Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

Using Problem Symmetry in Search Based Satisfiability Algorithms

Abstract: We introduce the notion of problem symmetry in search-based SAT algorithms. We develop a theory of essential points to formally characterize the potential search-space pruning that can be realized by exploiting problem symmetry. We unify several search-pruning techniques used in modern SAT solvers under a single framework, by showing them to be special cases of the general theory of essential points. We also propose a new pruning rule exploiting problem symmetry. Preliminary experimental results validate the efficacy of this rule in providing additional search-space pruning beyond the pruning realized by techniques implemented in leading-edge SAT solvers.

Deciding LTL over Mazurkiewicz traces

Abstract: Linear time temporal logic (LTL) has become a well established tool for specifying the dynamic behaviour of reactive systems with an interleaving semantics, and the automata-theoretic approach has proven to be a very useful mechanism for performing automatic verification in this setting. Especially alternating automata turned out to be a powerful tool in constructing efficient yet simple to understand decision procedures and directly yield further on-the-fly model checking procedures. In this paper we exhibit a decision procedure for LTL over Mazurkiewicz traces which generalises the classical automata-theoretic approach to a linear time temporal logic interpreted no longer over sequences but certain partial orders. Specifically, we construct a (linear) alternating Buchi automaton accepting the set of linearisations of traces satisfying the formula at hand. The salient point of our technique is to apply a notion of independence-rewriting to formulas of the logic. Furthermore, we show that the class of linear and trace-consistent alternating Buchi automata corresponds exactly to LTL formulas over Mazurkiewicz traces, lifting a similar result from Loding and Thomas formulated in the framework of LTL over words.

Shifts with Decidable Language and Non-Computable Entropy

Abstract: We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift

The quantum algorithm of Kieu does not solve the Hilbert's tenth problem

Abstract: Recently T Kieu (arXiv:quant-ph/0110136) claimed a quantum algorithm computing some functions beyond the Church-Turing class He notes that "it is in fact widely believed that quantum computation cannot offer anything new about computability" and claims the opposite However, his quantum algorithm does not work, which is the point of my short note I still believe that quantum computation leads to new complexity but retains the old computability

Computability of Operators on Continuous and Discrete Time Streams

Abstract: A stream is a sequence of data indexed by time. The behaviour of natural and artificial systems can be modelled by streams and stream transformations. There are two distinct types of data stream: streams based on continuous time and streams based on discrete time. Having investigated case studies of both kinds separately, we have begun to combine their study in a unified theory of stream transformers, specified by equations. Using only the standard mathematical techniques of topology, we have proved continuity properties of stream transformers. Here, in this sequel, we analyse their computability. We use the theory of computable functions on algebras to design two distinct methods for defining computability on continuous and discrete time streams of data from a complete metric space. One is based upon low-level concrete representations, specifically enumerations, and the other is based upon high-level programming, specifically while programs, over abstract data types. We analyse when these methods are equivalent. We demonstrate the use of the methods by showing the computability of an analog computing system. We discuss the idea that continuity and computability are important for models of physical systems to be “well-posed”.