The rho-calculi provide enlightening concepts for both computing and reasoning as well as their combination. They consist in the generalization of lambda-calculus to structures like terms, propositions or graphs and we will show how their interrelations with deduction provide powerful frameworks for the next generation of proof assistants.

In predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, deduction modulo allows us to make use of the computational part of the theory Th for true computations modulo which deductions are performed. Focusing on the sequent calculus, this paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. We call such a new deduction system "superdeduction ". We introduce a proof-term language and a cut-elimination procedure both based on Christian Urban's work on classical sequent calculus. Strong normalisation is proven under appropriate and natural hypothesis, therefore ensuring the consistency of the embedded theory and of the deduction system. The proofs obtained in such a new system are much closer to the human intuition and practice. We consequently sketch how superdeduction along with deduction modulo can be used to ground the formal foundations of new extendible proof assistants like lemuridae, our prototypal implementation of superdeduction modulo.

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).

When addressing the formal validation of generated software, two main alternatives consist either to prove the correctness of compilers or to directly validate the generated code. Here, we focus on directly proving the correctness of compiled code issued from powerful pattern matching constructions typical of ML like languages or rewrite based languages such as ELAN, Maude or Tom. In this context, our first contribution is to define a general framework for anchoring algebraic pattern-matching capabilities in existing languages like C, Java or ML. Then, using a just enough powerful intermediate language, we formalize the behavior of compiled code and define the correctness of compiled code with respect to pattern-matching behavior. This allows us to prove the equivalence of compiled code correctness with a generic first-order proposition whose proof could be achieved via a proof assistant or an automated theorem prover. We then extend these results to the multi-match situation characteristic of the ML like languages. The whole approach has been implemented on top of the Tom compiler and used to validate the syntactic matching code of the Tom compiler itself.

Solving goals - like deciding word problems or resolving constraints - is much easier in some theory presentations than in others. What have been called "completion processes", in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a given class of problems. We provide a general proof-theoretic setting within which completion-like processes can be modeled and studied. This framework centers around well-founded orderings of proofs. It allows for abstract definitions and very general characterizations of saturation processes and redundancy criteria.

International audience

In this paper we present a view of constraint programming based on the notion of rewriting controlled by strategies. We argue that this concept allows us to describe in a unified way the constraint solving mechanism as well as the meta-language neede

Higher-order unification is equational unification for /spl beta//spl eta/-conversion, but it is not first-order equational unification, as substitution has to avoid capture. In this paper higher-order unification is reduced to first-order equational unification in a suitable theory: the /spl lambda//spl sigma/-calculus of explicit substitutions.

Two main applications of term rewriting systems are equational reasoning in theorem provers and equational computation in programming languages. The present paper examines a number of term rewriting systems in terms of how rewriting is used in different implementations of theorem provers, the use of rewriting techniques in logic programming languages and the problem of efficiency. In addition a non-exhaustive catalogue of distributed implementations is presented.

An investigation is made of the relationship between unifiability of a general equation of the form f( nu /sub 1/, . . ., nu /sub n/)=/sup ?/g( nu /sub n+1/, . . ., nu /sub m/), and of the syntacticness of the theory. After introducing the concept of general equations and its basic properties, the precise definition of syntactic theories is given. It is proven that a theory is syntactic if and only is the general equations have finite complete set of E-solutions. The result is constructive in the sense that from the E-solutions of the general equations, a resolvent presentation is computed. This is applied to several theories: in particular, in order to show that distributivity is not syntactic. The authors also prove that the theory of associativity and commutativity is syntactic, which allows one, by the combination of Nipkow's results (ibid., p.278-88, 1990) and the authors', to infer a novel matching algorithm where there is no need to solve linear diophantine equations.>

Publie dans : Proceedings ISSAC'89 (International symposium on symbolic and algebraic computation), Portland (Oregon, USA), July 1989