E.-E. Doberkat | Dortmund University of Technology - Technische Universität Dortmund (original) (raw)

Papers by E.-E. Doberkat

Berichte des German Chapter of the ACM, 1993

In this paper we propose a multiparadigm approach for modeling tasks of systemdesign. The idea is... more In this paper we propose a multiparadigm approach for modeling tasks of systemdesign. The idea is to use executable task specifications which support analysis and understanding of complex design processes. Such specifications can act as a means for both rough and detailed planning of design tasks, providing system designers with proposals for instrumenting their activities and automatize appropriate work steps. The multiparadigm approach is based on a combination of an object-oriented language with high-level Petri nets and rules. The object-oriented language is used for modeling the characteristics of design-artifacts (e.g. design specifications, executable models, test plans, documentation). With high-level Petri nets the overall data and control flow in the design process is specified. Rules are used for the detailed specification and prototyping of design tasks.

Information and Control, 1980

If the state transitions of a nondeterministic or stochastic automaton are rewarded, the question... more If the state transitions of a nondeterministic or stochastic automaton are rewarded, the question arises whether or not the automaton can adopt a policy which makes sure that this return is maximal or nearly maximal. This problem is of interest, e.g., if one wants to find an optimal prediction for the next state of a stochastic automaton, or if optimal learning strategies are looked for, when optimality is measured in terms of a given goal of learning. It is shown in this paper that under some mildly restricted conditions such optimal or nearly optimal state transition policies exist. This is done for stochastic automata. By means of a representation of nondeterministic by stochastic automata-a result which seems to be of interest by itself-this carries over to the nondeterministic case. The methods and main auxiliary results come from the theory of set valued maps.

Monographs in Theoretical Computer Science, 2009

Morphisms relate stochastic relations in a way that preserves the probabilistic structure. We did... more Morphisms relate stochastic relations in a way that preserves the probabilistic structure. We did define a morphism f : K → L between stochastic relations K = ((X,A), (Y, B),K) and L = ((A,D), (B, e), L) as a pair (f, g) of surjective measurable maps so that 𝔖(g) ◦ K = L ◦ f. This means that for each measurable subset F of B, and for each x ∈ X the equality K(x)(g −1 [F])= L(f(x))(F) holds. Stochastic relations form a category with these morphisms, and the kernels of morphisms are exactly the congruences; see Section 1.7.3.

Annals of Pure and Applied Logic, 2012

We have a look at the set of congruences for a stochastic relation; conditions under which the in... more We have a look at the set of congruences for a stochastic relation; conditions under which the infimum or the supremum of two congruences is a congruence again are investigated. Congruences are based on smooth equivalence relations, and consequences of the observation that the supremum of two smooth relations may fail to be smooth are discussed: analytic spaces are not closed under pushouts, and the set of countably generated σ-algebras is not closed under finite intersections.

Article history: Received 23 November 2017 Received in revised form 13 March 2019 Accepted 30 May... more Article history: Received 23 November 2017 Received in revised form 13 March 2019 Accepted 30 May 2019 Available online 5 June 2019

The Journal of Logic and Algebraic Programming, 2005

Transition probabilities are proposed as the stochastic counterparts to set-based relations. We p... more Transition probabilities are proposed as the stochastic counterparts to set-based relations. We propose the construction of the converse of a stochastic relation. It is shown that two of the most useful properties carry over: the converse is idempotent as well as anticommutative. The nondeterminism associated with a stochastic relation is defined and briefly investigated. We define a bisimulation relation, and indicate conditions under which this relation is transitive; moreover it is shown that bisimulation and converse are compatible.

Monographs in Theoretical Computer Science, 2009

Lecture Notes in Computer Science, 2009

We show that logical and behavioral equivalence for stochastic Kripke models over general measura... more We show that logical and behavioral equivalence for stochastic Kripke models over general measurable spaces are the same. Usually, this requires some topological assumptions and includes bisimilarity; the results here indicate that a measurable structure on the state space of the Kripke model suffices. In contrast to a paper by Danos et al. we focus on the measurable structure of the factor space induced by the logic. This technique worked well in the analytic case, and it is shown to work here as well. The main contribution of the paper is methodological, since it provides a uniform framework for general measurable as well as more specialized analytic spaces.

