Showing papers on "Computability published in 2005"

Computability and Evolutionary Complexity: Markets as Complex Adaptive Systems (CAS)*

Abstract: Few will argue that the epi-phenomena of biological systems and socio-economic systems are anything but complex. The purpose of this Feature is to examine critically and contribute to the burgeoning multi-disciplinary literature on markets as complex adaptive systems (CAS). The new sciences of complexity, the principles of self-organisation and emergence along with the methods of evolutionary computation and artificially intelligent agent models have been developed in a multi-disciplinary fashion. The cognoscenti here consider that complex systems whether natural or artificial, physical, biological or socio-economic can be characterised by a unifying set of principles. Further, it is held that these principles mark a paradigm shift from earlier ways of viewing such phenomenon.

Computing over the Reals: Foundations for Scientific Computing

Abstract: We give a detailed treatment of the ``bit-model'' of computability and complexity of real functions and subsets of R^n, and argue that this is a good way to formalize many problems of scientific computation. In the introduction we also discuss the alternative Blum-Shub-Smale model. In the final section we discuss the issue of whether physical systems could defeat the Church-Turing Thesis.

Non-computable Julia sets

Abstract: Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most naive work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor's distance-estimator algorithm [Mil] uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show:

SAT-Based Complete Don't-Care Computation for Network Optimization

Abstract: The paper describes an improved approach to Boolean network optimization using internal don't-cares. The improvements concern the type of don't-cares computed, their scope, and the computation method. Instead of the traditionally used compatible observability don't-cares (CODCs), we introduce and justify the use of complete don't-cares (CDC). To ensure the robustness of the don't-care computation for very large industrial networks, an optional windowing scheme is implemented that computes substantial subsets of the CDCs in reasonable time. Finally, we give a SAT-based don't-care computation algorithm that is more efficient than BDD-based algorithms. Experimental results confirm that these improvements work well in practice. Complete don't-cares allow for a reduction in the number of literals compared to the CODCs. Windowing guarantees robustness, even for very large benchmarks on which previous methods could not be applied. SAT reduces the runtime and enhances robustness, making don't-cares affordable for a variety of other Boolean methods applied to the network.

Partition Theorems and Computability Theory

Abstract: The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, Konig’s Lemma and Ramsey’s Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω. We also identify each n ∈ ω with its set of predecessors, so n = {0, 1, 2, . . . , n− 1}.

Real Hypercomputation and Continuity

Abstract: By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.

PaMira - A Parallel SAT Solver with Knowledge Sharing

Abstract: In this paper we describe PaMira, a powerful distributed SAT solver. PaMira is based on the highly optimized, sequential SAT engine Mira, incorporating all essential optimization techniques modern algorithms utilize to maximize performance. For the distributed execution an efficient work stealing method has been implemented. PaMira also employs the exchange of conflict clauses between the processes to guide the search more efficiently. We provide experimental results showing linear speedup on a multiprocessor environment with four AMD Opteron processors

A Functional Approach to Computability on Real Numbers

Abstract: The aim of this thesis is to contribute to close the gap existing between the theory of computable analysis and actual computation. In order to study computability over real numbers we use several tools peculiar to the theory of programming languages. In particular we introduce a special kind of typed lambda calculus as an appropriate formalism for describing computations on real numbers. Furthermore we use domain theory, to give semantics to this typed lambda calculus and as a conseguence to give a notion of computability on real numbers. We discuss the adequacy of Scott-Domains as domains for representing real numbers. We relate the Scott topology on such domains to the euclidean topology on IR. Domain theory turns out to be useful also in the study of higher order functions. In particular one of the most important results contained in this thesis concerns the characterisation of the topological properties of the computable higher order functions on reals. Our approach allows moreover to phrase and discuss w.r.t. real numbers issues of programming languages. We address the problem of defining an implementation of real numbers as an abstract data type. Finally we investigate algorithms for carrying out efficient computations on reals.

Continuous semantics for strong normalization

Abstract: We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domain-theoretic semantics. The theorem applies to extensions of Godel's system T but also to various forms of bar recursion for which strong normalization was hitherto unknown.

Shortest path amidst disc obstacles is computable

Abstract: An open question in Exact Geometric Computation is whether there re transcendental computations that can be made "geometrically exact".Perhaps the simplest such problem in computational geometry is that of computing the shortest obstacle-avoiding path between two points p, q in the plane, where the obstacles re collection of n discs.This problem can be solved in O (n 2 log n)time in the Real RAM model, but nothing was known about its computability in the standard (Turing) model of computation. We first show the Turing-computability of this problem,provided the radii of the discs are rationally related. We make the usual assumption that the numerical input data are real algebraic numbers. By appealing to effective bounds from transcendental number theory, we further show single-exponential time upper bound when the input numbers are rational.Our result ppears to be the first example of non-algebraic combinatorial problem which is shown computable. It is also rare example of transcendental number theory yielding positive computational results.

Computability and continuity on the real arithmetic hierarchy and the power of type-2 nondeterminism

Abstract: The sometimes so-called Main Theorem of Recursive Analysis implies that any computable real function is necessarily continuous. We consider three relaxations of this common notion of real computability for the purpose of treating also discontinuous functions f: ℝ→ℝ: non-deterministic computation; relativized computation, specifically given access to oracles like ∅′ or ∅″; encoding input xeℝ and/or output y = f(x) in weaker ways according to the Real Arithmetic Hierarchy. It turns out that, among these approaches, only the first one provides the required power.

Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle

Abstract: We state a self-similar tiling principle which shows that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure. We use this method to get the optimal coverings and packings which give the exact value of the Hausdorff-type and packing measures. In particular, we show that the exact value of these measures coincides with the supremum or with the infimum of the inverse of the density of the natural probability measure on suitable classes of sets. This gives criteria for the numerical analysis of the measures, and allows us to compare their complexity in terms of computability.

Computability of finite-time reachable sets for hybrid systems

Abstract: In this paper we consider the computability of the evolution of hybrid systems, or equivalently, the computability of finite-time reachable sets. We use the framework of type-two computability theory and computable analysis, which gives a theory of computation for points, sets and maps by Turing machines, and is related to computable approximation. We show that, under suitable hypotheses, the system evolution may be lower or upper semicomputable, but cannot be both in the presence of grazing contact with the guard sets.

Towards computability of higher type continuous data

Abstract: This paper extends the logical approach to computable analysis via Σ–definability to higher type continuous data such as functionals and operators. We employ definability theory to introduce computability of functionals from arbitrary domain to the real numbers. We show how this concept works in particular cases.

Hilbert's tenth problem and paradigms of computation

Abstract: This is a survey of a century long history of interplay between Hilbert's tenth problem (about solvability of Diophantine equations) and different notions and ideas from the Computability Theory.

Computing the expected accumulated reward and gain for a subclass of infinite markov chains

Abstract: We consider the problem of computing the expected accumulated reward and the average gain per transition in a subclass of Markov chains with countable state spaces where all states are assigned a non-negative reward. We state several abstract conditions that guarantee computability of the above properties up to an arbitrarily small (but non-zero) given error. Finally, we show that our results can be applied to probabilistic lossy channel systems, a well-known model of processes communicating through faulty channels.

Computability in computational geometry

Abstract: We promote the concept of object directed computability in computational geometry in order to faithfully generalise the well-established theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative measure for the approximation. The domain-theoretic model of computational geometry provides such an object directed theory, which supports two such quantitative measures, one based on the Hausdorff metric and one on the Lebesgue measure. With respect to a new data type for the Euclidean space, given by its non-empty compact and convex subsets, we show that the convex hull, Voronoi diagram and Delaunay triangulation are Hausdorff and Lebesgue computable.

A reducibility for the dot-depth hierarchy

Abstract: Hierarchies considered in computability theory and in complexity theory are related to some reducibilities in the sense that levels of the hierarchies are downward closed and have complete sets. In this paper we propose a reducibility having similar relationship to the Brzozowski's dot-depth hierarchy and some its refinements. We prove some basic facts on the corresponding degree structure and discuss relationships of the reducibility to complexity theory (via the leaf-language approach).

On the Complexity of Real Functions

Abstract: We develop a notion of computability and complexity of functions over the reals, which seems to be very natural when one tries to determine just how "difficult" a certain function is. This notion can be viewed as an extension of both BSS computability [Blum, Cucker, Shub, Smale 1998], and bit computability in the tradition of computable analysis [Weihrauch 2000] as it relies on the latter but allows some discontinuities and multiple values.

On Automorphic Tuples of Elements in Computable Models

Abstract: A criterion is obtained for existence of two isomorphic but not hyperarithmetically isomorphic tuples in a hyperarithmetical model. This criterion is used to show that such a situation occurs in the models of well-known classes.

Simulating Turing machines on Maurer machines

Abstract: In a previous paper, we used Maurer machines to model and analyse micro-architectures. In the current paper, we investigate the connections between Turing machines and Maurer machines with the purpose to gain an insight into computability issues relating to Maurer machines. We introduce ways to simulate Turing machines on a Maurer machine which, dissenting from the classical theory of computability, take into account that in reality computations always take place on finite machines. In one of those ways, multi-threads and thread forking have an interesting theoretical application.

A Temporal Logic Specification Interface for Automata-Theoretic Finitary Control Synthesis

Abstract: This paper presents a correct and complete algorithm that translates propositional linear time temporal logic (PTL) formulae to event-based automata for finitary control synthesis. It proposes the translation algorithm as a specification interface to well-established automata-theoretic control synthesis methods for discrete-event systems (DES’s). This allows control requirements to be more easily described and understood in an expressive and readable language that temporal logic is widely recognized as. Adding such a translation interface effectively combines specifiability and readability in temporal logic with prescriptiveness and computability in automata. The former temporal logic features support specification while the latter automata features support the prescription of DES dynamics and algorithmic computations. A practical implementation of the interface has been developed, providing an enabling technology for writing more readable control specifications in PTL that it translates for finitary control synthesis in automata. A simple example illustrates the use of the proposed temporal logic interface. Practical implications of the complexity of the translation algorithm are discussed.