Capturing the polynomial hierarchy by second-order revised Krom logic

07/19/2022

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by   Shiguang Feng, et al.

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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≥ 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 of SO-EKROM and Π^1_2-EKROM capture co-NP on ordered finite structures.