Congruences for Contextual Graph-Rewriting
TL;DR: A comprehensive operational semantic theory of graph rewriting is introduced, recasting rewriting frameworks as Leifer and Milner's reactive systems, and the construction of groupoidal relative pushouts in suitable cospan categories over arbitrary adhesive categories is introduced.
Abstract: We introduce a comprehensive operational semantic theory of graph rewriting. The central idea is recasting rewriting frameworks as Leifer and Milner's reactive systems. Consequently, graph rewriting systems are associated with canonical labelled transition systems, on which bisimulation equivalence is a congruence with respect to arbitrary graph contexts (cospans of graphs). This construction is derived from a more general theorem of much wider applicability. Expressed in abstract categorical terms, the central technical contribution of the paper is the construction of groupoidal relative pushouts, introduced and developed by the authors in recent work, in suitable cospan categories over arbitrary adhesive categories. As a consequence, we both generalise and shed light on rewriting via borrowed contexts due to Ehrig and Konig.
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TL;DR: Adhesion categories are introduced, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular mon morphisms.
Abstract: We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved, as well as quasiadhesive categories which restrict attention to regular monomorphisms. Many examples of graphical structures used in computer science are shown to be examples of adhesive and quasiadhesive categories. Double-pushout graph rewriting generalizes well to rewriting on arbitrary adhesive and quasiadhesive categories.
234 citations
26 Jun 2005
TL;DR: A general construction of bicolimits in a class of bicategones of cospans is offered, which sheds light on as well as extends Ehrig and Konig's rewriting via borrowed contexts and opens the way to a unified treatment of several applications.
Abstract: The theory of reactive systems, introduced by Leifer and Milner and previously extended by the authors, allows the derivation of well-behaved labelled transition systems (LTS) for semantic models with an underlying reduction semantics. The derivation procedure requires the presence of certain colimits (or, more usually and generally, bicolimits) which need to be constructed separately within each model. In this paper, we offer a general construction of such bicolimits in a class of bicategones of cospans. The construction sheds light on as well as extends Ehrig and Konig's rewriting via borrowed contexts and opens the way to a unified treatment of several applications.
98 citations
TL;DR: The proof that the bisimilarity based on graph rewriting with borrowed contexts is a congruence relation is introduced and compared with the derivation of labelled transitions via relative pushouts is compared.
Abstract: Motivated by recent work on the derivation of labelled transitions and bisimulation congruences from unlabelled reaction rules, we show how to address this problem in the DPO (double-pushout) approach to graph rewriting. Unlike the case with previous approaches, we consider graphs as objects, rather than arrows, of the category under consideration. This allows us to present a very simple way of deriving labelled transitions (called rewriting steps with borrowed context), which integrates smoothly with the DPO approach, has a very constructive nature and requires only a minimum of category theory. The core part of this paper is the proof that the bisimilarity based on graph rewriting with borrowed contexts is a congruence relation. We will also introduce some proof techniques and compare our approach with the derivation of labelled transitions via relative pushouts.
81 citations
Cites background or methods from "Congruences for Contextual Graph-Re..."
...In fact, this requires corresponding noninjective extensions of Conditions A.1–A.8, which have been shown already in Sassone and Sobociński (2004) and Sobociński (2004), respectively....
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...† A more formal treatment can be found in Sassone and Sobociński (2004)....
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...(34) Moreover, it is shown in Sassone and Sobociński (2004) that the construction of rewriting steps based on their GRPOs corresponds exactly to rewriting with borrowed contexts in our sense (see Definition 3.1)....
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...Apart from this simplification, there are, in principle, two ways to force uniqueness: one can either add support to the category (Leifer 2001; Jensen and Milner 2003), that is, every graph item should have a unique unchangeable name; or one can work in a 2-categorical setting (Sassone and Sobociński 2004; Sassone and Sobociński 2003), where commuting cubes such as the one in Diagram (31) are additionally provided with an isomorphism giving a ‘proof of structural congruence’....
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...The alternative, which is pursued in Sassone and Sobociński (2003) and Sassone and Sobociński (2004), would be to fix a choice of pushout....
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TL;DR: A labelled transition system is derived for condition-event nets, corresponding to a natural notion of observable actions in Petri-net theory, and yields a congruential bisimilarity coinciding with one derived directly from the observable actions.
Abstract: A framework is defined within which reactive systems can be studied formally. The framework is based on s-categories, which are a new variety of categories within which reactive systems can be set up in such a way that labelled transition systems can be uniformly extracted. These lead in turn to behavioural preorders and equivalences, such as the failures preorder (treated elsewhere) and bisimilarity, which are guaranteed to be congruential. The theory rests on the notion of relative pushout, which was previously introduced by the authors.The framework is applied to a particular graphical model, known as link graphs, which encompasses a variety of calculi for mobile distributed processes. The specific theory of link graphs is developed. It is then applied to an established calculus, namely condition-event Petri nets.In particular, a labelled transition system is derived for condition-event nets, corresponding to a natural notion of observable actions in Petri-net theory. The transition system yields a congruential bisimilarity coinciding with one derived directly from the observable actions. This yields a calibration of the general theory of reactive systems and link graphs against known specific theories.
56 citations
Cites background or methods from "Congruences for Contextual Graph-Re..."
...Recent work (Ehrig 2002; Ehrig and König 2004; Sassone and Sobocinski 2004), however, bridges these gaps and therefore opens up the possibility for a formal comparison....
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...Recent work (Ehrig 2002; Ehrig and König 2004; Sassone and Sobocinski 2004), however, bridges these gaps and therefore opens up the possibility for a formal comparison....
