Capturing the polynomial hierarchy by second-order revised Krom logic
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910327608705024 |
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| author | Wang, Kexu Feng, Shiguang Zhao, Xishun |
| author_facet | Wang, Kexu Feng, Shiguang Zhao, Xishun |
| contents | 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 $Σ^1_1$-KROM$^r$ equals $Σ^1_1$-KROM, and captures NL. On all finite structures, for $k\geq 1$, we show that $Σ^1_{k}$ equals $Σ^1_{k+1}$-KROM$^r$ if $k$ is even, and $Π^1_{k}$ equals $Π^1_{k+1}$-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 $Π^{1}_{2}$-EKROM and equals $Π^1_1$. Both SO-EKROM and $Π^{1}_{2}$-EKROM capture co-NP on ordered finite structures. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_09226 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Capturing the polynomial hierarchy by second-order revised Krom logic Wang, Kexu Feng, Shiguang Zhao, Xishun Logic in Computer Science Computational Complexity 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 $Σ^1_1$-KROM$^r$ equals $Σ^1_1$-KROM, and captures NL. On all finite structures, for $k\geq 1$, we show that $Σ^1_{k}$ equals $Σ^1_{k+1}$-KROM$^r$ if $k$ is even, and $Π^1_{k}$ equals $Π^1_{k+1}$-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 $Π^{1}_{2}$-EKROM and equals $Π^1_1$. Both SO-EKROM and $Π^{1}_{2}$-EKROM capture co-NP on ordered finite structures. |
| title | Capturing the polynomial hierarchy by second-order revised Krom logic |
| topic | Logic in Computer Science Computational Complexity |
| url | https://arxiv.org/abs/2207.09226 |