Isomorphism versus commensurability for a class of finitely presented groups

We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem

A Primer on the Tools and Concepts of Computable Economics

Computability theory came into being as a result of Hilbert's attempts to meet Brouwer's challenges, from an intuitionistc and constructive standpoint, to formalism as a foundation for mathematical practice. Viewed this way, constructive mathematics should be one vision of computability theory. However, there are fundamental differences between computability theory and constructive mathematics: the Church-Turing thesis is a disciplining criterion in the former and not in the latter; and classical logic - particularly, the law of the excluded middle - is not accepted in the latter but freely invoked in the former, especially in proving universal negative propositions. In Computable Economic an eclectic approach is adopted where the main criterion is numerical content for economic entities. In this sense both the computable and the constructive traditions are freely and indiscriminately invoked and utilised in the formalization of economic entities. Some of the mathematical methods and concepts of computable economics are surveyed in a pedagogical mode. The context is that of a digital economy embedded in an information society

SOLVABILITY EXPLORATION OF SEGMENTATION PROBLEM WITH LINEAR CONVOLUTION ALGORITHMS

The paper is dedicated to decidability exploration of market segmentation problem with the help of linear convolution algorithms. Mathematical formulation of this problem represents interval task of bipartite graph cover by stars. Vertices of the first partition correspond to types of commodities, vertices of the second – to customers groups. Appropriate method is offered for interval problem reduction to two-criterion task that has one implemented linear convolution algorithm. Unsolvability with the help of linear convolution algorithm of multicriterion, and consequently interval, market segmentation problem is proved

Efficient algorithms for generalized Stable Marriage and Roommates problems

We consider a generalization of the Stable Roommates problem (SR), in which preference lists may be partially ordered and forbidden pairs may be present, denoted by SRPF. This includes, as a special case, a corresponding generalization of the classical Stable Marriage problem (SM), denoted by SMPF. By extending previous work of Feder, we give a two-step reduction from SRPF to 2-SAT. This has many consequences, including fast algorithms for a range of problems associated with finding "optimal" stable matchings and listing all solutions, given variants of SR and SM. For example, given an SMPF instance I, we show that there exists an O(m) "succinct" certificate for the unsolvability of I, an O(m) algorithm for finding all the super-stable pairs in I, an O(m+kn) algorithm for listing all the super-stable matchings in I, an O(m<sup>1.5</sup>) algorithm for finding an egalitarian super-stable matching in I, and an O(m) algorithm for finding a minimum regret super-stable matching in I, where n is the number of men, m is the total length of the preference lists, and k is the number of super-stable matchings in I. Analogous results apply in the case of SRPF

A Computable Economist’s Perspective on Computational Complexity

A computable economist.s view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called "Post's Program of Research for Higher Recursion Theory". Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix.

Uncomputability and Undecidability in Economic Theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory ([28]) and Algorithmic Statistics ([13]) are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory - and its applied branches, say computational complexity theory or algorithmic information theory - and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general - but concise - analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory, are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.General Equilibrium Theory, Game Theory, Recursive Macro-economics, (Un)computability, (Un)decidability, Constructivity