Pawel Sobocinski | University of Southampton (original) (raw)
Papers by Pawel Sobocinski
The exactness properties of coproducts in extensive categories and pushouts along monos in adhesi... more 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 \(\mathcal{S}pan_{\mathbb{C}}\) , provided that ℂ has pullbacks and enough colimits.
Rewriting systems over adhesive categories have been recently introduced as a general framework w... more 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.
submitted to calco
Abstract. We generalize the unfolding semantics, previously developed for concrete formalisms suc... more Abstract. We generalize the unfolding semantics, previously developed for concrete formalisms such as Petri nets and graph grammars, to the abstract setting of (single pushout) rewriting over adhesive categories. The unfolding construction is characterized as a coreflection, ie the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars.
… of Software Science and Computation Structures, Jan 1, 2004
We introduce adhesive categories, which are categories with structure ensuring that pushouts alon... more We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved. Many types of graphical structures used in computer science are shown to be examples of adhesive categories. Double-pushout graph rewriting generalises well to rewriting on arbitrary adhesive categories.
RAIRO-Theoretical Informatics and …, Jan 1, 2005
We introduce adhesive categories, which are categories with structure ensuring that pushouts alon... more 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.
Logic in Computer Science, 2005. …, Jan 1, 2005
The theory of reactive systems, introduced by Leifer and Milner and previously extended by the au... more 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 bicategories of cospans. The construction sheds light on as well as extends Ehrig and König's rewriting via borrowed contexts and opens the way to a unified treatment of several applications.
Nordic Journal of Computing, Jan 1, 2003
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-push... more 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.
… of the 6th International conference on …, Jan 1, 2003
G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Le... more G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner's approach to deriving labelled bisimulation congruences from reduction systems. This paper develops the theory of GRPOs further, arguing that they provide a simple and powerful basis towards a comprehensive solution. As an example, we construct GRPOs in a category of 'bunches and wirings.' We then examine the approach based on Milner's precategories and Leifer's functorial reactive systems, and show that it can be recast in a much simpler way into the 2-categorical theory of GRPOs.
This thesis is concerned with the development of a theory which, given a formalism with a reducti... more This thesis is concerned with the development of a theory which, given a formalism with a reduction semantics, allows the derivation of a canonical labelled transition system on which bisimilarity as well as other other equivalences are congruences; provided that the contexts of the formalism form a category which has certain colimits.
Foundations of Software Science …, Jan 1, 2007
We use the framework of biorthogonality to introduce a novel semantic definition of the concept o... more We use the framework of biorthogonality to introduce a novel semantic definition of the concept of barb (basic observable) for process calculi. We develop a uniform basic theory of barbs and demonstrate its robustness by showing that it gives rise to the correct observables in specific process calculi which model synchronous, asynchronous and broadcast communication regimes.
Electronic Notes in Theoretical Computer …, Jan 1, 2002
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-push... more 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.
Electronic Notes in Theoretical Computer …, Jan 1, 2005
We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategor... more We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategory of cospans over a category of p/t nets and embeddings. We derive a labelled transition system (LTS) semantics for such nets using GIPOs and characterise the resulting congruence. Technically, our results are similar to the recent work by Milner on applying the theory of bigraphs to Petri Nets. The two main differences are that we treat p/t nets instead of c/e nets and we deal directly with a category of nets instead of encoding them into bigraphs.
… of Software Science …, Jan 1, 2006
Rewriting systems over adhesive categories have been recently introduced as a general framework w... more 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-toone 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.
CONCUR 2008-Concurrency Theory, Jan 1, 2008
We present a new labelled transition system (lts) for the ambient calculus. Its most important pr... more We present a new labelled transition system (lts) for the ambient calculus. Its most important property is that ordinary (strong) bisimulation coincides with (strong) contextual equivalence. The lts is the outcome of the authors' ongoing work towards developing general techniques and systematic procedures for deriving ltss in the structural (sos) style from the underlying reduction semantics and observability.
Theoretical Computer Science, Jan 1, 2005
Groupoidal relative pushouts (GRPOs) have recently been proposed by the authors as a new foundati... more Groupoidal relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner's approach to deriving labelled bisimulation congruences from reduction systems. In this paper, we develop the theory of GRPOs further, proving that well-known equivalences, other than bisimulation, are congruences. To demonstrate the type of category theoretic arguments which are inherent in the 2-categorical approach, we construct GRPOs in a category of 'bunches and wirings.' Finally, we prove that the 2-categorical theory of GRPOs is a generalisation of the approaches based on Milner's precategories and Leifer's functorial reactive systems.
