Showing papers on "Computability published in 2000"
TL;DR: It is proved that the P systems with the possibility of objects to cooperate characterize the recursively enumerable sets of natural numbers; moreover, systems with only two membranes suffice.
2,327 citations
Book•
17 Jul 2000
TL;DR: This book discusses models for Context-Free Languages, properties of Regular Languages, and generalized models for Regular Languages.
Abstract: I Introduction.- Mathematical Background.- 1 Languages.- 2 Automata.- II Regular Languages.- 3 Models for Regular Languages.- 4 Properties of Regular Languages.- III Context-Free Languages.- 5 Models for Context-Free Languages.- 6 Properties of Context-Free Languages.- 7 Special Types of Context-Free Languages and Their Models.- IV Beyond Context-Free Languages.- 8 Generalized Models.- V Translations.- 9 Finite and Pushdown Transducers.- 10 Turing Transducers.- Indices.- Index to Special Symbols.- Index to Decision Problems.- Index to Algorithms.
116 citations
01 Jan 2000
TL;DR: In this article, the authors explore aspects of computable analysis and topology in the framework of relative realizability, and demonstrate how to develop computable topology and analysis in the logic of modest sets.
Abstract: In this dissertation, I explore aspects of computable analysis and topology in the framework of relative realizability. The computational models are partial combinatory algebras with subalgebras of computable elements, out of which categories of modest sets are constructed. The internal logic of these categories is suitable for developing a theory of computable analysis and topology, because it is equipped with a computability predicate and it supports many constructions needed in topology and analysis. In addition, a number of previously studied approaches to computable topology and analysis are special cases of the general theory of modest sets.
In the first part of the dissertation, I present categories of modest sets and axiomatize their internal logic, including the computability predicate. The logic is a predicative intuitionistic first-order logic with dependent types, subsets, quotients, inductive and coinductive types.
The second part of the dissertation investigates examples of categories of modest sets. I focus on equilogical spaces, and their relationship with domain theory and Type Two Effectivity (TTE). I show that domains with totality embed in equilogical spaces, and that the embedding preserves both simple and dependent types. I relate equilogical spaces and TTE in three ways: there is an applicative retraction between them, they share a common cartesian closed subcategory that contains all countably based T0-spaces, and they are related by a logical transfer principle. These connections explain why domain theory and TTE agree so well.
In the last part of the dissertation, I demonstrate how to develop computable analysis and topology in the logic of modest sets. The theorems and constructions performed in this logic apply to all categories of modest sets. Furthermore, by working in the internal logic, rather than directly with specific examples of modest sets, we argue abstractly and conceptually about computability in analysis and topology, avoiding the unpleasant details of the underlying computational models, such as Godel encodings and representations by sequences.
96 citations
TL;DR: Some of the main directions of research in concurrency theory are identified, and some of the major ideas on which all this is based are surveyed.
Abstract: Geometrical methods in concurrency theory (and in distributed systems theory) have appeared recently for modelling and analyzing the behaviour of systems and also for solving computability and complexity issues. We identify some of the main directions of research, and survey some of the major ideas on which all this is based (some of which are now more than thirty years old).
81 citations
01 Feb 2000
TL;DR: A global and comprehensive view of the properties of subclasses of two counters automata for which counters are only accessed through the following operations: increment (+1), decrement (-1), reset (c := 0), transfer (the whole content of counter c is transfered into counter c′), and testing for zero is presented.
Abstract: We present a global and comprehensive view of the properties of subclasses of two counters automata for which counters are only accessed through the following operations: increment (+1), decrement (-1), reset (c := 0), transfer (the whole content of counter c is transfered into counter c′), and testing for zero. We first extend Hopcroft-Pansiot's result (an algorithm for computing a finite description of the semilinear set post*) to two counters automata with only one test for zero (and one reset and one transfer operations). Then, we prove the semilinearity and the computability of pre* for the subclass of 2 counters automata with one test for zero on c1, two reset operations and one transfer from c1 to c2. By proving simulations between subclasses, we show that this subclass is the maximal class for which pre* is semilinear and effectively computable. All the (effective) semilinearity results are obtained with the help of a new symbolic reachability tree algorithm for counter automata using an Acceleration function. When Acceleration has the so-called stability property, the constructed tree computes exactly the reachability set.
46 citations
TL;DR: The authors detail the results of their prototype that results an order-of-magnitude speed-up in the execution of difficult satisfiability problems and suggest an ideal candidate for improvements based on instance-specific reconfiguration.
