8,001 research outputs found
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
Ludics and its Applications to natural Language Semantics
Proofs, in Ludics, have an interpretation provided by their counter-proofs,
that is the objects they interact with. We follow the same idea by proposing
that sentence meanings are given by the counter-meanings they are opposed to in
a dialectical interaction. The conception is at the intersection of a
proof-theoretic and a game-theoretic accounts of semantics, but it enlarges
them by allowing to deal with possibly infinite processes
Two loop detection mechanisms: a comparison
In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist, and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding emphet al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level
On paths-based criteria for polynomial time complexity in proof-nets
Girard's Light linear logic (LLL) characterized polynomial time in the
proof-as-program paradigm with a bound on cut elimination. This logic relied on
a stratification principle and a "one-door" principle which were generalized
later respectively in the systems L^4 and L^3a. Each system was brought with
its own complex proof of Ptime soundness.
In this paper we propose a broad sufficient criterion for Ptime soundness for
linear logic subsystems, based on the study of paths inside the proof-nets,
which factorizes proofs of soundness of existing systems and may be used for
future systems. As an additional gain, our bound stands for any reduction
strategy whereas most bounds in the literature only stand for a particular
strategy.Comment: Long version of a conference pape
Classical Structures Based on Unitaries
Starting from the observation that distinct notions of copying have arisen in
different categorical fields (logic and computation, contrasted with quantum
mechanics) this paper addresses the question of when, or whether, they may
coincide. Provided all definitions are strict in the categorical sense, we show
that this can never be the case. However, allowing for the defining axioms to
be taken up to canonical isomorphism, a close connection between the classical
structures of categorical quantum mechanics, and the categorical property of
self-similarity familiar from logical and computational models becomes
apparent.
The required canonical isomorphisms are non-trivial, and mix both typed
(multi-object) and untyped (single-object) tensors and structural isomorphisms;
we give coherence results that justify this approach.
We then give a class of examples where distinct self-similar structures at an
object determine distinct matrix representations of arrows, in the same way as
classical structures determine matrix representations in Hilbert space. We also
give analogues of familiar notions from linear algebra in this setting such as
changes of basis, and diagonalisation.Comment: 24 pages,7 diagram
Near-field properties of plasmonic nanostructures with high aspect ratio
Using the Green's dyad technique based on cuboidal meshing, we compute the
electromagnetic field scattered by metal nanorods with high aspect ratio. We
investigate the effect of the meshing shape on the numerical simulations. We
observe that discretizing the object with cells with aspect ratios similar to
the object's aspect ratio improves the computations, without degrading the
convergency. We also compare our numerical simulations to finite element method
and discuss further possible improvements
On Context Semantics and Interaction Nets
International audienceContext semantics is a tool inspired by Girard' s geometry of interaction. It has had many applications from study of optimal reduction to proofs of complexity bounds. Yet, context semantics have been defined only on -calculus and linear logic. In order to study other languages, in particular languages with more primitives (built-in arithmetic, pattern matching,...) we define a context semantics for a broader framework: interaction nets. These are a well-behaved class of graph rewriting systems. Here, two applications are explored. First, we define a notion of weight, based on context semantics paths, which bounds the length of reduction of nets. Then, we define a denotational semantics for a large class of interaction net systems
Logic Programming and Logarithmic Space
We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation
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
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