3,886 research outputs found
Cut Reduction in Linear Logic as Asynchronous Session-Typed Communication
Prior work has shown that intuitionistic linear logic can be seen as a
session-type discipline for the pi-calculus, where cut reduction in
the sequent calculus corresponds to synchronous process reduction. In
this paper, we exhibit a new process assignment from the asynchronous,
polyadic pi-calculus to exactly the same proof rules. Proof-theoretically, the difference between these interpretations can
be understood through permutations of inference rules that preserve
observational equivalence of closed processes in the synchronous case.
We also show that, under this new asynchronous interpretation, cut
reductions correspond to a natural asynchronous buffered session
semantics, where each session is allocated a separate communication
buffer
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
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
Intersection Logic in sequent calculus style
The intersection type assignment system has been designed directly as
deductive system for assigning formulae of the implicative and conjunctive
fragment of the intuitionistic logic to terms of lambda-calculus. But its
relation with the logic is not standard. Between all the logics that have been
proposed as its foundation, we consider ISL, which gives a logical
interpretation of the intersection by splitting the intuitionistic conjunction
into two connectives, with a local and global behaviour respectively, being the
intersection the local one. We think ISL is a logic interesting by itself, and
in order to support this claim we give a sequent calculus formulation of it,
and we prove that it enjoys the cut elimination property.Comment: In Proceedings ITRS 2010, arXiv:1101.410
The Lambek-Grishin calculus is NP-complete
The Lambek-Grishin calculus LG is the symmetric extension of the
non-associative Lambek calculus NL. In this paper we prove that the
derivability problem for LG is NP-complete
The Lambek calculus with iteration: two variants
Formulae of the Lambek calculus are constructed using three binary
connectives, multiplication and two divisions. We extend it using a unary
connective, positive Kleene iteration. For this new operation, following its
natural interpretation, we present two lines of calculi. The first one is a
fragment of infinitary action logic and includes an omega-rule for introducing
iteration to the antecedent. We also consider a version with infinite (but
finitely branching) derivations and prove equivalence of these two versions. In
Kleene algebras, this line of calculi corresponds to the *-continuous case. For
the second line, we restrict our infinite derivations to cyclic (regular) ones.
We show that this system is equivalent to a variant of action logic that
corresponds to general residuated Kleene algebras, not necessarily
*-continuous. Finally, we show that, in contrast with the case without division
operations (considered by Kozen), the first system is strictly stronger than
the second one. To prove this, we use a complexity argument. Namely, we show,
using methods of Buszkowski and Palka, that the first system is -hard,
and therefore is not recursively enumerable and cannot be described by a
calculus with finite derivations
Cut-Simulation and Impredicativity
We investigate cut-elimination and cut-simulation in impredicative
(higher-order) logics. We illustrate that adding simple axioms such as Leibniz
equations to a calculus for an impredicative logic -- in our case a sequent
calculus for classical type theory -- is like adding cut. The phenomenon
equally applies to prominent axioms like Boolean- and functional
extensionality, induction, choice, and description. This calls for the
development of calculi where these principles are built-in instead of being
treated axiomatically.Comment: 21 page
- …