Abstract: Reconfigurable accelerators can improve process time on combinatorial problems with fine-grained parallelism. Such problems contain a huge number of logical operations (NOT, AND and OR) that can evaluate simultaneously, a characteristic that varies considerably from problem to problem. Because of this variability, such combinatorial problems are approached using instance-specific reconfiguration-hardware tailored to a specific algorithm and a specific set of input data. Boolean satisfiability (SAT for short) is a common combinatorial problem that exhibits fine-grained parallelism. SAT varies considerably based on the situation. Its solution is thus an ideal candidate for improvements based on instance-specific reconfiguration. In fact, simulation of an instance-specific accelerator show potential speed-ups by a factor of up to 140,000 in execution time over the solution by a software solver. The authors detail the results of their prototype that results an order-of-magnitude speed-up in the execution of difficult satisfiability problems.
41 citations
TL;DR: In this paper, the authors introduce natural graded structures of finitely generated extension rings and present subclasses of such structures which allow uniform algorithmic solutions of the basic problems in the associated graded ring and, hence, of the computation of Grobner bases with respect to the graded structure.
40 citations
TL;DR: It is shown that to every recursive total continuous functional Φ there is a PCF-definable representative Ψ of Φ in the hierarchy of partial continuous functionals, where PCF is Plotkin's programming language for computable functionals.
Abstract: We show that to every recursive total continuous functional Φ there is a representative Ψ of Φ in the hierearchy of partial continuous functionals such that Ψ is S1 − S9 computable over the hierarchy of partial continuous functionals. Equivalently, the representative Ψ will be PCF -definable over the partial continuous functionals, where PCF is Plotkin’s programming language for computable functionals.
37 citations
Journal Article•
TL;DR: In this paper, the authors present a child's guide to computing in the context of games, and discuss the modern theory of induction, adaptive behaviour, and effective playability in arithmetical games.
Abstract: 1. Introduction and Outline 2. Computable Economics: Pioneers and Precursors 3. Computable Rationality 4. Adaptive Behaviour 5. The Modern Theory of Induction 6. Learning in a Computable Setting 7. Effective Playability in Arithmetical Games 8. Notes on Computational Complexity 9. Explorations in Computable Economics 10. Concluding Reflections Appendix: A Child's Guide to Computability
37 citations
TL;DR: A notion of counting problems over the real numbers is introduced and a complete characterization of such problems by logical means is given and the main emphasis of the results is model-theoretic.
37 citations
01 Jan 2000
TL;DR: This paper presents a method to simultaneously solve several closely related SAT instances using incremental satisfiability (ISAT), and shows that this methodology can achieve significant gains in total runtime when used as opposed to resetting the decision sequences and solving each instance from scratch.
Abstract: The Boolean satisfiability problem (SAT) has various applications in electronic design automation (EDA) fields such as testing, timing analysis and logic verification. SAT has been typically applied to EDA as follows: (1) formulation of the given problem as a SAT instance (2) solution of the SAT instance. In this paper we present a method to simultaneously solve several closely related SAT instances using incremental satisfiability (ISAT). In ISAT, the decision sequence made for a "prefix" function is used to solve another set of functions which have a number of new constraints (extensions) added to the prefix function. Our experiments show that we can achieve significant gains in total runtime when we use this methodology as opposed to resetting the decision sequences and solving each instance from scratch. Application of ISAT to delay fault testing is presented by formulating incremental path sensitization as an ISAT problem. Non-robust tests for the combinational portion of ISCAS 89 circuits are generated using this method.
08 Dec 2000
TL;DR: The notion of Limit-Computable Mathematics (LCM) will be introduced, a fragment of classical mathematics in which the law of excluded middle is restricted to ?
Abstract: The notion of Limit-Computable Mathematics (LCM) will be introduced. LCM is a fragment of classical mathematics in which the law of excluded middle is restricted to ?20-formulas. We can give an accountable computational interpretation to the proofs of LCM. The computational content of LCM-proofs is given by Gold's limiting recursive functions, which is the fundamentalno tion of learning theory. LCM is expected to be a right means for "Proof Animation", which was introduced by the first author [10]. LCM is related not only to learning theory and recursion theory, but also to many areas in mathematics and computer science such as computational algebra, computability theories in analysis, reverse mathematics, and many others.
01 Jan 2000
TL;DR: In this paper, the authors discuss two notions with regard to Penrose's criteria: one originated from computable analysis, and the other introduced by Blum, Shub and Smale.
Abstract: In his book \The Emperor’s New Mind" Roger Penrose implicitly denes some criteria which should be met by a reasonable notion of recursiveness for subsets of Euclidean space. We discuss two such notions with regard to Penrose’s criteria: one originated from computable analysis, and the one introduced by Blum, Shub and Smale.
