Computability

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

Integrating Temporal Logic as a State-Based Specification Language for Discrete-Event Control Design in Finite Automata

Abstract: This paper presents and analyzes a correct and complete translation algorithm that converts a class of propositional linear-time temporal-logic (PTL) formulae to deterministic finite (-trace) automata. The translation algorithm is proposed as a specification interface for finitary control design of discrete-event systems (DESs). While there has been a lot of computer science research that connects PTL formulae to omega-automata, there is relatively little prior work that translates state-based PTL formulae in the context of a finite-state DES model, to event-based finite automata-the formalism on which well-established control synthesis methods exist. The proposed translation allows control requirements to be more easily described and understood in temporal logic, widely recognized as a useful specification language for its intuitively appealing operators that provide the natural-language expressiveness and readability needed to express and explain these requirements. Adding such a translation interface could therefore effectively combine specifiability and readability in temporal logic with prescriptiveness and computability in finite 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 readable control specifications in PTL that it translates for discrete-event control synthesis in deterministic finite automata. Two application examples illustrate the use of the proposed temporal-logic interface. Practical implications of the complexity of the translation algorithm are discussed.

Relative Computability and Uniform Continuity of Relations

Abstract: A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multivalued functions f:[0,1]=>R (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y in f(x), new ways of (linearly) ordering quantifiers arise, yet none of them turn out as satisfactory. We are thus led to a notion of uniform continuity based on the Henkin Quantifier; and prove it necessary for relative computability. In fact iterating this condition yields a strict hierarchy of notions each necessary, and the omega-th level also sufficient, for relative computability.

Space-Time Universality of Field Calculus

Abstract: Recent work in the area of coordination models and collective adaptive systems promotes a view of distributed computations as functional blocks manipulating data structures spread over space and evolving over time. In this paper, we address expressiveness issues of such computations, and specifically focus on the field calculus, a prominent emerging language in this context. Based on the classical notion of event structure, we introduce the cone Turing machine as a ground for studying computability issues, and first use it to prove that field calculus is space-time universal. We then observe that, in the most general case, field calculus computations can be rather inefficient in the size of messages exchanged, but this can be remedied by an encoding to nearly similar computations with slower information speed. We capture this concept by a notion of delayed space-time universality, which we prove to hold for the set of message-efficient algorithms expressible by field calculus. As a corollary, it is derived that field calculus can implement with message-size efficiency all self-stabilising distributed algorithms.

Effective word-level interpolation for software verification

Abstract: We present an interpolation procedure for the theory of fixed-size bit-vectors, which allows to apply effective interpolation-based techniques for software verification without giving up the ability of handling precisely the word-level operations of typical programming languages. Our algorithm is based on advanced SMT techniques, and, although general, is optimized to exploit the structure of typical interpolation problems arising in software verification. We have implemented a prototype version of it within the MathSAT SMT solver, and we have integrated it into a software verification framework based on standard predicate abstraction. Our experimental results show that our new technique allows our prototype to significantly outperform other systems on programs requiring bit-precise modeling of word-level operations.

Distributed computability in Byzantine asynchronous systems

Abstract: In this work, we extend the topology-based approach for characterizing computability in asynchronous crash-failure distributed systems to asynchronous Byzantine systems. We give the first theorem with necessary and sufficient conditions to solve arbitrary tasks in asynchronous Byzantine systems where an adversary chooses faulty processes. For colorless tasks, an important subclass of distributed problems, the general result reduces to an elegant model that effectively captures the relation between the number of processes, the number of failures, as well as the topological structure of the task's simplicial complexes.