Dieter Spreen - Profile on Academia.edu (original) (raw)
Papers by Dieter Spreen
WORLD SCIENTIFIC eBooks, Apr 1, 2023
Working in a semi-constructive logical system that supports the extraction of concurrent programs... more Working in a semi-constructive logical system that supports the extraction of concurrent programs, we extract a program inverting nonsingular real valued matrices from a constructive proof based on Gaussian elimination. Concurrency is used for efficient pivoting, that is, for finding an entry that is apart from zero in a non-null vector of real numbers.
Dagstuhl Reports, 2015
This report documents the programme and outcomes of Dagstuhl Seminar 15441 'Duality in Computer S... more This report documents the programme and outcomes of Dagstuhl Seminar 15441 'Duality in Computer Science'. This seminar served as a follow-up seminar to the seminar 'Duality in Computer Science' (Dagstuhl Seminar 13311). In this seminar, we focused on applications of duality to semantics for probability in computation, to algebra and coalgebra, and on applications in complexity theory. A key objective of this seminar was to bring together researchers from these communities within computer science as well as from mathematics with the goal of uncovering commonalities, forging new collaborations, and sharing tools and techniques between areas based on their common use of topological methods and duality.
WORLD SCIENTIFIC eBooks, Apr 1, 2023
Information systems with witnesses have been introduced in [13] as a logic-style representation o... more Information systems with witnesses have been introduced in [13] as a logic-style representation of L-domains: The category of such information systems with approximable mappings as morphisms is equivalent to the category of L-domains with Scott continuous functions, which is known to be Cartesian closed. In the present paper a direct proof of the Cartesian closure of the category of information systems with witnesses and approximable mapppings is given. As is shown, the collection of approximable mappings between two information systems with witnesses comes with a natural information system structure.
arXiv (Cornell University), May 11, 2023
Partiality is a natural phenomenon in computability that we cannot get around. So, the question i... more Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where nontermination happens, more structure. In this paper we consider function classes which besides the total functions only contain finite functions whose domain of definition is an initial segment of the natural numbers. Such functions appear naturally in computation. We show that a rich computability theory can be developed for these functions classes which embraces the central results of classical computability theory, in which all partial (computable) functions are considered. To do so, the concept of a Gödel number is generalised, resulting in a broader class of numberings. The central algorithmic idea in this approach is to search in enumerated lists. In this way, function computability is reduced to set listability. Besides the development of a computability theory for the functions classes, the new numberings-called quasi-Gödel numberings-are studied from a numbering-theoretic perspective: they are complete, and each of the function classes numbered in this way is a retract of the Gödel numbered set of all partial computable functions. Moreover, the Rogers semi-lattice of all computable numberings of the considered function classes is studied and results as in the case of the computable numberings of the partial computable functions are obtained. The function classes are shown to be effectively given algebraic domains in the sense of Scott-Ershov. The quasi-Gödel numberings are exactly the admissible numberings of the computable elements of the domain. Moreover, the domain can be computably mapped onto every other effectively given one so that every admissible numbering of the computable domain elements is generated by a quasi-Gödel numbering via this mapping. Contents 9 p S p1q A as effectively given domain 43 10 Conclusion 51
arXiv (Cornell University), Apr 11, 2020
A uniform approach to computing with infinite objects like real numbers, tuples of these, compact... more A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets, and uniformly continuous maps is presented. In work of Berger it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch's Type-Two Theory of Effectivity. Contents 1 Introduction 2 Inductive and co-inductive definitions 3 D-Trees 4 Extended iterated function systems 5 Computable digit spaces 6 Extracting digital trees from co-inductive proofs 7 Equivalence with the Cauchy representation 8 Products 9 The hyperspace of non-empty compact subsets 10 Uniformly continuous functions 11 Compact-valued functions 12 Michael's Theorem 13 Conclusion * This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 731143.
On r.e. inseparability of CPO index sets
Springer eBooks, 1984
ABSTRACT
Dagstuhl Reports, 2013
Duality allows one to move between the two worlds: the world of certain algebras of properties an... more Duality allows one to move between the two worlds: the world of certain algebras of properties and a spacial world of individuals, thereby leading to a change of perspective that may, and often does, lead to new insights. Dualities have given rise to active research in a number of areas of theoretical computer science. Dagstuhl Seminar 13311 "Duality in Computer Science" was held to stimulate research in this area. This report collects the ideas that were presented and discussed during the course of the seminar.
A logic-oriented representation of L-domains in the style of Scott’s information sys-
Ìóôóðóóý Ò Óñôùøøö Ë Blockin
Archive for Mathematical Logic, 2004
A strong reducibility relation between partial numberings is introduced which is such that the re... more A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice. In addition, Ershov's completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic. This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices.
