A Restricted Second-Order Logic for Non-deterministic Poly-Logarithmic Time

TL;DR

This paper introduces a restricted second-order logic, $ ext{SO}^{plog}$, characterizing non-deterministic poly-logarithmic time complexity over ordered structures, and establishes a correspondence with the poly-logarithmic time hierarchy.

Contribution

It defines $ ext{SO}^{plog}$ and proves a Fagin-style theorem linking its existential fragment to non-deterministic poly-logarithmic time, extending descriptive complexity theory.

Findings

$ ext{SO}^{plog}$ captures problems in non-deterministic poly-logarithmic time.

Existential fragment of $ ext{SO}^{plog}$ corresponds exactly to NP in poly-logarithmic time.

Exact correspondence between quantifier prefix classes and the poly-logarithmic hierarchy.

Abstract

We introduce a restricted second-order logic $\mathrm{SO}^{\mathit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin's style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\mathrm{SO}^{\mathit{plog}}$ corresponds exactly to the class of decision problems that can be computed by a non-deterministic Turing machine with random access to the input in time $O((\log n)^k)$ for some $k \ge 0$, i.e., to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin's theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\mathrm{SO}^{\mathit{plog}}$ is too weak as to define a total order of the domain. Nevertheless $\mathrm{SO}^{\mathit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\mathrm{SO}^{\mathit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail.