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

Comparing and evaluating extended Lambek calculi

Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expressive logical calculus was needed for many other cases. Therefore, many extensions and variants of the Lambek calculus have been proposed, since the eighties and up until the present day. As a result, there is now a large class of calculi, each with its own empirical successes and theoretical results, but also each with its own logical primitives. This raises the question: how do we compare and evaluate these different logical formalisms? To answer this question, I present two unifying frameworks for these extended Lambek calculi. Both are proof net calculi with graph contraction criteria. The first calculus is a very general system: you specify the structure of your sequents and it gives you the connectives and contractions which correspond to it. The calculus can be extended with structural rules, which translate directly into graph rewrite rules. The second calculus is first-order (multiplicative intuitionistic) linear logic, which turns out to have several other, independently proposed extensions of the Lambek calculus as fragments. I will illustrate the use of each calculus in building bridges between analyses proposed in different frameworks, in highlighting differences and in helping to identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona, Spain. 201

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

Process algebra for performance evaluation

This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems – like large-scale computers, client–server architectures, networks – can accurately be described using such stochastic specification formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for different types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also briefly indicate how one can profit from the separation of time and actions when incorporating more general, non-Markovian distributions

Developments in the rewriting calculus

The theory of developments, originally developed for the Lambda calculus, has been successfully adapted to several other computational paradigms, like first- and higher-order term rewrite system. The main desirable results on developments are the fact that the complete development of a finite set of redexes always terminates (FD) and the fact that, for a given initial term, all complete developments of a fixed set of redexes end with the same term (FD!). Following the ideas in the Lambda calculus, in this paper, we present a notion of development and the proofs of theorems FD and FD! for the rewriting calculus, a framework embedding Lambda calculus and rewriting capabilities, by allowing abstraction not only on variables but also on patterns. As an additional contribution, a new proof of the confluence property for the rewriting calculus, is obtained as a consequence of the results on developments

The Dynamic Geometry of Interaction Machine: A Token-Guided Graph Rewriter

In implementing evaluation strategies of the lambda-calculus, both correctness and efficiency of implementation are valid concerns. While the notion of correctness is determined by the evaluation strategy, regarding efficiency there is a larger design space that can be explored, in particular the trade-off between space versus time efficiency. Aiming at a unified framework that would enable the study of this trade-off, we introduce an abstract machine, inspired by Girard's Geometry of Interaction (GoI), a machine combining token passing and graph rewriting. We show soundness and completeness of our abstract machine, called the \emph{Dynamic GoI Machine} (DGoIM), with respect to three evaluations: call-by-need, left-to-right call-by-value, and right-to-left call-by-value. Analysing time cost of its execution classifies the machine as ``efficient'' in Accattoli's taxonomy of abstract machines.Comment: arXiv admin note: text overlap with arXiv:1802.0649

Optimal Sharing Graphs for Substructural Higher-order Rewriting Systems

The notion of optimal reduction was introduced by Lévy (1980) in the context of the untyped ?-calculus, based on the concept of families of reducible expressions. It took more than a decade for an algorithm achieving this optimal reduction to be discovered, introduced by Lamping (1990) and then refined by Gonthier, Abadi & Lévy (1992). The existence of an analogous algorithm for higher-order term rewriting systems was later theorised by Van Oostrom (1996), but has of yet been unrealised. We provide such an algorithm by defining a class of higher-order rewriting systems having Intuitionistic Linear Logic (Benton, Bierman, de Paiva & Hyland 1992) as a substitution calculus, in the sense of Van Oostrom (1994), and introduce a method of translating terms and rules into equivalent Lamping-Gonthier sharing graphs. Our system thus offers a generalisation of the mechanism for optimal reduction from second- to higher-order term rewriting systems. Moreover, in the case of match-sequential systems, we provide a specific reduction strategy, as we are able to effectively identify needed redexes. Finally, we explore briefly the subtleties and complexities of applying the technique to various other term rewriting system, such as those with alternative substructural or polymorphic type systems, those with generalised patterns on the left-hand side, and those with rationally infinite terms. All these systems are built upon the same fundamental translation of ?-terms to sharing graphs

An Introduction to String Diagrams for Computer Scientists

This document is an elementary introduction to string diagrams. It takes a computer science perspective: rather than using category theory as a starting point, we build on intuitions from formal language theory, treating string diagrams as a syntax with its semantics. After the basic theory, pointers are provided to contemporary applications of string diagrams in various fields of science

Lexical and Derivational Meaning in Vector-Based Models of Relativisation

Sadrzadeh et al (2013) present a compositional distributional analysis of relative clauses in English in terms of the Frobenius algebraic structure of finite dimensional vector spaces. The analysis relies on distinct type assignments and lexical recipes for subject vs object relativisation. The situation for Dutch is different: because of the verb final nature of Dutch, relative clauses are ambiguous between a subject vs object relativisation reading. Using an extended version of Lambek calculus, we present a compositional distributional framework that accounts for this derivational ambiguity, and that allows us to give a single meaning recipe for the relative pronoun reconciling the Frobenius semantics with the demands of Dutch derivational syntax.Comment: 10 page version to appear in Proceedings Amsterdam Colloquium, updated with appendi