All of the reductions currently used in complexity theory (≤p, ≤γ, ≤R) have the property that they are honest. If A ≤ B then whatever machine M reduces A to B is such that: if on input x, M outputs y then x ε A ↔ y ε B. It would appear that this membership preserving property is intrinsic to the notion of reduction. We will see that it is not. We introduce reductions that lie and sometimes produce outputs y ε B when x ? A. We will use these reductions to further clarify the computational complexity of some problems raised by Gauss.

Abstract An infinitary characterisation of the first-order sentences true in all substructures of a structure M is used to obtain partial reduction of the decision problem for such sentences to that for Th(M). For the relational structure 〈R, ≤, + 〉 this gives a decision procedure for the ∃x∀y-part of the theory of all substructures, yet we show that the ∃x1x2∀y-part, and the entire theory, is Π11-complete. The theory of all ordered subsemigroups of 〈R, ≤, + 〉 is also shown Π11-complete. Applications in the philosophy of science are mentioned.

The method of showing a problem NP-complete by polynomial reduction is one of the most elegant and productive in our theory ([ 1 ], [ 3 ]). It is a means of providing compelling evidence that a problem in NP is not in P. In this paper we will demonstrate new methods for showing this.

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.