Equational reasoning with context-free families of string diagrams

String diagrams provide an intuitive language for expressing networks of interacting processes graphically. A discrete representation of string diagrams, called string graphs, allows for mechanised equational reasoning by double-pushout rewriting. However, one often wishes to express not just single equations, but entire families of equations between diagrams of arbitrary size. To do this we define a class of context-free grammars, called B-ESG grammars, that are suitable for defining entire families of string graphs, and crucially, of string graph rewrite rules. We show that the language-membership and match-enumeration problems are decidable for these grammars, and hence that there is an algorithm for rewriting string graphs according to B-ESG rewrite patterns. We also show that it is possible to reason at the level of grammars by providing a simple method for transforming a grammar by string graph rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-21145-9_

On Leśniewski’s Characteristica Universalis

Leśniewski's systems deviate greatly from standard logic in some basic features. The deviant aspects are rather well known, and often cited among the reasons why Leśniewski's work enjoys little recognition. This paper is an attempt to explain why those aspects should be there at all. Leśniewski built his systems inspired by a dream close to Leibniz's characteristica universalis: a perfect system of deductive theories encoding our knowledge of the world, based on a perfect language. My main claim is that Leśniewski built his characteristica universalis following the conditions of de Jong and Betti's Classical Model of Science (2008) to an astounding degree. While showing this I give an overview of the architecture of Leśniewski's systems and of their fundamental characteristics. I suggest among others that the aesthetic constraints Leśniewski put on axioms and primitive terms have epistemological relevance. © The Author(s) 2008

A method for finding new sets of axioms for classes of semigroups

We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices

Interacting Bialgebras are Frobenius

International audienceBialgebras and Frobenius algebras are different ways in which monoids and comonoids interact as part of the same theory. Such theories feature in many fields: e.g. quantum computing, compositional semantics of concurrency, network algebra and component-based programming. In this paper we study an important sub-theory of Coecke and Duncan's ZX-calculus, related to strongly-complementary observables, where two Frobenius algebras interact. We characterize its free model as a category of ℤ2-vector subspaces. Moreover, we use the framework of PROPs to exhibit the modular structure of its algebra via a universal construction involving span and cospan categories of ℤ2-matrices and distributive laws between PROPs. Our approach demonstrates that the Frobenius structures result from the interaction of bialgebras

JGXYZ: An ATP System for Gap and Glut Logics

This paper describes an ATP system, named JGXYZ, for some gap and glut logics. JGXYZ is based on an equi-provable translation to FOL, followed by use of an existing ATP system for FOL. A key feature of JGXYZ is that the translation to FOL is data-driven, in the sense that it requires only the addition of a new logic’s truth tables for the unary and binary connectives in order to produce an ATP system for the logic. Experimental results from JGXYZ illustrate the differences between the logics and translated problems, both technically and in terms of a quasi-real-world use case