TL;DR: In this article, the authors studied Σ-definability in hereditarily finite superstructures over algebraic systems, and proved the Σ definability criterion, which was then used as a basis for establishing the reduction theorem for regular theories and obtaining a characterization of simple theories.
Abstract: Σ-definability in hereditarily finite superstructures over algebraic systems is studied. We prove the Σ-definability criterion, which is then used as a basis for establishing the reduction theorem for regular theories and for obtaining a characterization of simple theories. The idea of a nonstandard recursion theory is developed using subfields of the field of reals as an example. A partial algebraic description is given for a distributive upper semilattice of mΣ-degrees in hereditarily finite superstructures over models of simple theories.
Book•
01 Jan 2000
TL;DR: In this paper, Cholak et al. considered the problem of counting the number of Turing degrees in a set and showed that it is NP-hard to find a solution.
Abstract: Randomness in computability theory by K. Ambos-Spies and A. Kucera Open questions about the $n$-c.e. degrees by M. Arslanov The theory of numberings: Open problems by S. Badaev and S. Goncharov $\mathrm{\Pi}^0_1$ classes -- Structure and applications by D. Cenzer and C. G. Jockusch, Jr. The global structure of computably enumerable sets by P. A. Cholak Computability theory in arithmetic: Provability, structure and techniques by C. T. Chong and Y. Yang How many Turing degrees are there? by R. Dougherty and A. S. Kechris Questions in computable algebra and combinatorics by R. Downey and J. B. Remmel Issues and problems in reverse mathematics by H. Friedman and S. G. Simpson Open problems in the theory of constructive algebraic systems by S. Goncharov and B. Khoussainov Independence results from ZFC in computability theory: Some open problems by M. Groszek Problems related to arithmetic by J. F. Knight Embeddings into the computably enumerable degrees by M. Lerman Definability in the c.e. degrees: Questions and results by A. Nies Strong reducibilities, again by P. Odifreddi Finitely axiomatizable theories and Lindenbaum algebras of semantic classes by M. Peretyat'kin Towards an analog of Hilbert's tenth problem for a number field by A. Shlapentokh Natural definability in degree structures by R. A. Shore Recursion theory in set theory by T. A. Slaman Extensions, automorphisms, and definability by R. I. Soare Open problems in the enumeration degrees by A. Sorbi.
14 Feb 2000
TL;DR: Two versions of a set theoretic Δ-language are considered as theoretical prototypes for "nested" data base query language where data base states and queries are represented, respectively, as hereditarily-finite (HF) sets and set theory operations.
Abstract: Two versions of a set theoretic Δ-language are considered as theoretical prototypes for "nested" data base query language where data base states and queries are represented, respectively, as hereditarily-finite (HF) sets and set theoretic operations. It is shown that these versions correspond exactly to (N/D) LOGSPACE computability over HF, respectively. Such languages over sets, capturing also PTIME, were introduced in previous works, however, descriptions of LOGSPACE over HF [A. Lisitsa and V. Sazonov, TCS (175) 1 (1997) pp. 183-222] were not completely satisfactory. Here we overcome the drawbacks of the previous approaches due to some new partial result on definability of a linear ordering over finite extensional acyclic graphs and present a unified and simplified approach.
07 Jul 2000
TL;DR: The resulting formalism enables one to reason about the change of sets in the course of (discrete) linear time and obtains a 'low' upper complexity bound of the corresponding satisfiability problem: NP.
Abstract: We provide an extension of topological nexttime logic by an operator expressing an increase of sets. The resulting formalism enables one to reason about the change of sets in the course of (discrete) linear time. We establish completeness and decidability of the new system, and determine its complexity. As for the latter, we obtain a 'low' upper complexity bound of the corresponding satisfiability problem: NP; this is due to the fact that the time operators involved in our logic are comparatively weak. It is intended that the system is applicable to diverse fields of temporal reasoning.
01 May 2000
TL;DR: This paper considers the problem of sorting such a distributed multiset and investigates its relationship with the election problem, providing strong characterizations, showing lower bounds, and establishing efficient upper bounds.
Abstract: An anonymous ring network is a ring where all processors (vertices) are totally indistinguishable except for their input value. Initially, to each vertex of the ring is associated a value from a totally ordered set; thus, forming a multiset. In this paper we consider the problem of sorting such a distributed multiset and we investigate its relationship with the election problem. We focus on the computability and the complexity of these problems, as well as on their interrelationship, providing strong characterizations, showing lower bounds, and establishing efficient upper bounds.
01 Jan 2000
TL;DR: The n-th Rademacher function n is a step function on the interval [0; 1), jumping at nitely many dyadic rationals of size 1 2 n and assuming values f1; 1g alternatingly.
