40 research outputs found
Characterising strongly normalising intuitionistic sequent terms
This paper gives a characterisation, via intersection types, of the strongly normalising terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing
system is reduced to that of a well known typing system with intersection types for the ordinary lambda-calculus. The completeness of the typing system is obtained from subject expansion at root position. This paper's sequent term calculus integrates smoothly the lambda-terms with generalised application or explicit substitution. Strong normalisability of these terms as
sequent terms characterises their typeability in certain "natural'' typing systems with intersection types. The latter are in the natural deduction format, like systems previously studied by Matthes and Lengrand et al., except that they do not contain any extra, exceptional rules for typing generalised applications or substitution
Characterising strongly normalising intuitionistic terms
This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the
typing system is reduced to that of a well known typing system with intersection types for the ordinary lambdal-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary lambda-terms, LambdaJ-terms (lambda-terms with generalised application),
and lambdax-terms (lambda-terms with explicit substitution). We explain via our system why the type systems iin the natural deduction format for LambdaJ and lambdax known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the beta-strongly normalising l-terms, as a corollary to a PSN-result, relating the lambda-calculus and the intuitionistic
sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known
characterisation of the beta-strongly normalising lambda-terms in terms of assignment of strict types follows as an easy corollary of our results.Fundação para a Ciência e Tecnologi
Resource control and strong normalisation
We introduce the \emph{resource control cube}, a system consisting of eight intuitionistic lambda calculi with either implicit or explicit control of resources and with either natural deduction or sequent calculus. The four calculi of the cube that correspond to natural deduction have been proposed by Kesner and Renaud and the four calculi that correspond to sequent lambda calculi are introduced in this paper. The presentation is parameterized with the set of resources (weakening or contraction), which enables a uniform treatment of the eight calculi of the cube. The simply typed resource control cube, on the one hand, expands the Curry-Howard correspondence to intuitionistic natural deduction and intuitionistic sequent logic with implicit or explicit structural rules and, on the other hand, is related to substructural logics. We propose a general intersection type system for the resource control cube calculi. Our main contribution is a characterisation of strong normalisation of reductions in this cube. First, we prove that typeability implies strong normalisation in the ''natural deduction base" of the cube by adapting the reducibility method. We then prove that typeability implies strong normalisation in the ''sequent base" of the cube by using a combination of well-orders and a suitable embedding in the ''natural deduction base". Finally, we prove that strong normalisation implies typeability in the cube using head subject expansion. All proofs are general and can be made specific to each calculus of the cube by instantiating the set of resources
A journey through resource control lambda calculi and explicit substitution using intersection types (an account)
In this paper we invite the reader to a journey through three lambda calculi with resource control: the lambda calculus, the sequent lambda calculus, and the lambda calculus with explicit substitution. All three calculi enable explicit control of resources due to the presence of weakening and contraction operators. Along this journey, we propose intersection type assignment systems for all three resource control calculi. We recognise the need for three kinds of variables all requiring different kinds of intersection types. Our main contribution is the characterisation of strong normalisation of reductions in all three calculi, using the techniques of reducibility, head subject expansion, a combination of well-orders and suitable embeddings of terms
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
Characterization of strong normalizability for a sequent lambda calculus with co-control
We study strong normalization in a lambda calculus of proof-terms
with co-control for the intuitionistic sequent calculus. In this sequent
lambda calculus, the management of formulas on the left hand
side of typing judgements is “dual" to the management of formulas
on the right hand side of the typing judgements in Parigot’s lambdamu
calculus - that is why our system has first-class “co-control".
The characterization of strong normalization is by means of intersection
types, and is obtained by analyzing the relationship with
another sequent lambda calculus, without co-control, for which a
characterization of strong normalizability has been obtained before.
The comparison of the two formulations of the sequent calculus,
with or without co-control, is of independent interest. Finally, since
it is known how to obtain bidirectional natural deduction systems
isomorphic to these sequent calculi, characterizations are obtained
of the strongly normalizing proof-terms of such natural deduction
systems.The authors would like to thank the anonymous
referees for their valuable comments and helpful suggestions.
This work was partly supported by FCT—Fundação para a Ciência
e a Tecnologia, within the project UID-MAT-00013/2013; by
COST Action CA15123 - The European research network on types
for programming and verification (EUTypes) via STSM; and by the
Ministry of Education, Science and Technological Development,
Serbia, under the projects ON174026 and III44006.info:eu-repo/semantics/publishedVersio
Resource control and intersection types: an intrinsic connection
In this paper we investigate the -calculus, a -calculus
enriched with resource control. Explicit control of resources is enabled by the
presence of erasure and duplication operators, which correspond to thinning and
con-traction rules in the type assignment system. We introduce directly the
class of -terms and we provide a new treatment of substitution by its
decompo-sition into atomic steps. We propose an intersection type assignment
system for -calculus which makes a clear correspondence between three
roles of variables and three kinds of intersection types. Finally, we provide
the characterisation of strong normalisation in -calculus by means of
an in-tersection type assignment system. This process uses typeability of
normal forms, redex subject expansion and reducibility method.Comment: arXiv admin note: substantial text overlap with arXiv:1306.228