302 research outputs found
Quantitative Models and Implicit Complexity
We give new proofs of soundness (all representable functions on base types
lies in certain complexity classes) for Elementary Affine Logic, LFPL (a
language for polytime computation close to realistic functional programming
introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs
are based on a common semantical framework which is merely instantiated in four
different ways. The framework consists of an innovative modification of
realizability which allows us to use resource-bounded computations as realisers
as opposed to including all Turing computable functions as is usually the case
in realizability constructions. For example, all realisers in the model for
LFPL are polynomially bounded computations whence soundness holds by
construction of the model. The work then lies in being able to interpret all
the required constructs in the model. While being the first entirely semantical
proof of polytime soundness for light logi cs, our proof also provides a
notable simplification of the original already semantical proof of polytime
soundness for LFPL. A new result made possible by the semantic framework is the
addition of polymorphism and a modality to LFPL thus allowing for an internal
definition of inductive datatypes.Comment: 29 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
Memoization for Unary Logic Programming: Characterizing PTIME
We give a characterization of deterministic polynomial time computation based
on an algebraic structure called the resolution semiring, whose elements can be
understood as logic programs or sets of rewriting rules over first-order terms.
More precisely, we study the restriction of this framework to terms (and logic
programs, rewriting rules) using only unary symbols. We prove it is complete
for polynomial time computation, using an encoding of pushdown automata. We
then introduce an algebraic counterpart of the memoization technique in order
to show its PTIME soundness. We finally relate our approach and complexity
results to complexity of logic programming. As an application of our
techniques, we show a PTIME-completeness result for a class of logic
programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201
A feasible algorithm for typing in Elementary Affine Logic
We give a new type inference algorithm for typing lambda-terms in Elementary
Affine Logic (EAL), which is motivated by applications to complexity and
optimal reduction. Following previous references on this topic, the variant of
EAL type system we consider (denoted EAL*) is a variant without sharing and
without polymorphism. Our algorithm improves over the ones already known in
that it offers a better complexity bound: if a simple type derivation for the
term t is given our algorithm performs EAL* type inference in polynomial time.Comment: 20 page
Soft linear logic and polynomial time
AbstractWe present a subsystem of second-order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms, and vice versa
A semantic account of strong normalization in Linear Logic
We prove that given two cut free nets of linear logic, by means of their
relational interpretations one can: 1) first determine whether or not the net
obtained by cutting the two nets is strongly normalizable 2) then (in case it
is strongly normalizable) compute the maximal length of the reduction sequences
starting from that net.Comment: 41 page
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