Insignificant Choice Polynomial Time: A Logic Capturing PTIME
Klaus-Dieter Schewe
TL;DR
This work addresses the question of whether $PTIME$ can be captured by a logic operating on finite structures. It introduces insignificant-choice polynomial time ($ICPT$), a restriction of nondeterministic Abstract State Machines that enforces atom-only nondeterminism, update-set isomorphisms, and a branching-consistency condition to guarantee global insignificance. The authors develop bounded-exploration witnesses and reduction lemmas to show that branching can be checked in polynomial time and that $ICPT$ both simulates any $PTIME$ computation and is simulatable in $PTIME$, yielding a true $PTIME$ logic on structures. Collectively, the results establish that $ICPT$ captures $PTIME$, thereby refuting Gurevich's conjecture and linking descriptive complexity with restricted nondeterministic ASM semantics in a structurally robust way.
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)