Decidable classes of recursive equations.
thesisposted on 19.11.2015, 08:55 by R. D. Lee
Many different formalisations of recursive arithmetic have been proposed, and this thesis is concerned mainly with the system proposed by R.L. Goodstein and known as the Axiom - Free Equation Calculus. As with all other formal systems of arithmetic with sufficient content, the system is incomplete and recursively undecidable. The interesting questions lie in the completeness and decidability, or otherwise, of fragments of the system. I attempt to answer some of these questions. It happens that some of the problems lead to well known questions in the theory of diophantine equations namely, Hilbert's 10th Problem, The Undecidability of Exponential Diophantine Equations, and the Integer Linear Programming Problem. In 1943 Kalmar proposed a set of functions called elementary functions, and Ilona Bereczki showed effectively that the class of equations F = 0, where F is any elementary function, is undecidable. The class of functions given by Kalmar was, variables, l,+,., |a - b|, [a/b], but it can easily be shown that this is the same as those formed by composition from +,.,? This latter definition is the one we use. In his paper, A Decidable Fragment of Recursive Arithmetic, Goodstein showed the class of equations F = 0 where F is any function formed by composition from x + y, x.y and 1 ? x is decidable. So I have attempted to extend Goodstein's result with the upper bound provided by the undecidability of the elementary equations. The main results I have obtained are 1. If F is any function formed by composition from x + y, x.y, 1 ? x, ? 1, E y=w, II y=w, then F = 0 is decidable, and furthermore the provability in the equation calculus of F = 0 is decidable and that this class of equations is complete. 2. If F,G are any functions formed from x + y, x.y, 1 ? x, x ? 1, by composition, then the class of equations F = G is decidable. 3. If F,G are any functions formed by composition from x + y, x ? y then the class of equations F = G is decidable. 4. If F.G are any functions formed by composition from x + y, x ? y, x.y, then the class of equations F = G is decidable if and only if Hilbert's 10th Problem is decidable. 5. If F,G are any functions formed by composition from x + y, x.y, II y=w then the class of equations F = G is undecidable. 6. Presburger's Algorithm can be used to solve the Integer Linear Programming Problem - the problem was not solved until 1958.