1,346 research outputs found
Light types for polynomial time computation in lambda-calculus
We propose a new type system for lambda-calculus ensuring that well-typed
programs can be executed in polynomial time: Dual light affine logic (DLAL).
DLAL has a simple type language with a linear and an intuitionistic type
arrow, and one modality. It corresponds to a fragment of Light affine logic
(LAL). We show that contrarily to LAL, DLAL ensures good properties on
lambda-terms: subject reduction is satisfied and a well-typed term admits a
polynomial bound on the reduction by any strategy. We establish that as LAL,
DLAL allows to represent all polytime functions. Finally we give a type
inference procedure for propositional DLAL.Comment: 20 pages (including 10 pages of appendix). (revised version; in
particular section 5 has been modified). A short version is to appear in the
proceedings of the conference LICS 2004 (IEEE Computer Society Press
Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence
It is well known that the length of a beta-reduction sequence of a simply
typed lambda-term of order k can be huge; it is as large as k-fold exponential
in the size of the lambda-term in the worst case. We consider the following
relevant question about quantitative properties, instead of the worst case: how
many simply typed lambda-terms have very long reduction sequences? We provide a
partial answer to this question, by showing that asymptotically almost every
simply typed lambda-term of order k has a reduction sequence as long as
(k-1)-fold exponential in the term size, under the assumption that the arity of
functions and the number of variables that may occur in every subterm are
bounded above by a constant. To prove it, we have extended the infinite monkey
theorem for strings to a parametrized one for regular tree languages, which may
be of independent interest. The work has been motivated by quantitative
analysis of the complexity of higher-order model checking
Verification of Ptime reducibility for system F terms via Dual Light Affine Logic.
Proceedings of Computer Science Logic 2006 (CSL'06), volume 4207 of Lecture Notes in Computer Science, pp.150-166. � SpringerIn a previous work we introduced Dual Light Affine Logic (DLAL) ([BaillotTerui04]) as a variant of Light Linear Logic suitable for guaranteeing complexity properties on lambda-calculus terms: all typable terms can be evaluated in polynomial time and all Ptime functions can be represented. In the present work we address the problem of typing lambda-terms in second-order DLAL. For that we give a procedure which, starting with a term typed in system F, finds all possible ways to decorate it into a DLAL typed term. We show that our procedure can be run in time polynomial in the size of the original Church typed system F term
Introduction to linear logic and ludics, part II
This paper is the second part of an introduction to linear logic and ludics,
both due to Girard. It is devoted to proof nets, in the limited, yet central,
framework of multiplicative linear logic and to ludics, which has been recently
developped in an aim of further unveiling the fundamental interactive nature of
computation and logic. We hope to offer a few computer science insights into
this new theory
The syntactic side of autonomous categories enriched over generalised metric spaces
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear λ-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear λ-theories based on this V-equational system form a category equivalent to a category of autonomous categories enriched over ‘generalised metric spaces’. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. Additionally, we show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.This work is financed by National Funds through FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within project IBEX, with reference PTDC/CCI-COM/4280/2021. We are also thankful for the reviewers’ helpful feedback
QPCF: higher order languages and quantum circuits
qPCF is a paradigmatic quantum programming language that ex- tends PCF with
quantum circuits and a quantum co-processor. Quantum circuits are treated as
classical data that can be duplicated and manipulated in flexible ways by means
of a dependent type system. The co-processor is essentially a standard QRAM
device, albeit we avoid to store permanently quantum states in between two
co-processor's calls. Despite its quantum features, qPCF retains the classic
programming approach of PCF. We introduce qPCF syntax, typing rules, and its
operational semantics. We prove fundamental properties of the system, such as
Preservation and Progress Theorems. Moreover, we provide some higher-order
examples of circuit encoding
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