Paul Ruet

Linear Concurrent Constraint Programming

In this paper we give a logical semantics for the class CC of concurrent constraint programming languages and for its extension LCC based on linear constraint systems. Besides the characterization in intuitionistic logic of the stores of CC computations, we show that both the stores and the successes of LCC computations can be characterized in intuitionistic linear logic. We illustrate the usefulness of these results by showing with examples how the phase semantics of linear logic can be used to give simple “semantical” proofs of safety properties of LCC programs.

Qualitative modelling of genetic networks: From logical regulatory graphs to standard Petri nets

In this paper, a systematic rewriting of logical genetic regulatory graphs in terms of standard Petri net models is proposed. We show that, in the Boolean case, the combination of the logical approach with the standard Petri net framework enables the analysis of isolated regulatory circuits, confirming their most fundamental dynamical properties. Furthermore, two more realistic applications are also presented, the first dealing with the control of the early cell cycles in the developing fly, the second dealing with flower morphogenesis.

From minimal signed circuits to the dynamics of Boolean regulatory networks

UNLABELLED It is acknowledged that the presence of positive or negative circuits in regulatory networks such as genetic networks is linked to the emergence of significant dynamical properties such as multistability (involved in differentiation) and periodic oscillations (involved in homeostasis). Rules proposed by the biologist R. Thomas assert that these circuits are necessary for such dynamical properties. These rules have been studied by several authors. Their obvious interest is that they relate the rather simple information contained in the structure of the network (signed circuits) to its much more complex dynamical behaviour. We prove in this article a nontrivial converse of these rules, namely that certain positive or negative circuits in a regulatory graph are actually sufficient for the observation of a restricted form of the corresponding dynamical property, differentiation or homeostasis. More precisely, the crucial property that we require is that the circuit be globally minimal. We then apply these results to the vertebrate immune system, and show that the two minimal functional positive circuits of the model indeed behave as modules which combine to explain the presence of the three stable states corresponding to the Th0, Th1 and Th2 cells. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

From logical regulatory graphs to standard petri nets: dynamical roles and functionality of feedback circuits

Logical modelling and Petri nets constitute two complementary approaches for the dynamical modelling of biological regulatory networks. Leaning on a translation of logical models into standard Petri nets, we propose a formalisation of the notion of circuit functionality in the Petri net framework. This approach is illustrated with the modelling and analysis of a molecular regulatory network involved in the control of Th-lymphocyte differentiation.

Regular ArticleLinear Concurrent Constraint Programming: Operational and Phase Semantics☆

In this paper we give a logical semantics for the class CC of concurrent constraint programming languages and for its extension LCC based on linear constraint systems. Besides the characterization in intuitionistic logic of the stores of CC computations, we show that both the stores and the successes of LCC computations can be characterized in intuitionistic linear logic. We illustrate the usefulness of these results by showing with examples how the phase semantics of linear logic can be used to give simple “semantical” proofs of safety properties of LCC programs.

On differentiation and homeostatic behaviours of boolean dynamical systems

We study rules proposed by the biologist R. Thomas relating the structure of a concurrent system of interacting genes (represented by a signed directed graph called a regulatory graph) with its dynamical properties. We prove that the results in [10] are stable under projection, and this enables us to relax the assumptions under which they are valid. More precisely, we relate here the presence of a positive (resp. negative) circuit in a regulatory graph to a more general form of biological differentiation (resp. of homeostasis).

Permutative logic

Recent work establishes a direct link between the complexity of a linear logic proof in terms of the exchange rule and the topological complexity of its corresponding proof net, expressed as the minimal rank of the surfaces on which the proof net can be drawn without crossing edges. That surface is essentially computed by sequentialising the proof net into a sequent calculus which is derived from that of linear logic by attaching an appropriate structure to the sequents. We show here that this topological calculus can be given a better-behaved logical status, when viewed in the variety-presentation framework introduced by the first author. This change of viewpoint gives rise to permutative logic, which enjoys cut elimination and focussing properties and comes equipped with new modalities for the management of the exchange rule. Moreover, both cyclic and linear logic are shown to be embedded into permutative logic. It provides the natural logical framework in which to study and constrain the topological complexity of proofs, and hence the use of the exchange rule.

Non-commutative logic III: focusing proofs

It is now well-established that the so-called focalization property plays a central role in the design of programming languages based on proof search, and more generally in the proof theory of linear logic. We present here a sequent calculus for non-commutative logic (NL) which enjoys the focalization property. In the multiplicative case, we give a focalized sequentialization theorem, and in the general case, we show that our focalized sequent calculus is equivalent to the original one by studying the permutabilities of rules for NL and showing that all permutabilities of linear logic involved in focalization can be lifted to NL permutabilities. These results are based on a study of the partitions of partially ordered sets modulo entropy.

Spatial Differentiation and Positive Circuits in a Discrete Framework

The biologist R. Thomas has enounced a rule relating multistationarity in a system of genes interacting in a single cell to the existence of a positive circuit in the regulatory graph of the system. In this paper, we address the question of a similar rule for spatial differentiation. We consider the interactions of genes in several biological cells located on a 1-dimensional infinite grid, and we assume that the expression levels of genes are Boolean. We show that the existence of a positive circuit is a necessary condition for a specific form of multistationarity, which naturally corresponds to spatial differentiation.

Non-commutative proof construction: A constraint-based approach

Abstract This work presents a computational interpretation of the construction process for cyclic linear logic (CyLL) and non-commutative logic (NL) sequential proofs. We assume a proof construction paradigm, based on a normalisation procedure known as focussing , which efficiently manages the non-determinism of the construction. Similarly to the commutative case, a new formulation of focussing for NL is used to introduce a general constraint-based technique in order to dealwith partial information during proof construction. In particular, the procedure develops through construction steps propagating constraints in intermediate objects called abstract proofs .