From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spatial Representation: Discrete vs. Contin... more From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spatial Representation: Discrete vs. Continuous Computational Models was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Infinite Gray code has been introduced by Tsuiki <cit.> as a redundancy-free representation... more Infinite Gray code has been introduced by Tsuiki <cit.> as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also <cit.>). Berger and Tsuiki <cit.> reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates and are defined and the inclusion ⊆ is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version _2 of S and ⊆_2 is s...
On the Equivalence Problem in Automata Theory: A Uniform Approach
Journal of Information Processing and Cybernetics, 1988
ABSTRACT An important problem in automata theory is the question whether the state equivalence is... more ABSTRACT An important problem in automata theory is the question whether the state equivalence is decidable. For some special types of automata a positive answer has been given to the question. All these results were proved separately. In this paper a uniform approach such that these results can be obtained as special cases is presented. Moreover, from the central result analogous results are derived for some classes of automata for which the decidability of the state equivalence has not yet been considered such as weighted automata, lattice automata, module automata, and linear stochastically independent automata. Furthermore, via this approach known bounds on the length of the input words which have to be considered in order to decide the state equivalence are improved. For Boolean automata the known bound is 2 2 n -1. The bound presented is 2 n -1. It cannot be improved.
Exact representations of real numbers such as the signed digit representation or more generally l... more Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first author it was shown how to extract certified algorithms working with the signed digit representations from constructive proofs. In this paper we lay the foundation for doing a similar thing with nonempty compact sets. It turns out that a representation by streams of finitely many digits is impossible and instead trees are needed.
The continuity problem, i.e., the question whether effective maps between effectively given topol... more The continuity problem, i.e., the question whether effective maps between effectively given topological spaces are effectively continuous, is reconsidered. In earlier work it was shown that this is always the case, if the effective map also has a witness for noninclusion. The extra condition does not have an obvious topological interpretation. As is shown in the present paper, it appears naturally where in the classical proof that sequentially continuous maps are continuous the Axiom of Choice is used. The question is therefore whether the witness condition appears in the general continuity theorem only for this reason, i.e., whether effective operators are effectively sequentially continuous. For two large classes of spaces covering all important applications it is shown that this is indeed the case. The general question, however, remains open. Spaces in this investigation are in general not required to be Hausdorff. They only need to satisfy the weaker T_0 separation condition.
Forensic computing (sometimes also called digital forensics, computer forensics or IT forensics) ... more Forensic computing (sometimes also called digital forensics, computer forensics or IT forensics) is a branch of forensic science pertaining to digital evidence, i.e., any legal evidence that is processed by digital computer systems or stored on digital storage media. Forensic computing is a new discipline evolving within the intersection of several established research areas such as computer science, computer engineering and law. Forensic computing is rapidly gaining importance since the amount of crime involving digital systems is steadily increasing. Furthermore, the area is still underdeveloped and poses many technical and legal challenges. This Dagstuhl seminar brought together researchers and practitioners from computer science and law covering the diverse areas of forensic computing. The goal of the seminar was to further establish forensic computing as a scientific research discipline, to identify the strengths and weaknesses of the research field, and to discuss the foundations of its methodology.
Paris-Diderot, FR, Jean-Eric.Pin@liafa.univ-paris-diderot.fr 3 A. P. Ershov Institute-Novosibirsk... more Paris-Diderot, FR, Jean-Eric.Pin@liafa.univ-paris-diderot.fr 3 A. P. Ershov Institute-Novosibirsk, RU, vseliv@iis.nsk.su 4 Universität Siegen, DE, spreen@math.uni-siegen.de Abstract Duality allows one to move between the two worlds: the world of certain algebras of properties and a spacial world of individuals, thereby leading to a change of perspective that may, and often does, lead to new insights. Dualities have given rise to active research in a number of areas of theoretical computer science. Dagstuhl Seminar 13311 "Duality in Computer Science" was held to stimulate research in this area. This report collects the ideas that were presented and discussed during the course of the seminar. Seminar License Creative Commons BY 3.0 Unported license © Mai Gehrke, Jean-Eric Pin, Victor Selivanov, and Dieter Spreen This seminar concentrated on applications of duality in computation, semantics, and formal languages. Duality and computation Consider the area of exact real number ...
There is a large gap between mathematical structures and the structures computer implementations ... more There is a large gap between mathematical structures and the structures computer implementations are based on. To stimulate research to overcome this-especially for infinitary structures-highly non-trivial problem the Dagstuhl Seminar 11411 "Computing with Infinite Data: Topolo-gical and Logical Foundations" was held. This report collects the ideas that were presented and discussed during the course of the seminar.
