Showing papers on "Computability published in 2009"

Computability and randomness

Abstract: The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts and methods originating from randomness enrich computability theory. Covering the basics as well as recent research results, this book provides a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory.

Computability In Context: Computation and Logic in the Real World

Abstract: Computability has played a crucial role in mathematics and computer science, leading to the discovery, understanding and classification of decidable/undecidable problems, paving the way for the modern computer era, and affecting deeply our view of the world. Recent new paradigms of computation, based on biological and physical models, address in a radically new way questions of efficiency and challenge assumptions about the so-called Turing barrier. This volume addresses various aspects of the ways computability and theoretical computer science enable scientists and philosophers to deal with mathematical and real-world issues, covering problems related to logic, mathematics, physical processes, real computation and learning theory. At the same time it will focus on different ways in which computability emerges from the real world, and how this affects our way of thinking about everyday computational issues. The list of contributors includes: S Abramsky, P Adriaans, M Agrawal, M Arslanov, G Ausiello, J Diaz, Y Ershov, G Longo, W Maass, I Nemeti, A Nerode, D Normann, G Odifreddi, M Rathjen, G Rozenberg, M Vardi, and P Welch.

Elementary Computable Topology

Abstract: We revise and extend the foundation of computable topology in the frame- work of Type-2 theory of effectivity, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations We start from a computable topo- logical space, which is a T0-space with a notation of a base such that intersection is computable, and define a number of multi-representations of the points and of the open, the closed and the compact sets and study their properties and relations We study computability of boolean operations By merely requiring "provability" of suitable re- lations (element, non-empty intersection, subset) we characterize in turn computability on the points, the open sets (!), computability on the open sets, computability on the closed sets, the compact sets(!), and computability on the compact sets We study modifications of the definition of a computable topological space that do not change the derived computability concepts We study subspaces and products and compare a number of representations of the space of partial continuous functions Since we are operating mainly with the base elements, which can be considered as regions for points ("pointless topology"), we study to which extent these regions can be filled with points (completions) We conclude with some simple applications including Dini's Theorem as an example

Exploration of Periodically Varying Graphs

Abstract: We study the computability and complexity of the exploration problem in a class of highly dynamic graphs: periodically varying (PV) graphs, where the edges exist only at some (unknown) times defined by the periodic movements of carriers. These graphs naturally model highly dynamic infrastructure-less networks such as public transports with fixed timetables, low earth orbiting (LEO) satellite systems, security guards' tours, etc. We establish necessary conditions for the problem to be solved. We also derive lower bounds on the amount of time required in general, as well as for the PV graphs defined by restricted classes of carriers movements: simple routes, and circular routes. We then prove that the limitations on computability and complexity we have established are indeed tight. We do so constructively presenting two worst case optimal solution algorithms, one for anonymous systems, and one for those with distinct nodes ids.

Exploration of Periodically Varying Graphs

Abstract: We study the computability and complexity of the exploration problem in a class of highly dynamic graphs: periodically varying (PV) graphs, where the edges exist only at some (unknown) times defined by the periodic movements of carriers. These graphs naturally model highly dynamic infrastructure-less networks such as public transports with fixed timetables, low earth orbiting (LEO) satellite systems, security guards' tours, etc. We establish necessary conditions for the problem to be solved. We also derive lower bounds on the amount of time required in general, as well as for the PV graphs defined by restricted classes of carriers movements: simple routes, and circular routes. We then prove that the limitations on computability and complexity we have established are indeed tight. In fact we prove that all necessary conditions are also sufficient and all lower bounds on costs are tight. We do so constructively presenting two worst case optimal solution algorithms, one for anonymous systems, and one for those with distinct nodes ids. An added benefit is that the algorithms are rather simple.

Secure Computability of Functions in the IT Setting with Dishonest Majority and Applications to Long-Term Security

Abstract: While general secure function evaluation (SFE) with information-theoretical (IT) security is infeasible in presence of a corrupted majority in the standard model, there are SFE protocols (Goldreich et al. [STOC'87]) that are computationally secure (without fairness) in presence of an actively corrupted majority of the participants. Now, computational assumptions can usually be well justified at the time of protocol execution. The concern is rather a potential violation of the privacy of sensitive data by an attacker whose power increases over time. Therefore, we ask which functions can be computed with long-term security, where we admit computational assumptions for the duration of a computation, but require IT security (privacy) once the computation is concluded. Towards a combinatorial characterization of this class of functions, we also characterize the classes of functions that can be computed IT securely in the authenticated channels model in presence of passive, semi-honest, active, and quantum adversaries.