Algebra and Coalgebra in Computer …, Jan 1, 2005
We consider open terms and parametric rules in the context of the systematic derivation of labell... more We consider open terms and parametric rules in the context of the systematic derivation of labelled transitions from reduction systems.
We introduce a comprehensive operational semantic theory of graphrewriting. Graph-rewriting here ... more We introduce a comprehensive operational semantic theory of graphrewriting. Graph-rewriting here is meant in a broad sense as we aim to cover and extend previous work based both on Milner's bigraphs and Ehrig and König's rewriting via borrowed contexts. 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). The central technical contribution of the paper is the construction of groupoidal relative pushouts, introduced and developed by the authors in recent work, in inputlinear cospan (bi)categories over arbitrary adhesive categories.
Fifth Ifip International Conference On Theoretical …, Jan 1, 2008
We re-examine the standard structural operational semantics of the π-calculus with the view that ... more We re-examine the standard structural operational semantics of the π-calculus with the view that both process structure and contextual observational power should play roles in describing the behavioural theory. To that end we provide a decomposition of the operational semantics of π which allows for a systematic definition of labelled transitions. These are derived from the calculus' underlying reduction rules by following the contexts-as-labels philosophy while being presented using the structural approach. Our novel transition system refines to a composite description of the standard early lts. We generalise our technique to higher-order and asynchronous variants.
Applied Categorical Structures, Jan 1, 2008
Subobject transformation systems (sts) are proposed as a novel formal framework for the analysis ... more Subobject transformation systems (sts) are proposed as a novel formal framework for the analysis of derivations of transformation systems based on the algebraic, double-pushout (dpo) approach. They can be considered as a simplified variant of dpo rewriting, acting in the distributive lattice of subobjects of a given object of an adhesive category. This setting allows for a direct analysis of all possible notions of causality between any two productions without requiring an explicit match. In particular several equivalent characterizations of independence of productions are proposed, as well as a local Church-Rosser theorem in the setting of sts. Finally, we show how any derivation tree in an ordinary dpo grammar leads to an sts via a process-like construction and show that relational reasoning in the resulting sts is sound and complete with respect to the independence in the original derivation tree.
CONCUR 2003-Concurrency Theory, Jan 1, 2003
The exactness properties of coproducts in extensive categories and pushouts along monos in adhesi... more 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 \(\mathcal{S}pan_{\mathbb{C}}\) , provided that ℂ has pullbacks and enough colimits.
Rewriting systems over adhesive categories have been recently introduced as a general framework w... more 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.
submitted to calco
Abstract. We generalize the unfolding semantics, previously developed for concrete formalisms suc... more Abstract. We generalize the unfolding semantics, previously developed for concrete formalisms such as Petri nets and graph grammars, to the abstract setting of (single pushout) rewriting over adhesive categories. The unfolding construction is characterized as a coreflection, ie the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars.
… of Software Science and Computation Structures, Jan 1, 2004
We introduce adhesive categories, which are categories with structure ensuring that pushouts alon... more We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved. Many types of graphical structures used in computer science are shown to be examples of adhesive categories. Double-pushout graph rewriting generalises well to rewriting on arbitrary adhesive categories.
RAIRO-Theoretical Informatics and …, Jan 1, 2005
We introduce adhesive categories, which are categories with structure ensuring that pushouts alon... more 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.
Logic in Computer Science, 2005. …, Jan 1, 2005
The theory of reactive systems, introduced by Leifer and Milner and previously extended by the au... more 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 bicategories of cospans. The construction sheds light on as well as extends Ehrig and König's rewriting via borrowed contexts and opens the way to a unified treatment of several applications.
Nordic Journal of Computing, Jan 1, 2003
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-push... more 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.
… of the 6th International conference on …, Jan 1, 2003
G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Le... more G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner's approach to deriving labelled bisimulation congruences from reduction systems. This paper develops the theory of GRPOs further, arguing that they provide a simple and powerful basis towards a comprehensive solution. As an example, we construct GRPOs in a category of 'bunches and wirings.' We then examine the approach based on Milner's precategories and Leifer's functorial reactive systems, and show that it can be recast in a much simpler way into the 2-categorical theory of GRPOs.
This thesis is concerned with the development of a theory which, given a formalism with a reducti... more This thesis is concerned with the development of a theory which, given a formalism with a reduction semantics, allows the derivation of a canonical labelled transition system on which bisimilarity as well as other other equivalences are congruences; provided that the contexts of the formalism form a category which has certain colimits.
