# Polynomial-time functions generate SAT: On P-splinters

## Abstract

*f*and an initial element

*x*

_{0}such that

We show that the answer is yes. Indeed, there is a single polynomial-time computable function *f* that generates both *SAT* and its complement \(\overline {SAT}\): There are elements *x*_{0} and *y*_{0} such that \(SAT = \left\{ {x_0 , f(x_0 ), .. } \right\} \overline {SAT} = \left\{ {y_0 , f(y_0 ), .. } \right\}\). Though the description of such functions as generators is by analogy with abstract algebra, such functions are polynomial-time analogues of the recursion-theoretic notion of splinters. Thus, *SAT* is a *P*-splinter, and in fact is a *P*-bisplinter.

Indeed, we show that all recursive *P*-cylinders are *P*-bi-splinters. We observe that the converse does not hold. Relatedly, we study honest *P*-splinters and conclude that, in a certain sense, *SAT* is arbitrarily close to being an honest *P*-bi-splinter. Nonetheless, we present strong structural evidence that many problems are not monotonic P-bi-splinters.

## Keywords

Polynomial Time Base Column Initial Element Polynomial Time Computa Target Column## Preview

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