Mathematics eJournal, 2003

We discuss congruences for stochastic relations, stressing the equivalence of smooth equivalence ... more We discuss congruences for stochastic relations, stressing the equivalence of smooth equivalence relations and countably generated σ-algebras. Factor spaces are constructed for congruences and for morphisms. Semi-pullbacks are needed when investigating the interplay between congruences and bisimulations, and it is shown that semi-pullbacks exist for stochastic relations over analytic spaces, generalizing a previous result and answering an open question. Equivalent congruences are investigated, and it is shown that stochastic relations that have equivalent congruences are bisimilar. The well-known equivalence relation coming from a Hennessy-Milner logic for labelled Markov transition systems is shown to be a special case in this development.

Lecture Notes in Computer Science, 2004

We investigate similarities between non-deterministic and probabilistic ways of describing a syst... more We investigate similarities between non-deterministic and probabilistic ways of describing a system in terms of computation trees. We first show that the construction of traces for both kinds of relations follow the same principles of construction (which could be described in terms of monads, but this does not happen here). Finally representations of measurable trees in terms of probabilistic relations are given.

The interpretation of propositional dynamic logic (PDL) through Kripke models requires the relati... more The interpretation of propositional dynamic logic (PDL) through Kripke models requires the relations constituting the interpreting Kripke model to closely observe the syntax of the modal operators. This poses a significant challenge for an interpretation of PDL through stochastic Kripke models, because the programs' operations do not always have a natural counterpart in the set of stochastic relations. We use rewrite rules for building up an interpretation of PDL. It is shown that each program corresponds to an essentially unique irreducible tree, which in turn is assigned a predicate lifting, serving as the program's interpretation. The paper establishes and studies this interpretation. It discusses the expressivity of probabilistic models for PDL and relates properties like logical and behavioral equivalence or bisimilarity to the corresponding properties of a Kripke model for a closely related non-dynamic logic of the Hennessy-Milner type.

Lecture Notes in Computer Science, 2002

The demonic product of two probabilistic relations is defined and investigated. It is shown that ... more The demonic product of two probabilistic relations is defined and investigated. It is shown that the product is stable under bisimulations when the mediating object is probabilistic, and that under some mild conditions the non-deterministic fringe of the probabilistic relations behaves properly: the fringe of the product equals the demonic product of the fringes.

Journal of Logical and Algebraic Methods in Programming, 2017

This is a short introduction to categories with some emphasis on coalgebras. We start from introd... more This is a short introduction to categories with some emphasis on coalgebras. We start from introducing basic notions (categories, functors, natural transformations), move to Kleisli tripels and monads, with a short discussion of monads in Haskell, and continue with displaying the interplay between algebras, adjunctions and monads. Coalgebras are discussed and applied to the semantics of modal logics, giving a brief introduction to coalgebraic logics as well. The development is illustrated through examples, usually taken from applications to computer science, with a certain predilection for stochastic systems.

We propose a coalgebraic interpretation of game logic, making the results of coalgebraic logic av... more We propose a coalgebraic interpretation of game logic, making the results of coalgebraic logic available for this context. We study some properties of a coalgebraic interpretation, showing among others that Aczel's Theorem on the characterization of bisimilar models through spans of morphisms is valid here. We investigate also congruences as those equivalences on the state space which preserve the structure of the model.

Electronic Notes in Theoretical Computer Science, 2008

Markov transition systems for interpreting a simple negation free Hennessy-Milner logic are calle... more Markov transition systems for interpreting a simple negation free Hennessy-Milner logic are called distributionally equivalent iff for each formula the probability for its extension in one model is matched probabilistically in the other one. This extends in a natural way the notion of logical equivalence which is defined on the states of a transition system to its subprobability distributions. It is known that logical equivalence is equivalent to bisimilarity, i.e., the existence of a span of Borel maps that act as morphisms. We show that distributional equivalence is equivalent to bisimilarity as well, using a characterization of distributional equivalent transition systems through ergodic morphisms. As an aside, we relate bisimilar transition systems to those systems, for which cospans-taken in the category of measurable maps resp. in the Kleisli category associated with the Giry monad-exist.

Information and Control, 1984

The expected number of interchanges and comparisons in Floyd's well-known algorithm to construct ... more The expected number of interchanges and comparisons in Floyd's well-known algorithm to construct heaps and derive the probability generating functions for these quantities are considered. From these functions the corresponding expected values are computed.

Journal of Logical and Algebraic Methods in Programming, 2014

Effectivity functions are the basic formalism for investigating the semantics game logic. We disc... more Effectivity functions are the basic formalism for investigating the semantics game logic. We discuss algebraic properties of stochastic effectivity functions, in particular the relationship to stochastic relations, morphisms and congruences are defined, and the relationship of abstract logical equivalence and behavioral equivalence is investigated.

