Pattern graph rewrite systems

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string diagrams, amenable to automated reasoning about diagrammatic theories via graph rewrite systems. In this extended abstract, we show how the power of such rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs, called !-boxes ("bang boxes"), on both sides of a rule can be copied any number of times or removed. After reviewing the string graph formalism, we show how string graphs can be extended to pattern graphs and how pattern graphs and pattern rewrite rules can be instantiated to concrete string graphs and rewrite rules. We then provide examples demonstrating the expressive power of pattern graphs and how they can be applied to study interacting algebraic structures that are central to categorical quantum mechanics.Comment: In Proceedings DCM 2012, arXiv:1403.757

On prefixal one-rule string rewrite systems

International audiencePrefixal one-rule string rewrite systems are one-rule string rewrite systems for which the left-hand side of the rule is a prefix of the right-hand side of the rule. String rewrite systems induce a transformation over languages: from a starting word, one can associate all its descendants. We prove, in this work, that the transformation induced by a prefixal one-rule rewrite system always transforms a finite language into a context-free language, a property that is surprisingly not satisfied by arbitrary one-rule rewrite systems. We also give here a decidable characterization of the prefixal one-rule rewrite systems whose induced transformation is a rational transduction

The Hydra Battle and Cichon’s Principle

Abstract In rewriting the Hydra battle refers to a term rewrite system H proposed by Dershowitz and Jouannaud. To date, H withstands any attempt to prove its termination automatically. This motivates our interest in term rewrite systems encoding the Hydra battle, as a careful study of such systems may prove useful in the design of automatic termination tools. Moreover it has been an open problem, whether any termination order compatible with H has to have the Howard-Bachmann ordinal as its order type, i.e., the proof theoretic ordinal of the theory of one inductive denition. We answer this question in the negative, by providing a reduction order compatible with H, whose order type is at most 0 , the proof theoretic ordinal of Peano arithmetic

Exploring linear size-change terminating programs

Ph.DDOCTOR OF PHILOSOPH