Computability

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

Computable Functions in ASP: Theory and Implementation

Abstract: Disjunctive Logic Programming (DLP) under the answer set semantics, often referred to as Answer Set Programming (ASP), is a powerful formalism for knowledge representation and reasoning (KRR). The latest years witness an increasing effort for embedding functions in the context of ASP. Nevertheless, at present no ASP system allows for a reasonably unrestricted use of function terms. Functions are either required not to be recursive or subject to severe syntactic limitations, if allowed at all in ASP systems. In this work we formally define the new class of finitely-ground programs, allowing for a powerful (possibly recursive) use of function terms in the full ASP language with disjunction and negation. We demonstrate that finitely-ground programs have nice computational properties: (i) both brave and cautious reasoning are decidable, and (ii) answer sets of finitely-ground programs are computable. Moreover, the language is highly expressive, as any computable function can be encoded by a finitely-ground program. Due to the high expressiveness, membership in the class of finitely-ground program is clearly not decidable (we prove that it is semi-decidable). We single out also a subset of finitely-ground programs, called finite-domain programs, which are effectively recognizable, while keeping computability of both reasoning and answer set computation. We implement all results in DLP, further extending the language in order to support list and set terms, along with a rich library of built-in functions for their manipulation. The resulting ASP system is very powerful: any computable function can be encoded in a rich and fully declarative KRR language, ensuring termination on every finitely-ground program. In addition, termination is "a priori" guaranteed if the user asks for the finite-domain check.

Local Computation: Lower and Upper Bounds

Abstract: The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a \emph{distributed} fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first poly-logarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.

Contextual Insertions/Deletions and Computability

Abstract: We investigate two generalizations of insertion and deletion of words, that have recently become of interest in the context of molecular computing. Given a pair of words (x, y), called a context, the (x, y)-contextual insertion of a wordvinto a worduis performed as follows. For each occurrence ofxyas a subword inu, we include in the result of the contextual insertion the words obtained by insertingvintou, betweenxandy. The (x, y)-contextual deletion operation is defined in a similar way. We study closure properties of the Chomsky families under the defined operations, contextual ins-closed and del-closed languages, and decidability of existence of solutions to equations involving these operations. Moreover, we prove that every Turing machine can be simulated by a system based entirely on contextual insertions and deletions

Computability and Complexity Results for a Spatial Assertion Language for Data Structures

Abstract: This paper studies a recently developed an approach to reasoning about mutable data structures, which uses an assertion language with spatial conjunction and implication connectives. We investigate computability and complexity properties of a subset of the language, which allows statements about the shape of pointer structures (such as "there is a link from x to y") to be made, but not statements about the data held in cells (such as "x is a prime number"). We show that validity, even for this restricted language, is not r.e., but that the quantifier-free sublanguage is decidable. We then consider the complexity of model checking and validity for several fragments.

A global optimization approach for architectural synthesis

Abstract: A relaxed linear programming model which simultaneously schedules and allocates functional units and registers is presented for synthesizing cost-constrained globally optimal architectures. This approach is important for industrial applications, because it provides exploration of optimal synthesized architectures and early architectural decisions have the greatest impact on the final design. An integer programming formulation of the architectural synthesis problem is transformed into the mode packing problem. Polyhedral theory is used to formulate constraints that decrease the size of the search space, thus improving solution efficiency. Execution times are an order of magnitude faster than for previous heuristic techniques. The present approach breaks new ground by (1) simultaneously scheduling and allocating in practical execution times, (2) guaranteeing globally optimal solutions for a specific objective function, and (3) providing a polynomial run-time algorithm for solving some instances of this NP-complete problem. >