45 results for Khoussainov, B, Report

  • Update Games and Update Networks

    Dinneen, Michael; Khoussainov, B (1999-06)

    Report
    The University of Auckland Library

    In this paper we model infinite processes with finite configurations as infinite games over finite graphs. We investigate those games, called update games, in which each configuration occurs an infinite number of times during a two-person play. We also present an efficient polynomial-time algorithm (and partial characterization) for deciding if a graph is an update network.

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  • Algebraic Constraints, Automata, and Regular Languages (Revised)

    Khoussainov, B (2001-11)

    Report
    The University of Auckland Library

    The paper studies classes of regular languages based on algebraic constraints imposed on transitions of automata and discusses issue related to specifications of these classes from algebraic, computational and logical points of view.

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  • Abstracts of the Workshop on Automata, Structures and Logic

    Khoussainov, B (2004-12)

    Report
    The University of Auckland Library

    [no abstract available]

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  • Automatic linear orders and trees (Revised)

    Khoussainov, B; Rubin, S; Stephan, F (2003-11)

    Report
    The University of Auckland Library

    We investigate partial orders that are computable, in a precise sense, by finite automata. Our emphasis is on trees and linear orders. We study the relationship between automatic linear orders and trees in terms of rank functions that are related to Cantor-Bendixson rank. We prove that automatic linear orders and automatic trees have finite rank. As an application we provide a procedure for deciding the isomorphism problem for automatic ordinals. We also investigate the complexity and definability of infinite paths in automatic trees. In particular we show that every infinite path in an automatic tree with countably many infinite paths is a regular language.

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  • Effective Model Theory: The Number of Models and Their Complexity

    Khoussainov, B; Shore, R.A (2000-03)

    Report
    The University of Auckland Library

    Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

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  • Scott Families and Computably Categorical Structures

    Khoussainov, B; Shore, R.A (1996-09)

    Report
    The University of Auckland Library

    Effective model theory is an area of logic that analyzes the effective content of the typical notions and results of model theory and universal algebra. Typical notions in model theory and universal algebra are languages and structures, theories and models, models and their submodels, automorphisms and isomorphisms, embeddings and elementary embeddings. In this paper languages, structures, and models are assumed to be countable. There are many ways to introduce considerations of effectiveness into the area of model theory or universal algebra. Here we will briefly explain considerations of effectiveness for theories and their models on the one hand, and for just structures on the other hand.

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  • Update Networks and Their Routing Strategies

    Dinneen, Michael; Khoussainov, B (2000-03)

    Report
    The University of Auckland Library

    We introduce the notion of update networks to model communication networks with infinite duration.In our formalization we use bipartite finite graphs and game-theoretic terminology as an underlying structure.F or these networks we exhibit a simple routing procedure to update information throughout the nodes of the network. We also introduce an hierarchy for the class of all update networks and discuss the complexity of some natural problems.

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  • Games Played on Finite Graphs and Temporal Logic

    Khoussainov, B (2002-01)

    Report
    The University of Auckland Library

    Our aim is to study reductive finite state systems (e.g. communication networks, banking systems, airtraffic control systems) by means of game-theoretic methods. A reactive system acts upon the inputs from environment by changing its states. The goal of the system is to satisfy given specifications no matter how environment behaves. We model this situation using games played on finite graphs first introduced by McNaughton [6].

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  • On Game-Theoretic Models of Networks

    Bodlaender, H.L; Dinneen, Michael; Khoussainov, B (2001-04)

    Report
    The University of Auckland Library

    In this paper, we study the complexity of deciding which player has a winning strategy in certain types of McNaughton games. These graph games can be used as models for computational problems and processes of infinite duration. We consider the cases (1) where the first player wins when vertices in a specified set are visited infinitely often and vertices in another specified set are visited finitely often, (2) where the first player wins when exactly those vertices in one of a number of specified disjoint sets are visited infinitely often, and (3) a generalization of these first two cases. We give polynomial time algorithms to determine which player has a winning strategy in each of the games considered.

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  • Recursively Enumerable Reals and Chaitin Omega Numbers

    Calude, C.S; Hertling, P.H; Khoussainov, B; Wang, Y (1997-10)

    Report
    The University of Auckland Library

    A real α is called recursively enumerable if it can be approximated by an increasing, recursive sequence of rationals. The halting probability of a universal self- delimiting Turing machine (Chaitin's Ω number, [10]) is a random r.e. real. Solovay's [25] Ω-like reals are also random r.e. reals. Solovay showed that any Chaitin Ω number is Ω-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine. Following Solovay [25] and Chaitin [11] we say that an r.e. real α dominates an r.e. real β if from a good approximation of α from below one can compute a good approximation of β from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Ω-like numbers are the maximal r.e. real numbers with respect to this order, that is, from a good approximation to an Ω-like real one can compute a good approximation for every r.e. real. This property shows the strength of Ω for approximation purposes. However, the situation is radically different if one wishes to compute digits of the binary expansion of an r.e. real: one cannot compute with a total recursive function the first n digits of the r.e. real 0:¬xK (the characteristic sequence of the halting problem) from the first g(n) digits of Ω, for any total recursive function g.