Foundations of Software Science …, Jan 1, 2007
We use the framework of biorthogonality to introduce a novel semantic definition of the concept o... more We use the framework of biorthogonality to introduce a novel semantic definition of the concept of barb (basic observable) for process calculi. We develop a uniform basic theory of barbs and demonstrate its robustness by showing that it gives rise to the correct observables in specific process calculi which model synchronous, asynchronous and broadcast communication regimes.
Electronic Notes in Theoretical Computer …, Jan 1, 2002
We introduce G-relative-pushouts (GRPO) which are a 2-categorical generalisation of relative-push... more 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.
Electronic Notes in Theoretical Computer …, Jan 1, 2005
We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategor... more We introduce a way of viewing Petri nets as open systems. This is done by considering a bicategory of cospans over a category of p/t nets and embeddings. We derive a labelled transition system (LTS) semantics for such nets using GIPOs and characterise the resulting congruence. Technically, our results are similar to the recent work by Milner on applying the theory of bigraphs to Petri Nets. The two main differences are that we treat p/t nets instead of c/e nets and we deal directly with a category of nets instead of encoding them into bigraphs.
… of Software Science …, Jan 1, 2006
Rewriting systems over adhesive categories have been recently introduced as a general framework w... more 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-toone 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.
CONCUR 2008-Concurrency Theory, Jan 1, 2008
We present a new labelled transition system (lts) for the ambient calculus. Its most important pr... more We present a new labelled transition system (lts) for the ambient calculus. Its most important property is that ordinary (strong) bisimulation coincides with (strong) contextual equivalence. The lts is the outcome of the authors' ongoing work towards developing general techniques and systematic procedures for deriving ltss in the structural (sos) style from the underlying reduction semantics and observability.
Theoretical Computer Science, Jan 1, 2005
Groupoidal relative pushouts (GRPOs) have recently been proposed by the authors as a new foundati... more Groupoidal relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner's approach to deriving labelled bisimulation congruences from reduction systems. In this paper, we develop the theory of GRPOs further, proving that well-known equivalences, other than bisimulation, are congruences. To demonstrate the type of category theoretic arguments which are inherent in the 2-categorical approach, we construct GRPOs in a category of 'bunches and wirings.' Finally, we prove that the 2-categorical theory of GRPOs is a generalisation of the approaches based on Milner's precategories and Leifer's functorial reactive systems.
Algebra and Coalgebra in Computer …, Jan 1, 2005
We consider open terms and parametric rules in the context of the systematic derivation of labell... more We consider open terms and parametric rules in the context of the systematic derivation of labelled transitions from reduction systems.
We introduce a comprehensive operational semantic theory of graphrewriting. Graph-rewriting here ... more We introduce a comprehensive operational semantic theory of graphrewriting. Graph-rewriting here is meant in a broad sense as we aim to cover and extend previous work based both on Milner's bigraphs and Ehrig and König's rewriting via borrowed contexts. 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). The central technical contribution of the paper is the construction of groupoidal relative pushouts, introduced and developed by the authors in recent work, in inputlinear cospan (bi)categories over arbitrary adhesive categories.
Fifth Ifip International Conference On Theoretical …, Jan 1, 2008
We re-examine the standard structural operational semantics of the π-calculus with the view that ... more We re-examine the standard structural operational semantics of the π-calculus with the view that both process structure and contextual observational power should play roles in describing the behavioural theory. To that end we provide a decomposition of the operational semantics of π which allows for a systematic definition of labelled transitions. These are derived from the calculus' underlying reduction rules by following the contexts-as-labels philosophy while being presented using the structural approach. Our novel transition system refines to a composite description of the standard early lts. We generalise our technique to higher-order and asynchronous variants.
Applied Categorical Structures, Jan 1, 2008
Subobject transformation systems (sts) are proposed as a novel formal framework for the analysis ... more Subobject transformation systems (sts) are proposed as a novel formal framework for the analysis of derivations of transformation systems based on the algebraic, double-pushout (dpo) approach. They can be considered as a simplified variant of dpo rewriting, acting in the distributive lattice of subobjects of a given object of an adhesive category. This setting allows for a direct analysis of all possible notions of causality between any two productions without requiring an explicit match. In particular several equivalent characterizations of independence of productions are proposed, as well as a local Church-Rosser theorem in the setting of sts. Finally, we show how any derivation tree in an ordinary dpo grammar leads to an sts via a process-like construction and show that relational reasoning in the resulting sts is sound and complete with respect to the independence in the original derivation tree.
CONCUR 2003-Concurrency Theory, Jan 1, 2003