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

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.Comment: In Proceedings ICE 2015, arXiv:1508.0459

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

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

Characterizing co-NL by a group action

International audienceIn 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

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.Comment: arXiv admin note: text overlap with arXiv:1402.432

Processes, Systems \& Tests: Defining Contextual Equivalences

In this position paper, we would like to offer and defend a new template to study equivalences between programs -- in the particular framework of process algebras for concurrent computation.We believe that our layered model of development will clarify the distinction that is too often left implicit between the tasks and duties of the programmer and of the tester. It will also enlighten pre-existing issues that have been running across process algebras as diverse as the calculus of communicating systems, the $\pi$-calculus -- also in its distributed version -- or mobile ambients.Our distinction starts by subdividing the notion of process itself in three conceptually separated entities, that we call \emph{Processes}, \emph{Systems} and \emph{Tests}.While the role of what can be observed and the subtleties in the definitions of congruences have been intensively studied, the fact that \emph{not every process can be tested}, and that \emph{the tester should have access to a different set of tools than the programmer} is curiously left out, or at least not often formally discussed.We argue that this blind spot comes from the under-specification of contexts -- environments in which comparisons takes place -- that play multiple distinct roles but supposedly always \enquote{stay the same}.We illustrate our statement with a simple Java example, the \enquote{usual} concurrent languages, but also back it up with $\lambda$-calculus and existing implementations of concurrent languages as well