Capturing the polynomial hierarchy by second-order revised Krom logic

TL;DR

This paper explores the expressive power of second-order revised Krom logic, showing it captures the polynomial hierarchy and related complexity classes on finite structures, with new variants collapsing to known logics.

Contribution

It introduces and analyzes second-order revised Krom logic and its extended version, demonstrating their ability to characterize the polynomial hierarchy and other complexity classes.

Findings

SO-KROM$^{r}$ captures NL on ordered structures.

Hierarchy collapses for even and odd levels of the polynomial hierarchy.

SO-EKROM collapses to $ ext{co-NP}$ and $ ext{Pi}^1_1$ on ordered structures.

Abstract

We study the expressive power and complexity of second-order revised Krom logic (SO-KROM$^{r}$). On ordered finite structures, we show that its existential fragment $\Sigma^1_1$-KROM$^r$ equals $\Sigma^1_1$-KROM, and captures NL. On all finite structures, for $k\geq 1$, we show that $\Sigma^1_{k}$ equals $\Sigma^1_{k+1}$-KROM$^r$ if $k$ is even, and $\Pi^1_{k}$ equals $\Pi^1_{k+1}$-KROM$^r$ if $k$ is odd. The result gives an alternative logic to capture the polynomial hierarchy. We also introduce an extended version of second-order Krom logic (SO-EKROM). On ordered finite structures, we prove that SO-EKROM collapses to $\Pi^{1}_{2}$-EKROM and equals $\Pi^1_1$. Both SO-EKROM and $\Pi^{1}_{2}$-EKROM capture co-NP on ordered finite structures.