An Application of Martin-Löf Randomness to Effective Probability Theory

Abstract: In this paper we provide a framework for computable analysis of measure, probability and integration theories. We work on computable metric spaces with computable Borel probability measures. We introduce and study the framework of layerwise computability which lies on Martin-Lof randomness and the existence of a universal randomness test. We then prove characterizations of effective notions of measurability and integrability in terms of layerwise computability. On the one hand it gives a simple way of handling effective measure theory, on the other hand it provides powerful tools to study Martin-Lof randomness, as illustrated in a sequel paper.

Ordinal Computability

Abstract: Ordinal computability uses ordinals instead of natural numbers in abstract machines like register or Turing machines. We give an overview of the computational strengths of *** -β -machines, where *** and β bound the time axis and the space axis of some machine model. The spectrum ranges from classical Turing computability to ***-***-computability which corresponds to Godel 's model of constructible sets. To illustrate some typical techniques we prove a new result on Infinite Time Register Machines (= ***-*** -register machines) which were introduced in [6]: a real number x *** *** 2 is computable by an Infinite Time Register Machine iff it is Turing computable from some finitely iterated hyperjump 0(n ).

Dynamics and abstract computability: computing invariant measures

Abstract: We consider the question of computing invariant measures from an abstract point of view. We work in a general framework (computable metric spaces, computable measures and functions) where this problem can be posed precisely. We consider invariant measures as fixed points of the transfer operator and give general conditions under which the transfer operator is (sufficiently) computable. In this case, a general result ensures the computability of isolated fixed points and hence invariant measures (in given classes of "regular" measures). This implies the computability of many SRB measures. On the other hand, not all computable dynamical systems have a computable invariant measure. We exhibit two interesting examples of computable dynamics, one having an SRB measure which is not computable and another having no computable invariant measure at all, showing some subtlety in this kind of problems.

On first-order definability and computability of progression for local-effect actions and beyond

Abstract: In a seminal paper, Lin and Reiter introduced the notion of progression for basic action theories in the situation calculus. Unfortunately, progression is not first-order definable in general. Recently, Vassos, Lakemeyer, and Levesque showed that in case actions have only local effects, progression is first-order representable. However, they could show computability of the first-order representation only for a restricted class. Also, their proofs were quite involved. In this paper, we present a result stronger than theirs that for local-effect actions, progression is always first-order definable and computable. We give a very simple proof for this via the concept of forgetting. We also show first-order definability and computability results for a class of knowledge bases and actions with non-local effects. Moreover, for a certain class of local-effect actions and knowledge bases for representing disjunctive information, we show that progression is not only first-order definable but also efficiently computable.

Applications of Effective Probability Theory to Martin-Löf Randomness

Abstract: We pursue the study of the framework of layerwise computability introduced in a preceding paper and give three applications (i) We prove a general version of Birkhoff's ergodic theorem for random points, where the transformation and the observable are supposed to be effectively measurable instead of computable This result significantly improves V'yugin and Nandakumar's ones (ii) We provide a general framework for deriving sharper theorems for random points, sensitive to the speed of convergence This offers a systematic approach to obtain results in the spirit of Davie's ones (iii) Proving an effective version of Prokhorov theorem, we positively answer a question recently raised by Fouche: can random Brownian paths reach any random number? All this shows that layerwise computability is a powerful framework to study Martin-Lof randomness, with a wide range of applications

Models of Computation: An Introduction to Computability Theory

Abstract: A Concise Introduction to Computation Models and Computability Theory provides an introduction to the essential concepts in computability, using several models of computation, from the standard Turing Machines and Recursive Functions, to the modern computation models inspired by quantum physics. An in-depth analysis of the basic concepts underlying each model of computation is provided. Divided into two parts, the first highlights the traditional computation models used in the first studies on computability: - Automata and Turing Machines; - Recursive functions and the Lambda-Calculus; - Logic-based computation models. and the second part covers object-oriented and interaction-based models. There is also a chapter on concurrency, and a final chapter on emergent computation models inspired by quantum mechanics. At the end of each chapter there is a discussion on the use of computation models in the design of programming languages.

