Is Church’s Thesis Still Relevant?

The article analyses the role of Church’s Thesis (hereinafter CT) in the context of the development of hypercomputation research. The text begins by presenting various views on the essence of computer science and the limitations of its methods. Then CT and its importance in determining the limits of methods used by computer science is presented. Basing on the above explanations, the work goes on to characterize various proposals of hypercomputation showing their relative power in relation to the arithmetic hierarchy.The general theme of the article is the analysis of mutual relations between the content of CT and the theories of hypercomputation. In the main part of the paper the arguments for abolition of CT caused by the introduction of hypercomputable methods in computer science are presented and critique of these views is presented. The role of the efficiency condition contained in the formulation of CT is stressed. The discussion ends with a summary defending the current status of Church’s thesis within the framework of philosophy and computer science as an important point of reference for determining what the notion of effective calculability really is. The considerations included in this article seem to be quite up-to-date relative to the current state of affairs in computer science.1

On the complexity of index sets for finite predicate logic programs which allow function symbols

We study the recognition problem in the metaprogramming of finite normal predicate logic programs. That is, let $\mathcal{L}$ be a computable first-order predicate language with infinitely many constant symbols and infinitely many $n$-ary predicate symbols and $n$-ary functions symbols for all $n \geq 1$. Then we can effectively list all the finite normal predicate logic programs $Q_0,Q_1,\ldots $ over $\mathcal{L}$. Given some property $\mathcal{P}$ of finite normal predicate logic programs over $\mathcal{L}$, we define the index set $I_{\mathcal{P}}$ to be the set of indices $e$ such that $Q_e$ has property $\mathcal{P}$. We classify the complexity of the index set $I_{\mathcal{P}}$ within the arithmetic hierarchy for various natural properties of finite predicate logic programs. For example, we determine the complexity of the index sets relative to all finite predicate logic programs and relative to certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having no recursive stable models, (iii) having at least one stable model, (iv) having at least one recursive stable model, (v) having exactly $c$ stable models for any given positive integer $c$, (vi) having exactly $c$ recursive stable models for any given positive integer $c$, (vii) having only finitely many stable models, (viii) having only finitely many recursive stable models, (ix) having infinitely many stable models and (x) having infinitely many recursive stable models.

LEARNING THEORY IN THE ARITHMETIC HIERARCHY

We consider the arithmetic complexity of index sets of uniformly computably enumerable families learnable under different learning criteria. We determine the exact complexity of these sets for the standard notions of finite learning, learning in the limit, behaviorally correct learning and anomalous learning in the limit. In proving the ${\rm{\Sigma }}_5^0$-completeness result for behaviorally correct learning we prove a result of independent interest; if a uniformly computably enumerable family is not learnable, then for any computable learner there is a ${\rm{\Delta }}_2^0$ enumeration witnessing failure.

A Novel Model for Multi-Parametric Load Balance in Grid Computing

Grid computing is a computing model in which computing resources are geographically dispersed. A computing node can share these resources as well as can transfer applications at other nodes to execute it. Due to the absence of centralized authority in grid, some resources may be overloaded and others may be under loaded. To obtain high performance, load balance strategy is necessarily needed. Load balance strategy can be affected by different parameters like network parameters, application characteristics, computing node capacity etc. In this paper, we consider using three parameters namely network parameters, computation node capacity and application characteristic to obtain effective load balance. All these parameters will pass to AHP (Arithmetic hierarchy process) for automatic decision making to select better resources for high performance service.

Some natural decision problems in automatic graphs

For automatic and recursive graphs, we investigate the following problems:(A) existence of a Hamiltonian path and existence of an infinite path in a tree(B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree(C) existence of an infinite clique and an infinite version of set coverThe complexity of these problems is determined for automatic graphs and. supplementing results from the literature, for recursive graphs. Our results show that these problems(A) are equally complex for automatic and for recursive graphs (-complete).(B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy),(C) are much simpler for automatic than for recursive graphs (decidable and -complete, resp.).

Approximate counting by hashing in bounded arithmetic

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.