Abstract not found

It is well-known that a term rewriting system can be faithfully described by a cartesian 2-category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2-categorical presentation for term graph rewriting. Building on a result presented in [8], which shows that term graphs over a given signature are in one-to-one correspondence with arrows of a gs-monoidal category freely generated from the signature, we associate with a term graph rewriting system a gs-monoidal 2-category, and show that cells faithfully represent its rewriting sequences. We exploit the categorical framework to relate term graph rewriting and term rewriting, since gs-monoidal (2-)categories can be regarded as "weak" cartesian (2-)categories, where certain (2-)naturality axioms have been dropped.

Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During the last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which can represent possibly infinite, rational terms. In [KKSdV94] the authors formalize the classical relationship between TGR and TR as an "adequate mapping" between rewriting systems, and extend it by proving that unraveling is an adequate mapping from cyclic TGR to rational, infinitary term rewriting: In fact, a single graph reduction may correspond to an infinite sequence of term reductions. Using the same notions, we propose a different adequacy result, showing that unraveling is an adequate mapping from cyclic TGR to rational parallel term rewriting, where at each reduction infinitely many rules can be applied in parallel. We also argue that our adequacy result is more natural...

. When evaluation in functional programming languages is explained using -calculus and/or term rewriting systems, expressions and function definitions are often defined as terms, that is as trees. Similarly, the collection of all terms is defined as a forest, that is a directed, acyclic graph where every vertex has at most one incoming edge. Concrete implementations usually drop the last restriction (and sometimes acyclicity as well), i.e. many terms can share a common subterm, meaning that different paths of subterm edges reach the same vertex in the graph. Any vertex in such a graph represents a term. A term is represented uniquely in such a graph if there are no two different vertices representing it. Such a representation can be established by using hash-consing for the creation of heap objects. We investigate the consequences of adopting uniqueness in this sense as a leitmotiv for implementation (called Unlimp), i.e. not allowing any two different vertices in a graph to represent ...