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High-level formalisms for reasoning about names and binding such as de Bruijn indices, various flavors of higher-order abstract syntax, the Theory of Contexts, and nominal abstract syntax address only one relatively restrictive form of scoping: namely, unary lexical scoping, in which the scope of a (single) bound name is a subtree of the abstract syntax tree (possibly with other subtrees removed due to shadowing). Many languages exhibit binding or renaming structure that does not fit this mold. Examples include binding transitions in the #-calculus; unique identifiers in contexts, memory heaps, and XML documents; declaration scoping in modules and namespaces; anonymous identifiers in automata, type schemes, and Horn clauses; and pattern matching and mutual recursion constructs in functional languages. In these cases, it appears necessary to either rearrange the abstract syntax so that lexical scoping can be used, or revert to first-order techniques. The purpose

We use the ρ-calculus as an intermediate language to compile functional languages with pattern-matching features, and give an interaction net encoding of the ρ-terms arising from the compilation. This encoding gives rise to new strategies of evaluation, where pattern-matching and ‘traditional ’ β-reduction can proceed in parallel without overheads.

Abstract Pure Pattern Type Systems (P 2 T S) combine in a unified setting the capabilities of rewriting and λ-calculus. Their type systems, adapted from Barendregt’s λ-cube, are especially interesting from a logical point of view. Strong normalization, an essential property for logical soundness, had only been conjectured so far: in this paper, we give a positive answer for the simply-typed system. The proof is based on a translation of terms and types from P 2 T S into the λ-calculus. First, we deal with untyped terms, ensuring that reductions are faithfully mimicked in the λ-calculus. For this, we rely on an original encoding of the pattern matching capability of P 2 T S into the λ-calculus. Then we show how to translate types: the expressive power of System Fω is needed in order to fully reproduce the original typing judgments of P 2 T S. We prove that the encoding is correct with respect to reductions and typing, and we conclude with the strong normalization of simply-typed P 2 T S terms. 1

Abstract. Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that uniformly integrates termrewriting and λ-calculus. The ρg-calculus has been recently introduced as an extension of the ρ-calculus, handling structures with cycles and sharing. The calculus over terms is naturally generalized by using unification constraints in addition to the standard ρ-calculus matching constraints. This leads to a term-graph representation in an equational style where terms consist of unordered lists of equations. In this paper we show that the (linear) ρg-calculus is confluent. The proof of this result is quite elaborated, due to the non-termination of the system and to the fact that we work on equivalence classes of terms. We also show that the ρg-calculus can be seen as a generalization of first-order term-graph rewriting, in the sense that for any term-graph rewrite step a corresponding sequence of rewritings can be found in the ρg-calculus. 1

Abstract Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalization of a proof term language that models appropriately superdeduction. We finaly examplify on several examples, including equality and noetherian induction, the usefulness of this approach which is implemented in the lemuridæ system, written in TOM. 1

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The rewriting calculus has been introduced as a general formalism that uniformly integrates rewriting and λ-calculus. In this calculus all the basic ingredients of rewriting such as rewrite rules, rule applications and results are first-class objects. The rewriting calculus has been originally designed and used for expressing the semantics of rule based as well as object oriented paradigms. We have previously shown that convergent term rewriting systems and classic strategies can be encoded naturally in the calculus. In this paper, we go a step further and we propose an extended version of the calculus that allows one to encode unrestricted term rewriting systems. This version of the calculus features a new evaluation rule describing the behavior of the result structures and a call-by-value evaluation strategy. We prove the confluence of the obtained calculus and the correctness and completeness of the proposed encoding.

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