Computability

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

A topological view on algebraic computation models

Abstract: We investigate the topological aspects of some algebraic computation models, in particular the BSS-model. Our results can be seen as bounds on how different BSS-computability and computability in the sense of computable analysis can be. The framework for this is Weihrauch reducibility. As a consequence of our characterizations, we establish that the solvability complexity index is (mostly) independent of the computational model, and that there thus is common ground in the study of non-computability between the BSS and TTE setting.

Certified undecidability of intuitionistic linear logic via binary stack machines and Minsky machines

Abstract: We formally prove the undecidability of entailment in intuitionistic linear logic in Coq. We reduce the Post correspondence problem (PCP) via binary stack machines and Minsky machines to intuitionistic linear logic. The reductions rely on several technically involved formalisations, amongst them a binary stack machine simulator for PCP, a verified low-level compiler for instruction-based languages and a soundness proof for intuitionistic linear logic with respect to trivial phase semantics. We exploit the computability of all functions definable in constructive type theory and thus do not have to rely on a concrete model of computation, enabling the reduction proofs to focus on correctness properties.

Randomness and Non-ergodic Systems

Abstract: We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi- computable functions. Random points are typical with respect to measure in that they have no measure-theoretically rare properties of a certain kind, while ergodic theo- rems describe regular measure-theoretic behavior. There has been a great deal of interest in the connection between these two kinds of regularity recently. We begin by defining the basic concepts in each field and then describe the ways in which they are related. Then we present our results on the relationship between algorithmic randomness and the satisfaction of Birkhoff's ergodic theorem for computable measure-preserving transforma- tions with respect to computable (and then lower semi-computable) func- tions. Those more familiar with ergodic theory than computability theory might find it useful to first read Section 6, a brief discussion of the notion of algorithmic randomness in the context of ergodic theory.

The computability of relaxed data structures: queues and stacks as examples

Abstract: Most concurrent data structures being designed today are versions of known sequential data structures. However, in various cases it makes sense to relax the semantics of traditional concurrent data structures in order to get simpler and possibly more efficient and scalable implementations. For example, when solving the classical producer-consumer problem by implementing a concurrent queue, it might be enough to allow the dequeue operation (by a consumer) to return and remove one of the two oldest values in the queue, and not necessarily the oldest one. We define infinitely many possible relaxations of several traditional data structures and objects: queues, stacks, multisets and registers, and examine their relative computational power.

Introduction to clarithmetic II

Abstract: The earlier paper "Introduction to clarithmetic I" constructed an axiomatic system of arithmetic based on computability logic, and proved its soundness and extensional completeness with respect to polynomial time computability. The present paper elaborates three additional sound and complete systems in the same style and sense: one for polynomial space computability, one for elementary recursive time (and/or space) computability, and one for primitive recursive time (and/or space) computability.