Equational term graph rewriting

We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the $mu$-rule, and translations are given between term graphs and $mu$-expressions. Using these, a proo

Confluent rewriting of bisimilar term graphs

AbstractWe present a survey of confluence properties of (acyclic) term graph rewriting. Results and counterexamples are given for four different kinds of term graph rewriting: besides plain applications of rewrite rules, extensions with the operations of collapsing and copying, and with both operations together are considered. Collapsing and copying together constitute bisimilarity of term graphs. We establish sufficient conditions for, and counterexamples to, confluence and confluence modulo bisimilarity of term graph rewriting over various classes of term rewriting systems

Syntactic definitions of undefined: On defining the undefined

In the lambda-calculus, there is a standard notion of what terms should be considered to be “undefined”: the unsolvable terms. There are various equivalent characterisations of this property of terms. We attempt to find a similar notion for orthogonal term rewrite systems. We find that in general the properties of terms analogous to the various characterisations of solvability differ. We give two axioms that a notion of undefinedness should satisfy, and explore some of their consequences. The axioms lead to a concept analogous to the Böhm trees of the λ-calculus. Each term has a unique B5hm tree, and the set of such trees forms a domain which provides a denotational semantics for the rewrite system. We consider several particular notions of undefinedness satisfying the axioms, and compare them

The graph rewriting calculus: confluence and expressiveness

Introduced at the end of the nineties, the Rewriting Calculus (rho-calculus, for short) is a simple calculus that uniformly integrates term-rewriting and lambda-calculus. The Rhog has been recently introduced as an extension of the rho-calculus, handling structures with cycles and sharing. The calculus over terms is naturally generalized by using unification constraints in addition to the standard rho-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) Rhog 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 Rhog 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 Rhog

Compilation of extended recursion in call-by-value functional languages

This paper formalizes and proves correct a compilation scheme for mutually-recursive definitions in call-by-value functional languages. This scheme supports a wider range of recursive definitions than previous methods. We formalize our technique as a translation scheme to a lambda-calculus featuring in-place update of memory blocks, and prove the translation to be correct.Comment: 62 pages, uses pi

Models of a Non-Associative Composition

International audienceWe characterise the polarised evaluation order through a categorical structure where the hypothesis that composition is associative is relaxed. Duploid is the name of the structure, as a reference to Jean-Louis Loday's duplicial algebras. The main result is a reflection Adj→Dupl where Dupl is a category of duploids and duploid functors, and Adj is the category of adjunctions and pseudo maps of adjunctions. The result suggests that the various biases in denotational semantics: indirect, call-by-value, call-by-name... are a way of hiding the fact that composition is not always associative.Nous caractérisons l'ordre d'évaluation polarisé à travers une structure catégorielle dont l'hypothèse que la composition est associative est relâchée. Duploïde est le nom de la structure, par référence aux algèbres dupliciales de Loday. Le résultat principal est une réflection Adj→Dupl où Dupl est une catégorie des duploïdes et des foncteurs de duploïdes, et Adj est la catégorie des adjonctions et des pseudo-morphismes d'adjonctions. Le résultat suggère que les biais des sémantiques dénotationnelles: indirectes, en appel par valeur, en appel par nom... sont des façons de cacher le fait que la composition n'est pas toujours associative

Sound and Complete Typing for lambda-mu

In this paper we define intersection and union type assignment for Parigot's calculus lambda-mu. We show that this notion is complete (i.e. closed under subject-expansion), and show also that it is sound (i.e. closed under subject-reduction). This implies that this notion of intersection-union type assignment is suitable to define a semantics.Comment: In Proceedings ITRS 2010, arXiv:1101.410