Choiceless polynomial time

TL;DR

This paper explores a new computational model based on choiceless polynomial time, aiming to characterize PTime without arbitrary choice, and introduces a more expressive logic that partially captures PTime but does not encompass it entirely.

Contribution

It proposes a novel choiceless polynomial time model using parallel execution in abstract state machines, expanding the landscape of PTime logics.

Findings

The new logic surpasses existing PTime logics in expressiveness.

The logic cannot fully capture all PTime properties.

Parallel execution replaces arbitrary choice in the model.

Abstract

Turing machines define polynomial time (PTime) on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time (without presuming the presence of a linear order)? Earlier, one of us conjectured the negative answer. The problem motivated a quest for stronger and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines (formerly called evolving algebras). The idea is to replace arbitrary choice with parallel execution. The resulting logic is more expressive than other PTime logics in the literature. A more difficult theorem shows that the logic does not capture all PTime.