198 research outputs found
Categorical Abstract Rewriting Systems and Functoriality of Graph Transformation
Rewriting systems are often defined as binary relations over a given set of
objects. This simple definition is used to describe various properties of
rewriting such as termination, confluence, normal forms etc. In this paper, we
introduce a new notion of abstract rewriting in the framework of categories.
Then, we define the functoriality property of rewriting systems. This property
is sometimes called vertical composition. We show that most of graph
transformation systems are functorial and provide a counter-example of graph
transformation systems which is not functorial
Rewriting Abstract Structures: Materialization Explained Categorically
The paper develops an abstract (over-approximating) semantics for
double-pushout rewriting of graphs and graph-like objects. The focus is on the
so-called materialization of left-hand sides from abstract graphs, a central
concept in previous work. The first contribution is an accessible, general
explanation of how materializations arise from universal properties and
categorical constructions, in particular partial map classifiers, in a topos.
Second, we introduce an extension by enriching objects with annotations and
give a precise characterization of strongest post-conditions, which are
effectively computable under certain assumptions
Diagrammatic Semantics for Digital Circuits
We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications
Deriving Bisimulation Congruences: A 2-Categorical Approach
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-pushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that sufficiently many GRPOs exist. The theory is applied to a simple subset of CCS and the resulting LTS is compared to one derived using a procedure proposed by Sewell
Deriving Bisimulation Congruences using 2-Categories
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-pushouts (RPO). They are suitable for deriving labelled transition systems (LTS) for process calculi where terms are viewed modulo structural congruence. We develop their basic properties and show that bisimulation on the LTS derived via GRPOs is a congruence, provided that sufficiently many GRPOs exist. The theory is applied to a simple subset of CCS and the resulting LTS is compared to one derived using a procedure proposed by Sewell
Two Algebraic Process Semantics for Contextual Nets
We show that the so-called 'Petri nets are monoids' approach initiated by Meseguer and Montanari can be extended from ordinary place/transition Petri nets to contextual nets by considering suitable non-free monoids of places. The algebraic characterizations of net concurrent computations we provide cover both the collective and the individual token philosophy, uniformly along the two interpretations, and coincide with the classical proposals for place/transition Petri nets in the absence of read-arcs
Coherent presentations of Artin monoids
We compute coherent presentations of Artin monoids, that is presentations by
generators, relations, and relations between the relations. For that, we use
methods of higher-dimensional rewriting that extend Squier's and Knuth-Bendix's
completions into a homotopical completion-reduction, applied to Artin's and
Garside's presentations. The main result of the paper states that the so-called
Tits-Zamolodchikov 3-cells extend Artin's presentation into a coherent
presentation. As a byproduct, we give a new constructive proof of a theorem of
Deligne on the actions of an Artin monoid on a category
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