Abstract: We will speculate on some computational properties of the system of Rademacher functions f n g. The n-th Rademacher function n is a step function on the interval [0; 1), jumping at nitely many dyadic rationals of size 1 2 n and assuming values f1; 1g alternatingly.
17 Sep 2000
TL;DR: In this article, the trajectories of continuous parts of hybrid systems can be represented by computable functionals, and a recursion scheme is proposed for formalizing complex systems, which is a suitable tool for formalisation of complex systems such as hybrid systems.
Abstract: Based on an effective theory of continuous domains, notions of computability for operators and real-valued functionals defined on the class of continuous functions are introduced. Definability and semantic characterisation of computable functionals are given. Also we propose a recursion scheme which is a suitable tool for formalisation of complex systems, such as hybrid systems. In this framework the trajectories of continuous parts of hybrid systems can be represented by computable functionals.
01 Nov 2000
TL;DR: An efficient tableau-based satisfiability checking procedure for a specification which consists of several modules which extracts reduced constraints from each module and verifies a property with them is introduced.
Abstract: We introduce an efficient tableau-based satisfiability checking procedure for a specification which consists of several modules. This method extracts reduced constraints from each module and verifies a property with them. We also show that this method is applicable to the decision procedure for strong satisfiability and stepwise satisfiability. Finally, we show the experimental results of the method.
01 Jan 2000
TL;DR: The proposed method not only allows the construction of any probabilistic automaton, but also increases the class of functions that can be computed by such networks.
Abstract: A new recognition algorithm to be used with a class of RAM-based neural networks or weightless neural networks, called general single-layer sequential weightless neural networks (GSSWNNs), is introduced. These networks are assumed to be implemented either with pRAM nodes or multi-valued probabilistic logic nodes. The new algorithm makes such networks behave as probabilistic automata. The computability of GSSWNNs is shown to be equivalent to that of probabilistic automata. Indeed, one of the proofs provides an algorithm to map any probabilistic automaton into a GSSWNN. In others words, the proposed method not only allows the construction of any probabilistic automaton, but also increases the class of functions that can be computed by such networks. For instance, these networks are not restricted to finite-state languages and can now deal with some context-free languages.
Book•
12 Jul 2000
17 Sep 2000
01 Jan 2000
TL;DR: This dissertation aims to provide a history of web exceptionalism from 1989 to 2002, a period chosen in order to explore its roots as well as specific cases up to and including the year in which descriptions of “Web 2.0” began to circulate.
Abstract: Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.
14 Feb 2000
TL;DR: It is proved that the computability power of RRMk machines is much bigger when working with classes of databases which are homogeneous, for three of the properties which are defined, and the strongest notion, pairwise k-homogeneity, allows RRMK machines to achieve completeness.
Abstract: We define four different properties of relational databases which are related to the notion of homogeneity in classical Model Theory. The main question for their definition is, for any given database, which is the minimum integer k, such that whenever two k-tuples satisfy the same properties which are expressible in First Order Logic with up to k variables (FOk), then there is an automorphism which maps each of these k-tuples onto each other? We study these four properties as a means to increase the computational power of sub-classes of Reflective Relational Machines (RRM) of bounded variable complexity. For this sake we give first a semantic characterization of the sub-classes of total RRM with variable complexity k, for every natural k, with the classes of queries which we denote as QCQk. We prove that these classes form a strict hierarchy in a strict sub-class of total(CQ). And it follows that it is orthogonal with the usual classification of computable queries in Time and Space complexity classes. We prove that the computability power of RRMk machines is much bigger when working with classes of databases which are homogeneous, for three of the properties which we define. As to the fourth one, we prove that the computability power of RRM with sub-linear variable complexity also increases when working on databases which satisfy that property. The strongest notion, pairwise k-homogeneity, allows RRMk machines to achieve completeness.
01 Jan 2000
TL;DR: Two new computing models based on information coding and chaotic dynamical systems are presented, which address the systematic solution of classical NP-complete problems such as the three colorability and the directed Hamiltonian path in addition to a new chaotic simulated annealing scheme.
Abstract: Two new computing models based on information coding and chaotic dynamical systems are presented. The novelty of these models lies on the blending of chaos theory and information coding to solve complex combinatorial problems. A unique feature of our computing models is that despite the nonpredictability property of chaos, it is possible to solve any combinatorial problem in a systematic way, and with only one dynamical system. This is in sharp contrast to methods based on heuristics employing an array of chaotic cells. To prove the computing power and versatility of our models, we address the systematic solution of classical NP-complete problems such as the three colorability and the directed Hamiltonian path in addition to a new chaotic simulated annealing scheme.