Tobias Heindel | University of Copenhagen (original) (raw)

Papers by Tobias Heindel

Lecture Notes in Computer Science, 2015

Corr, 2011

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show tha... more 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.

Electronic Proceedings in Theoretical Computer Science, 2010

ABSTRACT Inspired by decomposition problems in rule-based formalisms in Computational Systems Bio... more ABSTRACT Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation.

Lecture Notes in Computer Science, 2008

ABSTRACT In the area of specification and modelling of concurrent systems, Petri nets have become... more ABSTRACT In the area of specification and modelling of concurrent systems, Petri nets have become a standard tool, and they still work behind the scenes in tools for graph transformation systems (cf. [1]). Moreover there is still potential for crossfertilization between the graph transformation and Petri net community. Even a better understanding of adhesive categories [2] and the related concepts of [3] seems possible in light of the proposed notion of weakly adhesive categories [4], which has emerged during work on the generalization of the co-reflective semantics of Petri nets to the realm of graph transformation and adhesive rewriting systems.

It is well-known that the set of subobjects of an object in an adhesive category forms a distribu... more It is well-known that the set of subobjects of an object in an adhesive category forms a distributive lattice. This is a work-in-progress paper where we review the lattice-theoretic representation theorem for finite distributive lattices and show how it applies to subobject lattices. Furthermore we show that every arrow in an adhesive category can be interpreted as a lattice homomorphism and in addition we sketch some ideas about how to identify those homomorphisms between subobject lattices which arise from arrows. Adhesive categories (2) have been shown to provide a general categorical setting in which double-pushout rewriting can be defined in such a way that some fundamental results like the local Church-Rosser theorem and the concurrency theorem can be proved without the need for any additional conditions. The framework of adhesive categories encompasses graphs and several other graphical structures which play a role in the theory of concurrent and distributed systems. It is a ...

Lecture Notes in Computer Science, 2013

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.

Lecture Notes in Computer Science, 2014

ABSTRACT The paper gives a sufficient condition for the existence of all pushouts in an arbitrary... more ABSTRACT The paper gives a sufficient condition for the existence of all pushouts in an arbitrary category of partial maps ℂ *ℳ that is necessary whenever the category of total maps ℂ⊆ℂ *ℳ has cocones of spans; the latter is the case in all slice categories of ℂ and thus the condition is necessary locally. The main theorem is that, given an admissible class of monos ℳ in a category ℂ that has cocones of spans, the category of partial maps ℂ *ℳ has pushouts if and only if the category of total maps ℂ has hereditary pushouts and right adjoints to inverse image functors (where both properties are with respect to ℳ). This result clarifies previous work by Kennaway on graph rewriting in categories of partial maps that implicitly assumed existence of cocones of spans in the category of total maps.

Electronic Proceedings in Theoretical Computer Science, 2011

Process calculi and graph transformation systems provide models of reactive systems with labelled... more Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics (LTS). While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in general. Hence, the goal of this article is to obtain a compositional semantics for graph transformation system in analogy to the structural operational semantics (SOS) for Milner's Calculus of Communicating Systems (CCS).

Lecture Notes in Computer Science, 2010

ABSTRACT The introduction of adhesive categories revived interest in the study of properties of p... more ABSTRACT 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.

Lecture Notes in Computer Science, 2014

Mathematical Structures in Computer Science, 2014

ABSTRACT We generalise both the notion of a non-sequential process and the unfolding construction... more ABSTRACT We generalise both the notion of a non-sequential process and the unfolding construction (which was previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, that is, the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars.As the unfolding represents potentially infinite computations, we need to work in adhesive categories with ‘well-behaved’ colimits of ω-chains of monos. Compared with previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism.

Logical Methods in Computer Science, 2011

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show tha... more 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.

Electronic Notes in Theoretical Computer Science, 2009

Inspired by the scope extrusion phenomenon of name passing calculi that allow to reason about kno... more Inspired by the scope extrusion phenomenon of name passing calculi that allow to reason about knowledge of (secret) names, we propose an abstract formulation of the concept of secret in any weakly adhesive category. The guiding idea is to mark part of a system state as visible or publicly accessible; further, in principle, something that has become public knowledge will stay accessible indefinitely. The main technical contribution consists in providing a proof which shows that a recently proposed categorical construction, which produces a category having monomorphisms as objects and pullback squares as morphisms, preserves weak adhesivity. Finally we sketch how it is possible to verify certain secrecy properties using unfolding based verification approaches that lately have been generalized to rewriting systems in weakly adhesive categories.

Electronic Notes in Theoretical Computer Science, 2013

Verifying that a concurrent program satisfies a given property, such as deadlock-freeness, is com... more 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.

Lecture Notes in Computer Science, 2015

Corr, 2011

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show tha... more 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.

