalisation
properties without using the reducibility method. Our systems are more natural
than Valentini's ones and equivalent to the usual natural deduction style
systems. We prove the characterisation theorems of strong and weak
normalisation through the proposed systems, and, moreover, the approximation
theorem by means of direct inductive arguments. This provides in a uniform way
proofs of the normalisation and approximation theorems via type systems in
sequent calculus style.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
The polarized λ-calculus
A natural deduction system isomorphic to the focused sequent calculus for polarized intuitionistic logic is proposed. The
system comes with a language of proof-terms, named polarized λ-calculus, whose reduction rules express simultaneously a
normalization procedure and the isomorphic copy of the cut-elimination procedure pertaining to the focused sequent calculus.
Noteworthy features of this natural deduction system are: how the polarity of a connective determines the style of its
elimination rule; the existence of a proof-search strategy which is equivalent to focusing in the sequent calculus; the highlydisciplined
organization of the syntax - even atoms have introduction, elimination and normalization rules. The polarized
λ-calculus is a programming formalism close to call-by-push-value, but justified by its proof-theoretical pedigree.This research was financed by Portuguese Funds through FCT Fundac¸ao para a Ci ˜ encia ˆ
e a Tecnologia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio
Sequent Calculus and Equational Programming
Proof assistants and programming languages based on type theories usually
come in two flavours: one is based on the standard natural deduction
presentation of type theory and involves eliminators, while the other provides
a syntax in equational style. We show here that the equational approach
corresponds to the use of a focused presentation of a type theory expressed as
a sequent calculus. A typed functional language is presented, based on a
sequent calculus, that we relate to the syntax and internal language of Agda.
In particular, we discuss the use of patterns and case splittings, as well as
rules implementing inductive reasoning and dependent products and sums.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
Sequent calculi and decidability for intuitionistic hybrid logic
AbstractIn this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic (IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability
A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems
Basic proof-search tactics in logic and type theory can be seen as the
root-first applications of rules in an appropriate sequent calculus, preferably
without the redundancies generated by permutation of rules. This paper
addresses the issues of defining such sequent calculi for Pure Type Systems
(PTS, which were originally presented in natural deduction style) and then
organizing their rules for effective proof-search. We introduce the idea of
Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the
syntax of a permutation-free sequent calculus for propositional logic due to
Herbelin, which is strongly related to natural deduction and already well
adapted to proof-search. The operational semantics is adapted from Herbelin's
and is defined by a system of local rewrite rules as in cut-elimination, using
explicit substitutions. We prove confluence for this system. Restricting our
attention to PTSC, a type system for the ground terms of this system, we obtain
the Subject Reduction property and show that each PTSC is logically equivalent
to its corresponding PTS, and the former is strongly normalising iff the latter
is. We show how to make the logical rules of PTSC into a syntax-directed system
PS for proof-search, by incorporating the conversion rules as in
syntax-directed presentations of the PTS rules for type-checking. Finally, we
consider how to use the explicitly scoped meta-variables of PTSCalpha to
represent partial proof-terms, and use them to analyse interactive proof
construction. This sets up a framework PE in which we are able to study
proof-search strategies, type inhabitant enumeration and (higher-order)
unification
Proof-Theoretic Analysis of the Quantified Argument Calculus
This article investigates the proof theory of the Quantified Argument Calculus (Quarc) as developed and systematically studied by Hanoch Ben-Yami [3, 4]. Ben-Yami makes use of natural deduction (Suppes-Lemmon style), we, however, have chosen a sequent calculus presentation, which allows for the proofs of a multitude of significant meta-theoretic results with minor modifications to the Gentzen's original framework, i.e., LK. As will be made clear in course of the article LK-Quarc will enjoy cut elimination and its corollaries (including subformula property and thus consistency).Peer reviewe
Formalizing a lazy substitution proof system for \u3bc-calculus in the Calculus of Inductive Constructions
We present a Natural Deduction proof system for the pro- positional modal \u3bc-calculus, and its formalization in the Calculus of In- ductive Constructions. We address several problematic issues, such as the use of higher-order abstract syntax in inductive sets in presence of recursive constructors, the encoding of modal (sequent-style) rules and of context sensitive grammars. The formalization can be used in the sy- stem Coq, providing an experimental computer-aided proof environment for the interactive development of error-free proofs in the \u3bc-calculus. The techniques we adopt can be readily ported to other languages and proof systems featuring similar problematic issues. \ua9 Springer-Verlag Berlin Heidelberg 1999
Paraconsistent logic and query answering in inconsistent databases
This paper concerns the paraconsistent logic LPQ and
an application of it in the area of relational database theory. The notions of
a relational database, a query applicable to a relational database, and a
consistent answer to a query with respect to a possibly inconsistent relational
database are considered from the perspective of this logic. This perspective
enables among other things the definition of a consistent answer to a query
with respect to a possibly inconsistent database without resort to database
repairs. In a previous paper, LPQ is presented with a
sequent-style natural deduction proof system. In this paper, a sequent calculus
proof system is presented because it is common to use a sequent calculus proof
system as the basis of proof search procedures and such procedures may form the
core of algorithms for computing consistent answers to queries.Comment: 21 pages; revision of v4, some inaccuracies removed and material
streamlined at several place
Quantitative Types for Intuitionistic Calculi
We define quantitative type systems for two intuitionistic term languages. While the first language in natural deduction style is already known in the literature, the second one is one of the contributions of the paper, and turns out to be a natural computational interpretation of sequent calculus style by means of a non-idempotent type discipline. The type systems are able to characterize linear-head, weak and strong normalization sets of terms. All such characterizations are given by means of combinatorial arguments, i.e. there is a measure based on type derivations which is decreasing with respect to the different reduction relations considered in the paper
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