Abstract. From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spa-
WORLD SCIENTIFIC eBooks, Apr 1, 2023
Working in a semi-constructive logical system that supports the extraction of concurrent programs... more Working in a semi-constructive logical system that supports the extraction of concurrent programs, we extract a program inverting nonsingular real valued matrices from a constructive proof based on Gaussian elimination. Concurrency is used for efficient pivoting, that is, for finding an entry that is apart from zero in a non-null vector of real numbers.
Dagstuhl Reports, 2015
This report documents the programme and outcomes of Dagstuhl Seminar 15441 'Duality in Computer S... more This report documents the programme and outcomes of Dagstuhl Seminar 15441 'Duality in Computer Science'. This seminar served as a follow-up seminar to the seminar 'Duality in Computer Science' (Dagstuhl Seminar 13311). In this seminar, we focused on applications of duality to semantics for probability in computation, to algebra and coalgebra, and on applications in complexity theory. A key objective of this seminar was to bring together researchers from these communities within computer science as well as from mathematics with the goal of uncovering commonalities, forging new collaborations, and sharing tools and techniques between areas based on their common use of topological methods and duality.
WORLD SCIENTIFIC eBooks, Apr 1, 2023
Information systems with witnesses have been introduced in [13] as a logic-style representation o... more Information systems with witnesses have been introduced in [13] as a logic-style representation of L-domains: The category of such information systems with approximable mappings as morphisms is equivalent to the category of L-domains with Scott continuous functions, which is known to be Cartesian closed. In the present paper a direct proof of the Cartesian closure of the category of information systems with witnesses and approximable mapppings is given. As is shown, the collection of approximable mappings between two information systems with witnesses comes with a natural information system structure.
arXiv (Cornell University), May 11, 2023
Partiality is a natural phenomenon in computability that we cannot get around. So, the question i... more Partiality is a natural phenomenon in computability that we cannot get around. So, the question is whether we can give the areas where partiality occurs, that is, where nontermination happens, more structure. In this paper we consider function classes which besides the total functions only contain finite functions whose domain of definition is an initial segment of the natural numbers. Such functions appear naturally in computation. We show that a rich computability theory can be developed for these functions classes which embraces the central results of classical computability theory, in which all partial (computable) functions are considered. To do so, the concept of a Gödel number is generalised, resulting in a broader class of numberings. The central algorithmic idea in this approach is to search in enumerated lists. In this way, function computability is reduced to set listability. Besides the development of a computability theory for the functions classes, the new numberings-called quasi-Gödel numberings-are studied from a numbering-theoretic perspective: they are complete, and each of the function classes numbered in this way is a retract of the Gödel numbered set of all partial computable functions. Moreover, the Rogers semi-lattice of all computable numberings of the considered function classes is studied and results as in the case of the computable numberings of the partial computable functions are obtained. The function classes are shown to be effectively given algebraic domains in the sense of Scott-Ershov. The quasi-Gödel numberings are exactly the admissible numberings of the computable elements of the domain. Moreover, the domain can be computably mapped onto every other effectively given one so that every admissible numbering of the computable domain elements is generated by a quasi-Gödel numbering via this mapping. Contents 9 p S p1q A as effectively given domain 43 10 Conclusion 51
arXiv (Cornell University), Apr 11, 2020
A uniform approach to computing with infinite objects like real numbers, tuples of these, compact... more A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets, and uniformly continuous maps is presented. In work of Berger it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch's Type-Two Theory of Effectivity. Contents 1 Introduction 2 Inductive and co-inductive definitions 3 D-Trees 4 Extended iterated function systems 5 Computable digit spaces 6 Extracting digital trees from co-inductive proofs 7 Equivalence with the Cauchy representation 8 Products 9 The hyperspace of non-empty compact subsets 10 Uniformly continuous functions 11 Compact-valued functions 12 Michael's Theorem 13 Conclusion * This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 731143.
On r.e. inseparability of CPO index sets
Springer eBooks, 1984
ABSTRACT
Dagstuhl Reports, 2013
Duality allows one to move between the two worlds: the world of certain algebras of properties an... more Duality allows one to move between the two worlds: the world of certain algebras of properties and a spacial world of individuals, thereby leading to a change of perspective that may, and often does, lead to new insights. Dualities have given rise to active research in a number of areas of theoretical computer science. Dagstuhl Seminar 13311 "Duality in Computer Science" was held to stimulate research in this area. This report collects the ideas that were presented and discussed during the course of the seminar.
A logic-oriented representation of L-domains in the style of Scott’s information sys-
Ìóôóðóóý Ò Óñôùøøö Ë Blockin
Archive for Mathematical Logic, 2004
A strong reducibility relation between partial numberings is introduced which is such that the re... more A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice. In addition, Ershov's completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic. This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices.
