Knowledge in mathematics (definitions, theorems, proofs, etc.) is usually expressed in a way that combines natural language and mathematical expressions (e.g. equations). Using an ontology formalism such as OWL DL is well-suited for formalizing the natural language part, but complex mathematical expressions can be better handled by symbolic computation systems. We examine this representation issue and propose an original extension of OWL DL by call formulas, i.e., formulas from which assertions can be drawn thanks to calls to external functions. Using this formalism makes it possible to classify a mathematical problem defined by its relations to instances and classes and by some mathematical expressions: if a theorem for solving this problem is represented in the knowledge base, it can be retrieved, and thus, the problem can be solved by applying this theorem. We describe an inference algorithm and discuss its properties as well as its limitations. Indeed, the proposed extension, algorithm, and implementation represent a first step towards a combined formalism for representing mathematical knowledge, with some open issues regarding the representation of more complex problems: the resolution of multiscale, multiphysics cases in physics are foreseen.

Rewriting is a widely established formalism with major applications in computer science. It is indeed a staple of many formal verification applications as it is especially well suited to describe program semantics and transformations. In particular, constructor based term rewriting systems are generally used to illustrate the behaviour of functional programs.

Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages, with the negation over patterns. We propose in this paper a concise and clear algorithm computing the difference over patterns which can be used to define generic encodings of constructor term rewriting systems with negation and order into classical term rewriting systems. As a direct consequence, established methods used for term rewriting systems can be applied to analyze properties of the extended systems. The approach can also be seen as a generic compiler which targets any language providing basic pattern matching primitives. The formalism provides also a new method for deciding if a set of patterns subsumes a given pattern and thus, for checking the presence of useless patterns or the completeness of a set of patterns.

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and rewriting strategies are used to con- trol their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specific strategies have been studied. We propose in this paper a generic encoding of classic control and traversal strategies used in rewrite based languages such as Maude, Stratego and Tom into a plain term rewriting system. The encoding is proven sound and complete and, as a direct consequence, estab- lished termination methods used for term rewriting systems can be applied to analyze the termination of strategy controlled term rewriting systems. We show that the encoding of strategies into term rewriting systems can be easily adapted to handle many-sorted signa- tures and we use a meta-level representation of terms to reduce the size of the encodings. The corresponding implementation in Tom generates term rewriting systems compatible with the syntax of termination tools such as AProVE and TTT2, tools which turned out to be very effective in (dis)proving the termination of the generated term rewriting systems. The approach can also be seen as a generic strategy compiler which can be integrated into languages providing pattern matching primitives; experiments in Tom show that applying our encoding leads to performances comparable to the native Tom strategies.

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and declarative rewriting strategies are used to control their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specific strategies have been studied. We propose in this paper a generic encoding of classic control and traversal strategies used in rewrite based languages such as Maude, Stratego and Tom into a plain term rewriting system. The encoding is proven sound and complete and, as a direct consequence, established termination methods used for term rewriting systems can be applied to analyze the termination of strategy controlled term rewriting systems. The corresponding implementation in Tom generates term rewriting systems compatible with the syntax of termination tools such as AAProVE and TTT2, tools which turned out to be very effective in (dis)proving the termination of the generated term rewriting systems. The approach can also be seen as a generic strategy compiler which can be integrated into languages providing pattern matching primitives; this has been experimented for Tom and performances comparable to the native Tom strategies have been observed.

First designed to enable private networks to be opened up to the outside world in a secure way, the growing complexity of organizations make firewalls indispensable to control information flow within a company. The central role they hold in the security of the organization information make their management a critical task and that is why for years many works have focused on checking and analyzing firewalls. The composition of firewalls, taking into account routing rules, has nevertheless often been neglected. In this paper, we propose to specify all components of a firewall, ie filtering and translation rules, as a rewrite system. We show that such specifications allow us to handle usual problems such as comparison, structural analysis and query analysis. We also propose a formal way to describe the composition of firewalls (including routing) in order to build a whole network security policy. The properties of the obtained rewrite system are strongly related to the properties of the specified networks and thus, classical theoretical and practical tools can be used to obtain relevant security properties of the security policies.

The integration of first-order and higher-order paradigms has been one of the main challenges in the design of both declarative programming languages and proof environments. It has led to the development of new computation models and new logical frameworks, which have been obtained by enriching first-order rewriting with higher-order capabilities or by adding algebraic features to the λ-calculus.

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories. In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities. We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

International audience