Sequent Calculus For Linear Logic Research Articles

Focused Linear Logic and the λ-calculus

Linear logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus. We address this problem by providing a simple interpretation of focused proofs, a complete subclass of linear sequent proofs known to have a much stronger structure than the standard sequent calculus for linear logic. Despite the classical setting, the interpretation relates proofs to a refined linear λ-calculus, and we investigate its properties and relation to other calculi, such as the usual λ-calculus, the λμ-calculus, and their variants based on sequent calculi.

Real-time scheduling of batch systems using Petri nets and linear logic

This paper presents an approach to model, design and verify scenarios of real-time systems used in the scheduling and global coordination of batch systems. The initial requirements of a system specified with sequence diagrams are translated into a single p-time Petri net model representing the global behavior of the system. For the Petri net fragments involved in conflicts, symbolic production and consumption dates assigned to tokens are calculated based on the sequent calculus of linear logic. These dates are then used for off-line conflict resolution within a token player algorithm used for scenario verification of real-time specifications and which can be seen as a simulation tool for UML interaction diagrams.

Centralized Architecture for Real Time Scheduling of Batch Systems

The objective of this article is to present a centralized architecture based on a collaboration diagram and on a p-time Petri net model for real time scheduling of production systems. The main idea consists of proposing a software architecture which depends on an object whose purpose is to centralize all the interactions between the different software objects. The internal behaviour of the central object is represented by a p-time Petri net model which shows the global behaviour of the entire system. A specialized inference mechanism called token player is then applied to the p-time Petri net model for the real time scheduling. In particular, for the Petri net fragments involved in conflict situations, symbolic dates assigned to tokens are calculated using a non-conventional (max;+) algebra based on the sequent calculus of Linear Logic. These dates are used to solve conflict situations off-line on an implicit manner in such a way that the resource conflicts can be treated in real time at the central object level. The approach is illustrated through an example of Real Time System used at the global coordination level of a Batch System.

Evolvable concurrent processes

This paper presents a formal model of concurrent processes whose functions can be modified from the environments during execution. In many applications, it takes too much cost to stop the whole of the concurrent system working in distributed environments to update its components. Features of dynamic evolution that make it possible to modify functions of processes without termination are important. In this paper, a formal model of evolvable concurrent processes is presented using logical formulas of linear logic. The operational semantics is defined using a formal proof in a fragment of the sequent calculus of linear logic. A method for replacing the continuations of working processes by applications of inference rules of linear logic is presented.

Logic Programming with Focusing Proofs in Linear Logic

The deep symmetry of linear logic [18] makes it suitable for providing abstract models of computation, free from implementation details which are, by nature, oriented and non-symmetrical. I propose here one such model, in the area of logic programming, where the basic computational priciple is Computation = Proof search Proofs cinsidered here are those of the Gentzen style sequent calculus for linear logic. However, proofs in this system may be redundant, in that two proofs canbe syntactically different although identical up to some irrelevant reordering or simplification of the applications of the inferences rules. This leads to an untractable proof search where the search procedure is forced to make costly choices whch turn out to be irrelevant. To overcome this problem, a subclass of proofs, called the ‘focusing’ proofs, which is both complete (any derivable formla in linear logic has a focusing proof) and tractable (many irrelevant choices in the search are eliminated when aimed at focusing proofs) is identified. The main constraint underlying the specificatuon of focusing proofs has been to preserve the symmetry of linear logic, which is its most salient feature. In particular, dual connectives have dual properties with respect to focusing proofs Then, a progrmming language, called LinLog, consisting of a fragment of linear logic, in which focussing proofs have a more compact form, is presented. Linlog deals with formulae which have a syntax similar to the of the definite clauses and goals of Horn logic, but the crucial difference here is that it allows clauses with multiple atoms in the head, connected by the ‘par’ (multiplicative disjuction). It is then shown that the syntyactic restriction induced by LinLog is not performed at the cost of any expressive power: a mapping from full linear logic to LinLog, preserving focusing proofs, and analogous to the normalization to clausal form for classical logic, is presented.