57 research outputs found
An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus
This paper brings together two lines of research: implicit characterization
of complexity classes by Linear Logic (LL) on the one hand, and computation
over an arbitrary ring in the Blum-Shub-Smale (BSS) model on the other. Given a
fixed ring structure K we define an extension of Terui's light affine
lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K. We
show that this calculus captures the polynomial time function class FP(K):
every typed term can be evaluated in polynomial time and conversely every
polynomial time BSS machine over K can be simulated in this calculus.Comment: 11 pages. A preliminary version appeared as Research Report IAC CNR
Roma, N.57 (11/2004), november 200
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
A type system for PSPACE derived from light linear logic
We present a polymorphic type system for lambda calculus ensuring that
well-typed programs can be executed in polynomial space: dual light affine
logic with booleans (DLALB).
To build DLALB we start from DLAL (which has a simple type language with a
linear and an intuitionistic type arrow, as well as one modality) which
characterizes FPTIME functions. In order to extend its expressiveness we add
two boolean constants and a conditional constructor in the same way as with the
system STAB.
We show that the value of a well-typed term can be computed by an alternating
machine in polynomial time, thus such a term represents a program of PSPACE
(given that PSPACE = APTIME).
We also prove that all polynomial space decision functions can be represented
in DLALB.
Therefore DLALB characterizes PSPACE predicates.Comment: In Proceedings DICE 2011, arXiv:1201.034
Light Logics and the Call-by-Value Lambda Calculus
The so-called light logics have been introduced as logical systems enjoying
quite remarkable normalization properties. Designing a type assignment system
for pure lambda calculus from these logics, however, is problematic. In this
paper we show that shifting from usual call-by-name to call-by-value lambda
calculus allows regaining strong connections with the underlying logic. This
will be done in the context of Elementary Affine Logic (EAL), designing a type
system in natural deduction style assigning EAL formulae to lambda terms.Comment: 28 page
Linear Logic by Levels and Bounded Time Complexity
We give a new characterization of elementary and deterministic polynomial
time computation in linear logic through the proofs-as-programs correspondence.
Girard's seminal results, concerning elementary and light linear logic, achieve
this characterization by enforcing a stratification principle on proofs, using
the notion of depth in proof nets. Here, we propose a more general form of
stratification, based on inducing levels in proof nets by means of indexes,
which allows us to extend Girard's systems while keeping the same complexity
properties. In particular, it turns out that Girard's systems can be recovered
by forcing depth and level to coincide. A consequence of the higher flexibility
of levels with respect to depth is the absence of boxes for handling the
paragraph modality. We use this fact to propose a variant of our polytime
system in which the paragraph modality is only allowed on atoms, and which may
thus serve as a basis for developing lambda-calculus type assignment systems
with more efficient typing algorithms than existing ones.Comment: 63 pages. To appear in Theoretical Computer Science. This version
corrects minor fonts problems from v
Elementary linear logic revisited for polynomial time and an exponential time hierarchy (extended version)
Nombre de pages: 20. Une version courte de ce travail est à paraître dans les actes de: Asian Symposium on Programming Languages and Systems (APLAS 2011).Elementary linear logic is a simple variant of linear logic, introduced by Girard and which characterizes in the proofs-as-programs approach the class of elementary functions (computable in time bounded by a tower of exponentials of fixed height). Our goal here is to show that despite its simplicity, elementary linear logic can nevertheless be used as a common framework to characterize the different levels of a hierarchy of deterministic time complexity classes, within elementary time. We consider a variant of this logic with type fixpoints and weakening. By selecting specific types we then characterize the class P of polynomial time predicates and more generally the hierarchy of classes k-EXP, for k>=0, where k-EXP is the union of DTIME(2_k^{n^i}), for i>=1
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