Electronic Proceedings in Theoretical Computer Science, 2010

ABSTRACT Inspired by decomposition problems in rule-based formalisms in Computational Systems Bio... more ABSTRACT Inspired by decomposition problems in rule-based formalisms in Computational Systems Biology and recent work on compositionality in graph transformation, this paper proposes to use arbitrary colimits to "deconstruct" models of reactions in which states are represented as objects of adhesive categories. The fundamental problem is the decomposition of complex reactions of large states into simpler reactions of smaller states. The paper defines the local decomposition problem for transformations. To solve this problem means to "reconstruct" a given transformation as the colimit of "smaller" ones where the shape of the colimit and the decomposition of the source object of the transformation are fixed in advance. The first result is the soundness of colimit decomposition for arbitrary double pushout transformations in any category, which roughly means that several "local" transformations can be combined into a single "global" one. Moreover, a solution for a certain class of local decomposition problems is given, which generalizes and clarifies recent work on compositionality in graph transformation.

Lecture Notes in Computer Science, 2008

ABSTRACT In the area of specification and modelling of concurrent systems, Petri nets have become... more ABSTRACT In the area of specification and modelling of concurrent systems, Petri nets have become a standard tool, and they still work behind the scenes in tools for graph transformation systems (cf. [1]). Moreover there is still potential for crossfertilization between the graph transformation and Petri net community. Even a better understanding of adhesive categories [2] and the related concepts of [3] seems possible in light of the proposed notion of weakly adhesive categories [4], which has emerged during work on the generalization of the co-reflective semantics of Petri nets to the realm of graph transformation and adhesive rewriting systems.

It is well-known that the set of subobjects of an object in an adhesive category forms a distribu... more It is well-known that the set of subobjects of an object in an adhesive category forms a distributive lattice. This is a work-in-progress paper where we review the lattice-theoretic representation theorem for finite distributive lattices and show how it applies to subobject lattices. Furthermore we show that every arrow in an adhesive category can be interpreted as a lattice homomorphism and in addition we sketch some ideas about how to identify those homomorphisms between subobject lattices which arise from arrows. Adhesive categories (2) have been shown to provide a general categorical setting in which double-pushout rewriting can be defined in such a way that some fundamental results like the local Church-Rosser theorem and the concurrency theorem can be proved without the need for any additional conditions. The framework of adhesive categories encompasses graphs and several other graphical structures which play a role in the theory of concurrent and distributed systems. It is a ...

Lecture Notes in Computer Science, 2013

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.

Lecture Notes in Computer Science, 2014

ABSTRACT The paper gives a sufficient condition for the existence of all pushouts in an arbitrary... more ABSTRACT The paper gives a sufficient condition for the existence of all pushouts in an arbitrary category of partial maps ℂ *ℳ that is necessary whenever the category of total maps ℂ⊆ℂ *ℳ has cocones of spans; the latter is the case in all slice categories of ℂ and thus the condition is necessary locally. The main theorem is that, given an admissible class of monos ℳ in a category ℂ that has cocones of spans, the category of partial maps ℂ *ℳ has pushouts if and only if the category of total maps ℂ has hereditary pushouts and right adjoints to inverse image functors (where both properties are with respect to ℳ). This result clarifies previous work by Kennaway on graph rewriting in categories of partial maps that implicitly assumed existence of cocones of spans in the category of total maps.

Electronic Proceedings in Theoretical Computer Science, 2011

Process calculi and graph transformation systems provide models of reactive systems with labelled... more Process calculi and graph transformation systems provide models of reactive systems with labelled transition semantics (LTS). While the semantics for process calculi is compositional, this is not the case for graph transformation systems, in general. Hence, the goal of this article is to obtain a compositional semantics for graph transformation system in analogy to the structural operational semantics (SOS) for Milner's Calculus of Communicating Systems (CCS).

Lecture Notes in Computer Science, 2010

ABSTRACT The introduction of adhesive categories revived interest in the study of properties of p... more ABSTRACT 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.

Lecture Notes in Computer Science, 2014

Mathematical Structures in Computer Science, 2014

ABSTRACT We generalise both the notion of a non-sequential process and the unfolding construction... more ABSTRACT We generalise both the notion of a non-sequential process and the unfolding construction (which was previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, that is, the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars.As the unfolding represents potentially infinite computations, we need to work in adhesive categories with ‘well-behaved’ colimits of ω-chains of monos. Compared with previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism.

Logical Methods in Computer Science, 2011

Colimits that satisfy the Van Kampen condition have interesting exactness properties. We show tha... more 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.

Electronic Notes in Theoretical Computer Science, 2009

Inspired by the scope extrusion phenomenon of name passing calculi that allow to reason about kno... more Inspired by the scope extrusion phenomenon of name passing calculi that allow to reason about knowledge of (secret) names, we propose an abstract formulation of the concept of secret in any weakly adhesive category. The guiding idea is to mark part of a system state as visible or publicly accessible; further, in principle, something that has become public knowledge will stay accessible indefinitely. The main technical contribution consists in providing a proof which shows that a recently proposed categorical construction, which produces a category having monomorphisms as objects and pullback squares as morphisms, preserves weak adhesivity. Finally we sketch how it is possible to verify certain secrecy properties using unfolding based verification approaches that lately have been generalized to rewriting systems in weakly adhesive categories.

Electronic Notes in Theoretical Computer Science, 2013

Verifying that a concurrent program satisfies a given property, such as deadlock-freeness, is com... more 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.