Tobias Heindel

Van Kampen colimits as bicolimits in span

The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in C that induce bicolimit diagrams in the bicategory of spans SpanC, provided that C has pullbacks and enough colimits.

Sesqui-pushout rewriting

Sesqui-pushout (SqPO) rewriting-sesqui means one and a half in Latin-is a new algebraic approach to abstract rewriting in any category. SqPO rewriting is a deterministic and conservative extension of double-pushout (DPO) rewriting, which allows to model deletion in unknown context, a typical feature of single-pushout (SPO) rewriting, as well as cloning. After illustrating the expressiveness of the proposed approach through a case study modelling an access control system, we discuss sufficient conditions for the existence of final pullback complements and we analyze the relationship between SqPO and the classical DPO and SPO approaches.

BEING VAN KAMPEN IS A UNIVERSAL PROPERTY

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show that the elementary presentation of the Van Kampen condition is actually a characterisation of a universal property in the associated bicategory of spans. The main theorem states that Van Kampen cocones are precisely those diagrams in a category that induce bicolimit diagrams in its associated bicategory of spans, provided that the category has pullbacks and enough colimits.

Hereditary pushouts reconsidered

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks, which started over thirty years ago in the category of graphs. Adhesive categories provide a single property of pushouts that suffices to derive lemmas that are essential for central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that the same lemmas already hold for pushouts that are hereditary, i.e. those pushouts that remain pushouts when they are embedded into the associated category of partial maps. Hereditary pushouts - a twenty year old concept - induce a generalization of adhesive categories, which will be dubbed partial map adhesive. An application relevant category that does not fit the framework of adhesive categories and its variations in the literature will serve as an illustrating example of a partial map adhesive category.

Processes for adhesive rewriting systems

Rewriting systems over adhesive categories have been recently introduced as a general framework which encompasses several rewriting-based computational formalisms, including various modelling frameworks for concurrent and distributed systems. Here we begin the development of a truly concurrent semantics for adhesive rewriting systems by defining the fundamental notion of process, well-known from Petri nets and graph grammars. The main result of the paper shows that processes capture the notion of true concurrency—there is a one-to-one correspondence between concurrent derivations, where the sequential order of independent steps is immaterial, and (isomorphism classes of) processes. We see this contribution as a step towards a general theory of true concurrency which specialises to the various concrete constructions found in the literature.

Reversible Sesqui-Pushout Rewriting

The paper proposes a variant of sesqui-pushout rewriting (SqPO) that allows one to develop the theory of nested application conditions (NACs) for arbitrary rule spans; this is a considerable generalisation compared with existing results for NACs, which only hold for linear rules (w.r.t. a suitable class of monos). Besides this main contribution, namely an adapted shifting construction for NACs, the paper presents a uniform commutativity result for a revised notion of independence that applies to arbitrary rules; these theorems hold in any category with (enough) stable pushouts and a class of monos rendering it weak adhesive HLR. To illustrate results and concepts, we use simple graphs, i.e. the category of binary endorelations and relation preserving functions, as it is a paradigmatic example of a category with stable pushouts; moreover, using regular monos to give semantics to NACs, we can shift NACs over arbitrary rule spans.

Approximations for Stochastic Graph Rewriting

In this note we present a method to compute approximate descriptions of a class of stochastic systems. For the method to apply, the system must be presented as a Markov chain on a state space consisting in graphs or graph-like objects, and jumps must be described by transformations which follow a finite set of local rules.

A lattice-theoretical perspective on adhesive categories

It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category can be obtained as the colimit of its irreducible subobjects. Furthermore we show that every arrow between finite objects in an adhesive category can be interpreted as a lattice homomorphism between subobject lattices and, conversely, we characterize those homomorphisms between subobject lattices which can be seen as arrows.

A Geometric View of Partial Order Reduction

Verifying that a concurrent program satisfies a given property, such as deadlock-freeness, is computationally difficult. Naive exploration techniques are facing the state space explosion problem: they consider an exponential number of interleavings of parallel threads (relative to the program size). Partial order reduction is a standard method to address this difficulty. It is based on the observation that certain sets of instructions, called persistent sets, are not affected by other concurrent instructions and can thus always be explored first when searching for deadlocks. More recent models of concurrent processes use directed topological spaces: states are points, computations are paths, and equivalent interleavings are homotopic. This geometric approach applies theoretical results of algebraic topology to improve verification. Despite the very different origin of the approaches, the paper compares partial-order reduction with a construction of the geometric approach, the category of future components. The main result, which shows that the two techniques make essentially the same use of persistent transitions, is of foundational interest and aims for cross-fertilization of the two approaches to improve verification methods for concurrent programs.

Adhesivity with Partial Maps instead of Spans

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks that started over thirty years ago for the category of graphs. Adhesive categories - of which graphs are the “archetypal” example - are defined by a single property of pushouts along monos that implies essential lemmas and central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that a strictly weaker condition on pushouts suffices to obtain essentially the same results: it suffices to require pushouts to be hereditary, i.e. they have to remain pushouts when they are embedded into the associated category of partial maps. This fact however is not the only reason to introduce partial map adhesive categories as categories with pushouts along monos (of a certain stable class) that are hereditary. There are two equally important motivations: first, there is an application relevant example category that cannot be captured by the more established variations of adhesive categories; second, partial map adhesive categories are “conceptually similar” to adhesive categories as the latter can be characterized as those categories with pushout along monos that remain bi-pushouts when they are embedded into the associated bi-category of spans. Thus, adhesivity with partial maps instead of spans appears to be a natural candidate for a general rewriting framework.