Insignificant Choice Polynomial Time: A Logic Capturing PTIME

Abstract

In this article choiceless polynomial time (CPT) is extended using non-determini\-stic Abstract State Machines (ASMs), which are restricted by three conditions: (1) choice is restricted to choice among atoms; (2) update sets in a state must be isomorphic; (3) for any two isomorphic update sets on states $S$ and $S^\prime$, respectively, the sets of update sets of the corresponding successor states are isomorphic. The restrictions can be incorporated into the semantics of ASM rules such that update sets are only yielded, if the conditions are satisfied. Furthermore, the conditions can be checked in polynomial time on a simulating Turing machine. Finally, the conditions imply global insignificance, i.e. the final result is independent from the choices. These properties suffice to show that the ASMs restricted this way define a logic capturing PTIME, which we call insignificant choice polynomial time (ICPT)