155 research outputs found
A New Proof of P-time Completeness of Linear Lambda Calculus
We give a new proof of P-time completeness of Linear Lambda Calculus, which
was originally given by H. Mairson in 2003. Our proof uses an essentially
different Boolean type from the type Mairson used. Moreover the correctness of
our proof can be machined-checked using an implementation of Standard ML
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
Soft Linear Logic and Polynomial Complexity Classes
AbstractWe describe some results inspired to Lafont's Soft Linear Logic (SLL) which is a subsystem of second-order linear logic with restricted rules for exponentials, correct and complete for polynomial time computations. SLL is the basis for the design of type assignment systems for lambda-calculus, characterizing the complexity classes PTIME, PSPACE and NPTIME. PTIME is characterized by a type assignments system where types are a proper subset of SLL formulae. The characterization consists in the fact that a well typed term can be reduced to normal form by a number of beta-reductions polynomial in its lenght, and moreover all polynomial time functions can be computed by well typed terms. PSPACE is characterized by a type assignment system obtained from the previous one, by extending the set of types by a type for booleans, and the lambda-calculus by two boolean constants and a conditional constructor. The system assigns types to terms in such a way that the evaluation of programs (closed terms of type boolean) can be performed carefully in polynomial space. Moreover all polynomial space decision problems can be computed by terms typable in this system. In order to characterize NPTIME we extend the lambda-calculus by a nondeterministic choice operator, and the system by a rule for dealing with this new term constructor
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
Linear lambda terms as invariants of rooted trivalent maps
The main aim of the article is to give a simple and conceptual account for
the correspondence (originally described by Bodini, Gardy, and Jacquot) between
-equivalence classes of closed linear lambda terms and isomorphism
classes of rooted trivalent maps on compact oriented surfaces without boundary,
as an instance of a more general correspondence between linear lambda terms
with a context of free variables and rooted trivalent maps with a boundary of
free edges. We begin by recalling a familiar diagrammatic representation for
linear lambda terms, while at the same time explaining how such diagrams may be
read formally as a notation for endomorphisms of a reflexive object in a
symmetric monoidal closed (bi)category. From there, the "easy" direction of the
correspondence is a simple forgetful operation which erases annotations on the
diagram of a linear lambda term to produce a rooted trivalent map. The other
direction views linear lambda terms as complete invariants of their underlying
rooted trivalent maps, reconstructing the missing information through a
Tutte-style topological recurrence on maps with free edges. As an application
in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent
maps as linear lambda terms containing no closed proper subterms, and conclude
by giving a natural reformulation of the Four Color Theorem as a statement
about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio
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
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