Hypercomputation by definition

Hypercomputation refers to computation surpassing the Turing model, not just exceeding the von Neumann architecture. Algebraic constructions yield a $nitely based pseudorecursive equational theory (Internat. J. Algebra Comput. 6 (1996) 457–510). It is not recursive, although for each given number n, its equations in n variables form a recursive set. Hypercomputation is therefore required for an algorithmic answer to the membership problem of such a theory. Yet Alfred Tarski declared these theories to be decidable. The dilemma of a decidable but not recursive set presents an impasse to standard computability theory. One way to break the impasse is to predicate that the theory is computable—in other words, hypercomputation by de$nition. c © 2003 Elsevier B.V. All rights reserved. MSC: primary 03D35; secondary 03C05; 03D15; 03D40; 20M07; 68Q05; 68Q10; 68Q15