Polynomial-time functions generate SAT: On P-splinters

Abstract

We ask if there is a polynomial-time computable function f and an initial element x ₀ such that

$$SAT = \left\{ {x_0 , f(x_0 ), f(f(x_0 )), ...} \right\}.$$

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.

This work was done in part while the first author was at Columbia University, and in part while the authors visited Gerd Wechsung in Jena.

Research supported by NSF grants CCR-8809174 and CCR-8996198, a Hewlett-Packard Corporation equipment grant, and NSF Presidential Young Investigator Award CCR-8957604.