arXiv subjects

Clément Aubert

Publications and source records attributed to Clément Aubert.

31 records · Page 2Linked to original sources

arXiv2020

Structural Equivalences for Reversible Calculi of Communicating Systems (Oral communication)

The formalization of process algebras usually starts with a minimal core of operators and rules for its transition system, and then relax the system to improve its usability and ease the proofs. In the calculus of communicating systems (CCS), the structural congruence plays this role by making e.g. parallel composition commutative and associative: without it, the system would be cumbersome to use and reason about, and it can be proven that this change is innocuous in a precise technical sense. For the two reversible calculi extending CCS, the situation is less clear: CCS with Communication Keys (CCSK) was first defined without any structural congruence, and then was endowed with a fragment of CCS's congruence. Reversible CCS (RCCS) made the choice of "backing in" the structural equivalence, that became part of the "minimal core" of the system. In this short oral communication, we would like to re-consider the status and role of the structural congruence in general, to question its role in RCCS in particular, and to ask the more general question of the structural equivalences legitimacy.

cs.FL↗

arXiv2019

Categories for Me, and You?

A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the pseudo-cartesian structure on Eilenberg-Moore categories. References and proofs are provided, sometimes, to my knowledge, for the first time.

math.CT↗

arXiv2019

Logarithmic Space and Permutations

In a recent work, Girard proposed a new and innovative approach to computational complexity based on the proofs-as-programs correspondence. In a previous paper, the authors showed how Girard proposal succeeds in obtaining a new characterization of co-NL languages as a set of operators acting on a Hilbert Space. In this paper, we extend this work by showing that it is also possible to define a set of operators characterizing the class L of logarithmic space languages.

cs.LO↗

arXiv2018

Unification and Logarithmic Space

We present an algebraic characterization of the complexity classes Logspace and Nlogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is rooted in proof theory and more specifically linear logic and geometry of interaction. We show how to build a model of computation in the unification algebra and then, by means of a syntactic representation of finite permutations in the algebra, we prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. Finally, we show that the construction naturally corresponds to pointer machines, a convenient way of understanding logarithmic space computation.

cs.LO↗

arXiv2018

History-Preserving Bisimulations on Reversible Calculus of Communicating Systems

History-and hereditary history-preserving bisimulation (HPB and HHPB) are equivalences relations for denotational models of concurrency. Finding their counterpart in process algebras is an open problem, with some partial successes: there exists in calculus of communicating systems (CCS) an equivalence based on causal trees that corresponds to HPB. In Reversible CSS (RCCS), there is a bisimulation that corresponds to HHPB, but it considers only processes without auto-concurrency. We propose equivalences on CCS with auto-concurrency that correspond to HPB and HHPB, and their so-called "weak" variants. The equivalences exploit not only reversibility but also the memory mechanism of RCCS.

cs.LO↗

arXiv2015

Contextual equivalences in configuration structures and reversibility

Contextual equivalence equate terms that have the same observable behaviour in any context. A standard contextual equivalence for CCS is the strong barbed congruence. Configuration structures are a denotational semantics for processes in which one define equivalences that are more discriminating, i.e. that distinguish the denotation of terms equated by barbed congruence. Hereditary history preserving bisimulation (HHPB) is such a relation. We define a strong back-and-forth barbed congruence on RCCS, a reversible variant of CCS. We show that the relation induced by the back-and-forth congruence on configuration structures is equivalent to HHPB, thus providing a contextual characterization of HHPB.

cs.LO↗

arXiv2015

Reversible Barbed Congruence on Configuration Structures

A standard contextual equivalence for process algebras is strong barbed congruence. Configuration structures are a denotational semantics for processes in which one can define equivalences that are more discriminating, i.e. that distinguish the denotation of terms equated by barbed congruence. Hereditary history preserving bisimulation (HHPB) is such a relation. We define a strong back and forth barbed congruence using a reversible process algebra and show that the relation induced by the back and forth congruence is equivalent to HHPB, providing a contextual characterization of HHPB.

cs.LO↗

arXiv2015

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.

cs.LO↗

arXiv2015

An in-between "implicit" and "explicit" complexity: Automata

Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model.

cs.LO↗

arXiv2015

Characterizing co-NL by a group action

In a recent paper, Girard proposes to use his recent construction of a geometry of interaction in the hyperfinite factor in an innovative way to characterize complexity classes. We begin by giving a detailed explanation of both the choices and the motivations of Girard's definitions. We then provide a complete proof that the complexity class co-NL can be characterized using this new approach. We introduce as a technical tool the non-deterministic pointer machine, a concrete model to computes algorithms.

cs.LO↗

arXiv2014

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.

cs.LO↗

arXiv2014

Unification and Logarithmic Space

We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof theory and more specifically linear logic and Geometry of Interaction. We show how unification can be used to build a model of computation by means of specific subalgebras associated to finite permutations groups. We then prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. We also show that the construction can naturally represent pointer machines, an intuitive way of understanding logarithmic space computing.

cs.LO↗

arXiv2012

Sublogarithmic uniform Boolean proof nets

Using a proofs-as-programs correspondence, Terui was able to compare two models of parallel computation: Boolean circuits and proof nets for multiplicative linear logic. Mogbil et. al. gave a logspace translation allowing us to compare their computational power as uniform complexity classes. This paper presents a novel translation in AC0 and focuses on a simpler restricted notion of uniform Boolean proof nets. We can then encode constant-depth circuits and compare complexity classes below logspace, which were out of reach with the previous translations.

cs.CC↗