Effective computability of solutions of differential inclusions --- the ten thousand monkeys approach

Abstract: In this paper we consider the computability of the solution of the initial-value problem for differential equations and for differential inclusions with semicontinuous right-hand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of a locally Lipschitz differential equation is computable even if the function is not effectively locally Lipschitz, and recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable. We give an example of a computable locally Lipschitz function which is not effectively locally Lipschitz. We also show that the solutions of a convex-valued upper-semicontinuous differential inclusion are upper-semicomputable, and the solutions of a lower-semicontinuous one-sided Lipschitz differential inclusion are lower-semicomputable.

Abstract geometrical computation 3: black holes for classical and analog computing

Abstract: The so-called Black Hole model of computation involves a non Euclidean space-time where one device is infinitely "accelerated" on one world-line but can send some limited information to an observer working at "normal pace". The key stone is that after a finite duration, the observer has received the information or knows that no information was ever sent by the device which had an infinite time to complete its computation. This allows to decide semi-decidable problems and clearly falls out of classical computability. A setting in a continuous Euclidean space-time that mimics this is presented. Not only is Zeno effect possible but it is used to unleash the black hole power. Both discrete (classical) computation and analog computation (in the understanding of Blum, Shub and Smale) are considered. Moreover, using nested singularities (which are built), it is shown how to decide higher levels of the corresponding arithmetical hierarchies.

Static Dependency Pair Method Based on Strong Computability for Higher-Order Rewrite Systems

Abstract: Higher-order rewrite systems (HRSs) and simply-typed term rewriting systems (STRSs) are computational models of functional programs. We recently proposed an extremely powerful method, the static dependency pair method, which is based on the notion of strong computability, in order to prove termination in STRSs. In this paper, we extend the method to HRSs. Since HRSs include λ-abstraction but STRSs do not, we restructure the static dependency pair method to allow λ-abstraction, and show that the static dependency pair method also works well on HRSs without new restrictions.

O-Minimal Hybrid Reachability Games

Abstract: In this paper, we consider reachability games over general hybrid systems, and distinguish between two possible observation frameworks for those games: either the precise dynamics of the system is seen by the players (this is the perfect observation framework), or only the starting point and the delays are known by the players (this is the partial observation framework). In the first more classical framework, we show that time-abstract bisimulation is not adequate for solving this problem, although it is sufficient in the case of timed automata . That is why we consider an other equivalence, namely the suffix equivalence based on the encoding of trajectories through words. We show that this suffix equivalence is in general a correct abstraction for games. We apply this result to o-minimal hybrid systems, and get decidability and computability results in this framework. For the second framework which assumes a partial observation of the dynamics of the system, we propose another abstraction, called the superword encoding, which is suitable to solve the games under that assumption. In that framework, we also provide decidability and computability results.

A Computability Perspective on Self-Modifying Programs

Abstract: In order to increase their stealth, malware commonly use the self-modification property of programs. By doing so, programs can hide their real code so that it is difficult to define a signature for it. But then, what is the meaning of those programs: the obfuscated form, or the hidden one? Furthermore, from a computability perspective, it becomes hard to speak about the program since, its own code varies over time. To cope with these issues, we provide an operational semantics for self-modifying programs and we show that they can be constructively rewritten to a non-modifying program.

Dependency Pairs and Polynomial Path Orders

Abstract: We show how polynomial path orders can be employed efficiently in conjunction with weak innermost dependency pairs to automatically certify polynomial runtime complexity of term rewrite systems and the polytime computability of the functions computed. The established techniques have been implemented and we provide ample experimental data to assess the new method.

Sequences close to periodic

Abstract: This paper is a survey of concepts and results connected with generalizations of the notion of a periodic sequence, both classical and new. The topics discussed relate to almost periodicity in such areas as combinatorics on words, symbolic dynamics, expressibility in logical theories, computability, Kolmogorov complexity, and number theory. Bibliography: 124 titles.