Berichte des German Chapter of the ACM, 1993

In this paper we propose a multiparadigm approach for modeling tasks of systemdesign. The idea is... more In this paper we propose a multiparadigm approach for modeling tasks of systemdesign. The idea is to use executable task specifications which support analysis and understanding of complex design processes. Such specifications can act as a means for both rough and detailed planning of design tasks, providing system designers with proposals for instrumenting their activities and automatize appropriate work steps. The multiparadigm approach is based on a combination of an object-oriented language with high-level Petri nets and rules. The object-oriented language is used for modeling the characteristics of design-artifacts (e.g. design specifications, executable models, test plans, documentation). With high-level Petri nets the overall data and control flow in the design process is specified. Rules are used for the detailed specification and prototyping of design tasks.

Information and Control, 1980

If the state transitions of a nondeterministic or stochastic automaton are rewarded, the question... more If the state transitions of a nondeterministic or stochastic automaton are rewarded, the question arises whether or not the automaton can adopt a policy which makes sure that this return is maximal or nearly maximal. This problem is of interest, e.g., if one wants to find an optimal prediction for the next state of a stochastic automaton, or if optimal learning strategies are looked for, when optimality is measured in terms of a given goal of learning. It is shown in this paper that under some mildly restricted conditions such optimal or nearly optimal state transition policies exist. This is done for stochastic automata. By means of a representation of nondeterministic by stochastic automata-a result which seems to be of interest by itself-this carries over to the nondeterministic case. The methods and main auxiliary results come from the theory of set valued maps.

Monographs in Theoretical Computer Science, 2009

Morphisms relate stochastic relations in a way that preserves the probabilistic structure. We did... more Morphisms relate stochastic relations in a way that preserves the probabilistic structure. We did define a morphism f : K → L between stochastic relations K = ((X,A), (Y, B),K) and L = ((A,D), (B, e), L) as a pair (f, g) of surjective measurable maps so that 𝔖(g) ◦ K = L ◦ f. This means that for each measurable subset F of B, and for each x ∈ X the equality K(x)(g −1 [F])= L(f(x))(F) holds. Stochastic relations form a category with these morphisms, and the kernels of morphisms are exactly the congruences; see Section 1.7.3.

Annals of Pure and Applied Logic, 2012

We have a look at the set of congruences for a stochastic relation; conditions under which the in... more We have a look at the set of congruences for a stochastic relation; conditions under which the infimum or the supremum of two congruences is a congruence again are investigated. Congruences are based on smooth equivalence relations, and consequences of the observation that the supremum of two smooth relations may fail to be smooth are discussed: analytic spaces are not closed under pushouts, and the set of countably generated σ-algebras is not closed under finite intersections.

Article history: Received 23 November 2017 Received in revised form 13 March 2019 Accepted 30 May... more Article history: Received 23 November 2017 Received in revised form 13 March 2019 Accepted 30 May 2019 Available online 5 June 2019

The Journal of Logic and Algebraic Programming, 2005

Transition probabilities are proposed as the stochastic counterparts to set-based relations. We p... more Transition probabilities are proposed as the stochastic counterparts to set-based relations. We propose the construction of the converse of a stochastic relation. It is shown that two of the most useful properties carry over: the converse is idempotent as well as anticommutative. The nondeterminism associated with a stochastic relation is defined and briefly investigated. We define a bisimulation relation, and indicate conditions under which this relation is transitive; moreover it is shown that bisimulation and converse are compatible.

Monographs in Theoretical Computer Science, 2009

Lecture Notes in Computer Science, 2009

We show that logical and behavioral equivalence for stochastic Kripke models over general measura... more We show that logical and behavioral equivalence for stochastic Kripke models over general measurable spaces are the same. Usually, this requires some topological assumptions and includes bisimilarity; the results here indicate that a measurable structure on the state space of the Kripke model suffices. In contrast to a paper by Danos et al. we focus on the measurable structure of the factor space induced by the logic. This technique worked well in the analytic case, and it is shown to work here as well. The main contribution of the paper is methodological, since it provides a uniform framework for general measurable as well as more specialized analytic spaces.

