Capturing the polynomial hierarchy by second-order revised Krom logic
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Published:2023-07-14
Issue:
Volume:Volume 19, Issue 3
Page:
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ISSN:1860-5974
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Container-title:Logical Methods in Computer Science
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language:en
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Short-container-title:
Author:
Wang Kexu,Feng Shiguang,Zhao Xishun
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.
Publisher
Centre pour la Communication Scientifique Directe (CCSD)
Subject
General Computer Science,Theoretical Computer Science