Axiomatic Theory of Algorithms: Computability and Decidability in Algorithmic Classes

Computability logic is a formal theory of (interactive) computability in the same sense as classical logic is a formal theory of truth. This approach was initiated very recently in "Introduction to computability logic" (Annals of Pure and Applied Logic 123 (2003), pp.1-99). The present paper reintroduces computability logic in a more compact and less technical way. It is written in a semitutorial style with a general computer science, logic or mathematics audience in mind. An...

Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are defined as a certain sort games between a machine and its environment, with logical operators standing for operations on such games. Within the ambitious program of finding axiomatizations for incrementally rich fragments of this semantica...

One of the fundamental results in computability is the existence of well-defined functions that cannot be computed. In this paper we study the effects of data representation on computability; we show that, while for each possible way of representing data there exist incomputable functions, the computability of a specific abstract function is never an absolute property, but depends on the representation used for the function domain. We examine the scope of this dependency and ...

This paper proposes a new approach to defining and expressing algorithms: the notion of {\it task logical} algorithms. This notion allows the user to define an algorithm for a task $T$ as a set of agents who can collectively perform $T$. This notion considerably simplifies the algorithm development process and can be seen as an integration of the sequential pseudocode and logical algorithms. This observation requires some changes to algorithm development process. We propose a...

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century. Second, building on the theory of functional relations in mathematics, it provides a theoretical framework where we can rigorously distinguish two pairs of concepts: Between solving a problem and verifying the solution to a problem. Between a d...

This article discusses what can be proved about the foundations of mathematics using the notions of algorithm and information. The first part is retrospective, and presents a beautiful antique, Godel's proof, the first modern incompleteness theorem, Turing's halting problem, and a piece of postmodern metamathematics, the halting probability Omega. The second part looks forward to the new century and discusses the convergence of theoretical physics and theoretical computer sci...

We discuss the legacy of Alan Turing and his impact on computability and analysis.

The starting point of this paper is a collection of properties of an algorithm that have been distilled from the informal descriptions of what an algorithm is that are given in standard works from the mathematical and computer science literature. Based on that, the notion of a proto-algorithm is introduced. The thought is that algorithms are equivalence classes of proto-algorithms under some equivalence relation. Three equivalence relations are defined. Two of them give bound...

After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.

This volume contains the papers presented at the 6th conference on Machines, Computations and Universality (MCU 2013). MCU 2013 was held in Zurich, Switzerland, September 9-11, 2013. The MCU series began in Paris in 1995 and has since been concerned with gaining a deeper understanding of computation through the study of models of general purpose computation. This volume continues in this tradition and includes new simple universal models of computation, and other results that...