Mathematics eJournal, 2003

We discuss congruences for stochastic relations, stressing the equivalence of smooth equivalence ... more We discuss congruences for stochastic relations, stressing the equivalence of smooth equivalence relations and countably generated σ-algebras. Factor spaces are constructed for congruences and for morphisms. Semi-pullbacks are needed when investigating the interplay between congruences and bisimulations, and it is shown that semi-pullbacks exist for stochastic relations over analytic spaces, generalizing a previous result and answering an open question. Equivalent congruences are investigated, and it is shown that stochastic relations that have equivalent congruences are bisimilar. The well-known equivalence relation coming from a Hennessy-Milner logic for labelled Markov transition systems is shown to be a special case in this development.

Lecture Notes in Computer Science, 2004

We investigate similarities between non-deterministic and probabilistic ways of describing a syst... more We investigate similarities between non-deterministic and probabilistic ways of describing a system in terms of computation trees. We first show that the construction of traces for both kinds of relations follow the same principles of construction (which could be described in terms of monads, but this does not happen here). Finally representations of measurable trees in terms of probabilistic relations are given.

The interpretation of propositional dynamic logic (PDL) through Kripke models requires the relati... more The interpretation of propositional dynamic logic (PDL) through Kripke models requires the relations constituting the interpreting Kripke model to closely observe the syntax of the modal operators. This poses a significant challenge for an interpretation of PDL through stochastic Kripke models, because the programs' operations do not always have a natural counterpart in the set of stochastic relations. We use rewrite rules for building up an interpretation of PDL. It is shown that each program corresponds to an essentially unique irreducible tree, which in turn is assigned a predicate lifting, serving as the program's interpretation. The paper establishes and studies this interpretation. It discusses the expressivity of probabilistic models for PDL and relates properties like logical and behavioral equivalence or bisimilarity to the corresponding properties of a Kripke model for a closely related non-dynamic logic of the Hennessy-Milner type.

Lecture Notes in Computer Science, 2002

The demonic product of two probabilistic relations is defined and investigated. It is shown that ... more The demonic product of two probabilistic relations is defined and investigated. It is shown that the product is stable under bisimulations when the mediating object is probabilistic, and that under some mild conditions the non-deterministic fringe of the probabilistic relations behaves properly: the fringe of the product equals the demonic product of the fringes.

Journal of Logical and Algebraic Methods in Programming, 2017

This is a short introduction to categories with some emphasis on coalgebras. We start from introd... more This is a short introduction to categories with some emphasis on coalgebras. We start from introducing basic notions (categories, functors, natural transformations), move to Kleisli tripels and monads, with a short discussion of monads in Haskell, and continue with displaying the interplay between algebras, adjunctions and monads. Coalgebras are discussed and applied to the semantics of modal logics, giving a brief introduction to coalgebraic logics as well. The development is illustrated through examples, usually taken from applications to computer science, with a certain predilection for stochastic systems.

We propose a coalgebraic interpretation of game logic, making the results of coalgebraic logic av... more We propose a coalgebraic interpretation of game logic, making the results of coalgebraic logic available for this context. We study some properties of a coalgebraic interpretation, showing among others that Aczel's Theorem on the characterization of bisimilar models through spans of morphisms is valid here. We investigate also congruences as those equivalences on the state space which preserve the structure of the model.

Electronic Notes in Theoretical Computer Science, 2008

Markov transition systems for interpreting a simple negation free Hennessy-Milner logic are calle... more Markov transition systems for interpreting a simple negation free Hennessy-Milner logic are called distributionally equivalent iff for each formula the probability for its extension in one model is matched probabilistically in the other one. This extends in a natural way the notion of logical equivalence which is defined on the states of a transition system to its subprobability distributions. It is known that logical equivalence is equivalent to bisimilarity, i.e., the existence of a span of Borel maps that act as morphisms. We show that distributional equivalence is equivalent to bisimilarity as well, using a characterization of distributional equivalent transition systems through ergodic morphisms. As an aside, we relate bisimilar transition systems to those systems, for which cospans-taken in the category of measurable maps resp. in the Kleisli category associated with the Giry monad-exist.

Information and Control, 1984

The expected number of interchanges and comparisons in Floyd's well-known algorithm to construct ... more The expected number of interchanges and comparisons in Floyd's well-known algorithm to construct heaps and derive the probability generating functions for these quantities are considered. From these functions the corresponding expected values are computed.

Journal of Logical and Algebraic Methods in Programming, 2014

Effectivity functions are the basic formalism for investigating the semantics game logic. We disc... more Effectivity functions are the basic formalism for investigating the semantics game logic. We discuss algebraic properties of stochastic effectivity functions, in particular the relationship to stochastic relations, morphisms and congruences are defined, and the relationship of abstract logical equivalence and behavioral equivalence is investigated.