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Polynomial-time functions generate SAT: On P-splinters

  • Lane A. Hemachandra
  • Albrecht Hoene
  • Dirk Siefkes
Communications
Part of the Lecture Notes in Computer Science book series (LNCS, volume 379)

Abstract

We ask if there is a polynomial-time computable function f and an initial element x0 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 x0 and y0 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Lane A. Hemachandra
    • 1
  • Albrecht Hoene
    • 2
  • Dirk Siefkes
    • 2
  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA
  2. 2.Technische Universität BerlinBerlin 10

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