Sequences close to periodic

Abstract: The paper is a survey of notions and results related to classical and new generalizations of the notion of a periodic sequence. The topics related to almost periodicity in combinatorics on words, symbolic dynamics, expressibility in logical theories, algorithmic computability, Kolmogorov complexity, number theory, are discussed.

An Order on Sets of Tilings Corresponding to an Order on Languages

Abstract: Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view, leading to the notion of subshift. In this article we establish a correspondence between an order on subshifts based on dynamical transformations on them and an order on languages of forbidden patterns based on computability properties.

Computability on linear orderings enriched with predicates

Abstract: Let L be a quasidiscrete linear ordering. We specify some conditions for the existence of a computable presentation for L or for the structure (L, adj), where adj(x, y) is a predicate distinguishing adjacent elements.

Testing Computability by Width Two OBDDs

Abstract: Property testing is concerned with deciding whether an object (e.g. a graph or a function) has a certain property or is "far" (for some definition of far) from every object with that property. In this paper we give lower and upper bounds for testing functions for the property of being computable by a read-once width-2 Ordered Binary Decision Diagram (OBDD), also known as a branching program , where the order of the variables is known. Width-2 OBDDs generalize two classes of functions that have been studied in the context of property testing - linear functions (over GF (2)) and monomials. In both these cases membership can be tested in time that is linear in 1/*** . Interestingly, unlike either of these classes, in which the query complexity of the testing algorithm does not depend on the number, n , of variables in the tested function, we show that (one-sided error) testing for computability by a width-2 OBDD requires ***(log(n )) queries, and give an algorithm (with one-sided error) that tests for this property and performs $\tilde{O}(\log(n)/\epsilon)$ queries.

Unification with Singleton Tree Grammars

Abstract: First-order term unification is an essential concept in areas like functional and logic programming, automated deduction, deductive databases, artificial intelligence, information retrieval, compiler design, etc. We build upon recent developments in general grammar-based compression mechanisms for terms, which are more general than dags and investigate algorithms for first-order unification of compressed terms. We prove that the first-order unification of compressed terms is decidable in polynomial time, and also that a compressed representation of the most general unifier can be computed in polynomial time. We use several known results on the used tree grammars, called singleton tree grammars (STG)s, like polynomial time computability of several subalgorithmms: certain grammar extensions, deciding equality of represented terms, and generating the preorder traversal. An innovation is a specialized depth of an STG that shows that unifiers can be represented in polynomial space.

The standard parts problem and the complexity of control communication

Abstract: The objective of the standard parts optimal control problem is to find a number, m, of control inputs to a given input-output system that can be used in different combinations to achieve a certain number, n, of output objectives and to do this in such a way that a specified figure-of-merit measuring the average cost of control is minimized. The problem is especially interesting when m is significantly less than n. Distributed optimization problems of this type arise naturally in connection with recent work on control communication complexity. In what follows a general formulation of the standard parts optimization problem is given along with some simple illustrative examples. Control communication complexity is defined, and it is shown how one measure of this complexity naturally leads to a standard parts optimization problem. The entire circle of ideas is explored in the context of quadratic optimal control of the Heisenberg system, and recent results on computability using simple closed curve inputs are presented.

Learning Context Free Grammars by Using SAT Solvers

Abstract: In this paper, we propose a novel approach for learning context free grammars (CFGs) from positive and negative samples by solving a Boolean satisfiability problem (SAT). We encode the set of samples, together with limits on the sizes of rule sets to be synthesized as a Boolean expression. An assignment satisfying the Boolean expression contains a minimal set of rules that derives all positive samples and no negative samples. A feature of this approach is that we can synthesize the minimal set of rules in Chomsky normal form. The other feature is that our learning method reflects any improvements of SAT solvers. We present experimental results on learning CFGs for fundamental context free languages, including a set of strings composed of the equal numbers of a's and b's and the set of strings over {a, b}* not of the form ww.

On universal computably enumerable prefix codes

Abstract: We study computably enumerable (c.e.) prefix codes that are capable of coding all positive integers in an optimal way up to a fixed constant: these codes will be called universal. We prove various characterisations of these codes, including the following one: a c.e. prefix code is universal if and only if it contains the domain of a universal self-delimiting Turing machine. Finally, we study various properties of these codes from the points of view of computability, maximality and density.