Open Graphs and Monoidal Theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits "enough adhesivity" from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are "type-safe" in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discreet, finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parametrised by graphical signatures, similar to the monoidal signatures of Joyal and Street, which define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs, and more general monoidal theories. Thus open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting review

Modular Graph Rewriting to Compute Semantics

International audienceTaking an asynchronous perspective on the syntax-semantics interface, we propose to use modular graph rewriting systems as the model of computation. We formally define them and demonstrate their use with a set of modules which produce underspecified semantic representations from a syntactic dependency graph. We experimentally validate this approach on a set of sentences. The results open the way for the production of underspecified semantic dependency structures from corpora annotated with syntactic dependencies and, more generally, for a broader use of modular rewriting systems for computational linguistics

On Constructor Rewrite Systems and the Lambda Calculus

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by- value beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in a term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.Comment: 27 pages. arXiv admin note: substantial text overlap with arXiv:0904.412

A Graph Grammar for Modelling RNA Folding

We propose a new approach for modelling the process of RNA folding as a graph transformation guided by the global value of free energy. Since the folding process evolves towards a configuration in which the free energy is minimal, the global behaviour resembles the one of a self-adaptive system. Each RNA configuration is a graph and the evolution of configurations is constrained by precise rules that can be described by a graph grammar.Comment: In Proceedings GaM 2016, arXiv:1612.0105

General Ramified Recurrence is Sound for Polynomial Time

Leivant's ramified recurrence is one of the earliest examples of an implicit characterization of the polytime functions as a subalgebra of the primitive recursive functions. Leivant's result, however, is originally stated and proved only for word algebras, i.e. free algebras whose constructors take at most one argument. This paper presents an extension of these results to ramified functions on any free algebras, provided the underlying terms are represented as graphs rather than trees, so that sharing of identical subterms can be exploited

Higher-order port-graph rewriting

The biologically inspired framework of port-graphs has been successfully used to specify complex systems. It is the basis of the PORGY modelling tool. To facilitate the specification of proof normalisation procedures via graph rewriting, in this paper we add higher-order features to the original port-graph syntax, along with a generalised notion of graph morphism. We provide a matching algorithm which enables to implement higher-order port-graph rewriting in PORGY, thus one can visually study the dynamics of the systems modelled. We illustrate the expressive power of higher-order port-graphs with examples taken from proof-net reduction systems.Comment: In Proceedings LINEARITY 2012, arXiv:1211.348

Circuit Complexity Meets Ontology-Based Data Access

Ontology-based data access is an approach to organizing access to a database augmented with a logical theory. In this approach query answering proceeds through a reformulation of a given query into a new one which can be answered without any use of theory. Thus the problem reduces to the standard database setting. However, the size of the query may increase substantially during the reformulation. In this survey we review a recently developed framework on proving lower and upper bounds on the size of this reformulation by employing methods and results from Boolean circuit complexity.Comment: To appear in proceedings of CSR 2015, LNCS 9139, Springe

Intensional properties of polygraphs

We present polygraphic programs, a subclass of Albert Burroni's polygraphs, as a computational model, showing how these objects can be seen as first-order functional programs. We prove that the model is Turing complete. We use polygraphic interpretations, a termination proof method introduced by the second author, to characterize polygraphic programs that compute in polynomial time. We conclude with a characterization of polynomial time functions and non-deterministic polynomial time functions.Comment: Proceedings of TERMGRAPH 2007, Electronic Notes in Computer Science (to appear), 12 pages, minor changes from previous versio