From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spatial Representation: Discrete vs. Contin... more From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spatial Representation: Discrete vs. Continuous Computational Models was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Infinite Gray code has been introduced by Tsuiki <cit.> as a redundancy-free representation... more Infinite Gray code has been introduced by Tsuiki <cit.> as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also <cit.>). Berger and Tsuiki <cit.> reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates and are defined and the inclusion ⊆ is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version _2 of S and ⊆_2 is s...
On the Equivalence Problem in Automata Theory: A Uniform Approach
Journal of Information Processing and Cybernetics, 1988
ABSTRACT An important problem in automata theory is the question whether the state equivalence is... more ABSTRACT An important problem in automata theory is the question whether the state equivalence is decidable. For some special types of automata a positive answer has been given to the question. All these results were proved separately. In this paper a uniform approach such that these results can be obtained as special cases is presented. Moreover, from the central result analogous results are derived for some classes of automata for which the decidability of the state equivalence has not yet been considered such as weighted automata, lattice automata, module automata, and linear stochastically independent automata. Furthermore, via this approach known bounds on the length of the input words which have to be considered in order to decide the state equivalence are improved. For Boolean automata the known bound is 2 2 n -1. The bound presented is 2 n -1. It cannot be improved.
Exact representations of real numbers such as the signed digit representation or more generally l... more Exact representations of real numbers such as the signed digit representation or more generally linear fractional representations or the infinite Gray code represent real numbers as infinite streams of digits. In earlier work by the first author it was shown how to extract certified algorithms working with the signed digit representations from constructive proofs. In this paper we lay the foundation for doing a similar thing with nonempty compact sets. It turns out that a representation by streams of finitely many digits is impossible and instead trees are needed.
The continuity problem, i.e., the question whether effective maps between effectively given topol... more The continuity problem, i.e., the question whether effective maps between effectively given topological spaces are effectively continuous, is reconsidered. In earlier work it was shown that this is always the case, if the effective map also has a witness for noninclusion. The extra condition does not have an obvious topological interpretation. As is shown in the present paper, it appears naturally where in the classical proof that sequentially continuous maps are continuous the Axiom of Choice is used. The question is therefore whether the witness condition appears in the general continuity theorem only for this reason, i.e., whether effective operators are effectively sequentially continuous. For two large classes of spaces covering all important applications it is shown that this is indeed the case. The general question, however, remains open. Spaces in this investigation are in general not required to be Hausdorff. They only need to satisfy the weaker T_0 separation condition.
Forensic computing (sometimes also called digital forensics, computer forensics or IT forensics) ... more Forensic computing (sometimes also called digital forensics, computer forensics or IT forensics) is a branch of forensic science pertaining to digital evidence, i.e., any legal evidence that is processed by digital computer systems or stored on digital storage media. Forensic computing is a new discipline evolving within the intersection of several established research areas such as computer science, computer engineering and law. Forensic computing is rapidly gaining importance since the amount of crime involving digital systems is steadily increasing. Furthermore, the area is still underdeveloped and poses many technical and legal challenges. This Dagstuhl seminar brought together researchers and practitioners from computer science and law covering the diverse areas of forensic computing. The goal of the seminar was to further establish forensic computing as a scientific research discipline, to identify the strengths and weaknesses of the research field, and to discuss the foundations of its methodology.
Paris-Diderot, FR, Jean-Eric.Pin@liafa.univ-paris-diderot.fr 3 A. P. Ershov Institute-Novosibirsk... more Paris-Diderot, FR, Jean-Eric.Pin@liafa.univ-paris-diderot.fr 3 A. P. Ershov Institute-Novosibirsk, RU, vseliv@iis.nsk.su 4 Universität Siegen, DE, spreen@math.uni-siegen.de Abstract Duality allows one to move between the two worlds: the world of certain algebras of properties and a spacial world of individuals, thereby leading to a change of perspective that may, and often does, lead to new insights. Dualities have given rise to active research in a number of areas of theoretical computer science. Dagstuhl Seminar 13311 "Duality in Computer Science" was held to stimulate research in this area. This report collects the ideas that were presented and discussed during the course of the seminar. Seminar License Creative Commons BY 3.0 Unported license © Mai Gehrke, Jean-Eric Pin, Victor Selivanov, and Dieter Spreen This seminar concentrated on applications of duality in computation, semantics, and formal languages. Duality and computation Consider the area of exact real number ...
There is a large gap between mathematical structures and the structures computer implementations ... more There is a large gap between mathematical structures and the structures computer implementations are based on. To stimulate research to overcome this-especially for infinitary structures-highly non-trivial problem the Dagstuhl Seminar 11411 "Computing with Infinite Data: Topolo-gical and Logical Foundations" was held. This report collects the ideas that were presented and discussed during the course of the seminar.
Abstract. From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spa-