Matrix equations and Hilbert's tenth problem

We show a reduction of Hilbert's tenth problem to the solvability of the matrix equation Xi1 1 Xi2 2 Xik k = Z over non-commuting integral matrices, where Z is the zero matrix, thus proving that the solvability of the equation is undecidable. This is in contrast to the case whereby the matrix semigroup is commutative in which the solvability of the same equation was shown to be decidable in general. The restricted problem where k = 2 for commutative matrices is known as the \A-B-C Problem" and we show that this problem is decidable even for a pair of non-commutative matrices over an algebraic number field

The undecidability of simultaneous rigid E-unification with two variables

Abstract. Recently it was proved that the problem of simultaneous rigid E-unification, or SREU, is undecidable. Here we show that 4 rigid equations with ground left-hand sides and 2 variables already imply undecidability. As a corollary we improve the undecidability result of the 3*-fragment of intuitionistic logic with equality. Our proof shows undecidability of a very restricted subset of the 33-fragment. Together with other results, it contributes to a complete characterization of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. 1 I n t r o d u c t i o n Recently it was proved that the problem of simultaneous rigid E-unification (SREU) is undecidable Background of S R E U Simultaneous rigid E-unification was proposed by Ga~er, Raatz and Snyder 1 It has been noted by Gurevich and Veanes that 3 rigid equations suffice

Quanta, Computations, and Extended Everett Concept

ABSTRACT In the Extended Everett Concept, proposed by M.B. Mensky, is assumed, that numerous quantum alternatives that are included in a superposition of many-world Everett universe, are available for the analysis to consciousness, being in some threshold states like sleep or meditation. Since this possibility is in the direct contradiction to no-cloning theorem of quantum mechanics, the question arises how this can be. The proposed solution is based on an analogy with the 'miracle of cloning of states', which can be implemented for an observer living in the quantum world, simulated on a classical computer. The basis of the hypothesis is the algorithmic solvability of the quantum theory

Finitism--an essay on Hilbert's programme

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1991.Includes bibliographical references (p. 213-219).by David Watson Galloway.Ph.D