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  • A Quest For Algorithmically Random Infinite Structures, II

    Khoussainov, B (2014)

    Report
    The University of Auckland Library

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  • Automata with Equational Constraints

    Dinneen, Michael; Khoussainov, B (1999-08)

    Report
    The University of Auckland Library

    We introduce the concept of nite automata with algebraic constraints. We show that the languages accepted by these automata are closed under the Boolean operations. We give efficient polynomial-time algorithms for some decision problems related to these automata and their languages, including sufficient conditions for when we can determinize automata in polynomial time.

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  • Computably Categorical Structures and Expansions by Constants

    Cholak, P; Goncharov, S.S; Khoussainov, B; Shore, R.A (1996-11)

    Report
    The University of Auckland Library

    Effective model theory is the subject that analyzes the typical notions and results of model theory to determine their effective content and counterparts The subject has been developed both in the former Soviet Union and in the west with various names (recursive model theory, constructive model theory, etc.) and divergent terminology. (We use “effective model theory” as the most general and descriptive designation. Harizanov [6] is an excellent introduction to the subject as is Millar [14]. The basic subjects of model theory include languages, structures, theories, models and various types of maps between these objects. There are many ways to introduce considerations of effectiveness into the area. The two most prominent derive from starting, on the one hand, with the notion of a theory and its models or, on the other with just structures. --from Introduction

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  • Finite Nondeterministic Automata: Simulation and Minimality

    Calude, C.S; Calude, E; Khoussainov, B (1997 -09)

    Report
    The University of Auckland Library

    Motivated by recent applications of finite automata to theoretical physics, we study the minimization problem for nondeterministic automata (with outputs, but no initial states). We use Ehrenfeucht-Fraïsse-like games to model automata responses and simulations. The minimal automaton is constructed and, in contrast with the classical case, proved to be unique up to an isomorphism. Finally, we investigate the partial ordering induced by automata simulations. For example, we prove that, with respect to this ordering, the class of deterministic automata forms an ideal in the class of all automata.

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  • Algebraic Constraints, Automata, and Regular Languages

    Khoussainov, B (2000-03)

    Report
    The University of Auckland Library

    A class of decision problems is Boolean if it is closed under the set{theoretic operations of union, intersection and complementation. The paper introduces new Boolean classes of decision problems based on algebraic constraints imposed on transitions of nite automata. We discuss issues related to speci cations of these classes from algebraic, computational and proof{theoretic points of view.

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  • On Isomorphism Invariants of Some Automatic Structures

    Ishihara, H; Khoussainov, B; Rubin, S (2002-01)

    Report
    The University of Auckland Library

    In this paper we study structures defined by finite automata, called automatic structures. We provide a method that reduces the study of automatic structures to the study of automatic graphs. We investigate isomorphism invariants of automatic structures with an emphasis to equivalence relation structures, linearly ordered sets, and permutation structures.

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  • Deciding Parity Games in Quasipolynomial Time

    Calude, CS; Jain, S; Khoussainov, B; Li, W; Stephan, F (2016)

    Report
    The University of Auckland Library

    It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game (with n nodes and m distinct values) is proven to be in the class of fixed parameter tractable (FPT) problems (when parameterised over m).

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  • Semiautomatic Structures

    Jain, S; Khoussainov, B; Stephan, F; Teng, D; Zou, S (2014)

    Report
    The University of Auckland Library

    Semiautomatic structures generalise automatic structures in the sense that for some of the relations and functions in the structure one only requires the derived relations and structures are automatic when all but one input are filled with constants. One can also permit that this applies to equality in the structure so that only the sets of representatives equal to a given element of the structure are regular while equality itself is not an automatic relation on the domain of representatives. It is shown that one can find semiautomatic representations for the field of rationals and also for finite algebraic field extensions of it. Furthermore, one can show that infinite algebraic extensions of finite fields have semiautomatic representations in which the addition and equality are both automatic. Further prominent examples of semiautomatic structures are term algebras, any relational structure over a countable domain with a countable signature and any permutation algebra with a countable domain. Furthermore, examples of structures which fail to be semiautomatic are provided.

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  • Decidable Kripke Models of Intuitionistic Theories

    Ishihara, H; Khoussainov, B; Nerode, A (1997-01)

    Report
    The University of Auckland Library

    [no abstract available]

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  • Degree-Theoretic Aspects of Computably Enumerable Reals

    Calude, C.S; Coles, R.J; Hertling, P.H; Khoussainov, B (1998-09)

    Report
    The University of Auckland Library

    A real α is computable if its left cut, L(α); is computable. If (qi)i is a computable sequence of rationals computably converging to α, then {qi}, the corresponding set, is always computable. A computably enumerable (c.e.) real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. reals, that is the degrees of increasing computable sequences converging to α. For example, every representation A of α is Turing reducible to L(α). Every noncomputable c.e. real has both a computable and noncomputable representation. In fact, the representations of noncomputable c.e. re- als are dense in the c.e. Turing degrees, and yet not every c.e. Turing degree below degT L(α) necessarily contains a representation of α.

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