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...This led to the paper Ehrig and König (2004), which was further generalised by Sassone and Sobocinski (2004), in which the RPO technique is transferred to graph-embedding categories....
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Dissertation•
01 Jan 2006
TL;DR: Higher level tools for quantum mechanics will be better suited to computation than those presently employed in the field, and this work connects quantum mechanics to mainstream areas of computer science such as categorical logic, type theory, program language semantics, and rewriting.
Abstract: This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in low-level formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly important in this treatment, as is appropriate, since entanglement is a fundamental driver of quantum computation. The benefits two-fold: as well as producing foundational tools for the study of quantum computation this work also connects quantum mechanics to mainstream areas of computer science such as categorical logic, type theory, program language semantics, and rewriting. We describe, following Abramsky and Coecke, how quantum mechanics may be carried out without reference to Hilbert space, in a strongly compact closed category. In particular we show how to freely construct a categorical model of abstract quantum mechanics from an arbitrary category. We introduceMultiplicative Categorical Quantum Logic (mCQL), a sequent calculus whose proof rules capture the structure of compact closed categories. This sequent calculus is interpreted in a freely generated compact closed category, and its semantics is sound with respect to cut elimination. We define an equivalent graphical syntax, similar to linear logic’s proof-nets, and prove that these proof-nets provide a full and faithful representation of any freely generated compact closed category. Further analysis of the structure of quantum states which correspond to mCQL proofs using multiplicative linear logic shows that the linear type system describes the quantum entanglement found in such states. We show that the entanglement present in these states is always of a particularly simple form: collections of entangled pairs. In order to tackle arbitrary entanglement, we generalise the work of Kelly and Laplaza to give a representation theorem for the free compact closed category by a polycategory. Such categories are shown to be equivalent to a generalised system of proof-nets whose axioms may have more than one premise
51 citations
References
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Book•
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
9,254 citations
15 Oct 1973
TL;DR: An algebraic theory of graph-grammars is presented using homomorphisms and pushout-constructions to specify embeddings and direct derivations constructively and allows simplification of the proofs and pregnant formulation of concepts like "parallel composition" and "translation of grammars".
Abstract: The paper presents an algebraic theory of graph-grammars using homomorphisms and pushout-constructions to specify embeddings and direct derivations constructively. We consider the case of arbitrary directed graphs permitting loops and parallel edges. The gluing of two arbitrary labeled graphs (push-out) is defined allowing a strictly symmetric definition of direct derivations and the embedding of derivations into a common frame. A two-dimensional hierarchy of graph-grammars is given including the classical case of Chomsky-grammars and several other graphgrammar constructions as special types. The use of well-known categorical constructions and results allows simplification of the proofs and pregnant formulation of concepts like "parallel composition" and "translation of grammars".
540 citations
"Congruences for Contextual Graph-Re..." refers background in this paper
...Introduced in [6], it has recently been generalised in [3, 11, 4]....
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22 Aug 2000
TL;DR: In this paper, the notion of relative pushout (RPO) is introduced to ensure that bisimilarity is a congruence when sufficient RPOs exist, and two examples -a simplified form of action calculi and term-rewriting -are given.
Abstract: The dynamics of reactive systems, e.g. CCS, has often been defined using a labelled transition system (LTS). More recently it has become natural in defining dynamics to use reaction rules - i.e. unlabelled transition rules - together with a structural congruence. But LTSs lead more naturally to behavioural equivalences. So one would like to derive from reaction rules a suitable LTS.
This paper shows how to derive an LTS for a wide range of reactive systems. A label for an agent a is defined to be any context F which intuitively is just large enough so that the agent Fa ("a in context F") is able to perform a reaction. The key contribution of this paper is a precise definition of "just large enough", in terms of the categorical notion of relative pushout (RPO), which ensures that bisimilarity is a congruence when sufficient RPOs exist. Two examples - a simplified form of action calculi and term-rewriting - are given, for which it is shown that sufficient RPOs indeed exist. The thrust of this paper is, therefore, towards a general method for achieving useful behavioural congruence relations.
221 citations
TL;DR: In this article, the authors introduce basic notions like productions, derivations, parellel and sequential independence for high-level replacement syetms leading to Church-Rosser, parallelism and concurrency Theorems.
Abstract: High-level replacement systems are formulated in an axiomatic algebraic framework based on categories pushouts. This approach generalizes the well-known algebraic approach to graph grammars and several other types of replacement systems, especially the replacement of algebraic specifications which was recently introduced for a rule-based approach to modular system design.in this paper basic notions like productions, derivations, parellel and sequential independence are introduced for high-level replacement syetms leading to Church-Rosser, Parallelism and concurrency Theorems previously shown in the literature for special cases only. In the general case of high-level replacement systems specific conditions, called HLR1- and HLR2-conditions, are formulated in order to obtain these results.Several examples of high-level replacement systems are discussed and classified w.r.t. HLR1- and HLR2-conditions showing which of the results are valid in each case.
177 citations
"Congruences for Contextual Graph-Re..." refers background in this paper
...Introduced in [6], it has recently been generalised in [3, 11, 4]....
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TL;DR: This paper considers calculi with arbitrary reduction semantics of three simple classes, firstly groundterm rewriting, then left-linear term rewriting, and then a class which is essentially the action calculi lacking substantive name binding, which gives rise to bisimulation congruences.
118 citations
"Congruences for Contextual Graph-Re..." refers background in this paper
...The labels of the lts are the ‘smallest’ contexts which allow reactions to occur ‐ an idea due to Sewell [ 18 ]....
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