An analytic condition for P ⊂ NP Jos´ e F´ elix Costa * Jerzy Mycka † March 23, 2006 Abstract In this paper, we prove that there exists some condition, involving real functions, which implies P = NP . 1 Introduction and motivation The theory of analog computation, where the internal states of a computer are continuous rather than discrete, has enjoyed a recent resurgence of interest. In this historical framework we go back to Claude Shannon’s (see [17]) so-called General Purpose Analog Computer (GPAC). This was defined as a mathemati- cal model of an analog device, the Differential Analyser, the fundamental prin- ciples of which were described by Lord Kelvin in 1876. We have been working recently towards inductive definitions of generalized GPAC functions over R. The first presentation of such a theory, analogous to Kleene’s classical theory of recursive functions over N, was attempted by Cristopher Moore [10]. Real recursive functions are generated by a fundamental operator, called differential recursion. (The other fundamental operator is the taking of infinite limits, introduced in [11].) Let us recall the concept of a real recursive function, and the corresponding class REC(R), as was introduced in [12]. The class REC(R) of real recursive vectors is generated from the real recursive scalars 0, 1, and −1, and the real recursive projections, by the following operators: composition, differential recursion (the solution of a Cauchy problem or a initial value problem in mathematical analysis), and infinite limits. If we consider a wider context for solving differential equations, then it is possible to justify the concept of generalized solution of a given differential equa- tion. We present in this paper the concept of (restricted) real recursive function and the corresponding class REC R (R), which is based on the above definition of real recursive function, without infinite limits, with a new understanding of differential recursion. * Department of Mathematics, I.S.T., Universidade T´ ecnica de Lisboa, Lisboa, Portugal, fgc@math.ist.utl.pt † Institute of Mathematics, University of Maria Curie-Sklodowska, Lublin, Poland, Jerzy.Mycka@umcs.lublin.pl, corresponding author 1