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From QBFs to MALL and Back via Focussing
Journal of Automated Reasoning ( IF 1.1 ) Pub Date : 2020-05-22 , DOI: 10.1007/s10817-020-09564-x
Anupam Das

In this work we investigate how to extract alternating time bounds from ‘focussed’ proof systems. Our main result is the obtention of fragments of $$\mathsf {MALL} {\mathsf {w} }$$ MALL w ( $$\mathsf {MALL} $$ MALL with weakening) complete for each level of the polynomial hierarchy. In one direction we encode QBF satisfiability and in the other we encode focussed proof search, and we show that the composition of the two encodings preserves quantifier alternation, yielding the required result. By carefully composing with well-known embeddings of $$\mathsf {MALL} {\mathsf {w} }$$ MALL w into $$\mathsf {MALL} $$ MALL , we obtain a similar delineation of $$\mathsf {MALL} $$ MALL formulas, again carving out fragments complete for each level of the polynomial hierarchy. This refines the well-known results that both $$\mathsf {MALL} {\mathsf {w} }$$ MALL w and $$\mathsf {MALL} $$ MALL are $$\mathbf {PSPACE}$$ PSPACE -complete. A key insight is that we have to refine the usual presentation of focussing to account for deterministic computations in proof search, which correspond to invertible rules that do not branch. This is so that we may more faithfully associate phases of focussed proof search to their alternating time complexity. This presentation seems to uncover further dualities, at the level of proof search, than usual presentations, so could be of proof theoretic interest in its own right.

中文翻译:

从 QBF 到 MALL 再通过 Focussing 返回

在这项工作中,我们研究如何从“聚焦”证明系统中提取交替时间界限。我们的主要结果是获得了 $$\mathsf {MALL} {\mathsf {w} }$$ MALL w ( $$\mathsf {MALL} $$ MALL with weaking) 的片段,对于多项式层次结构的每个级别都是完整的。在一个方向上,我们对 QBF 可满足性进行编码,而在另一个方向上,我们对聚焦证明搜索进行编码,并且我们表明这两种编码的组合保留了量词交替,从而产生了所需的结果。通过将 $$\mathsf {MALL} {\mathsf {w} }$$ MALL w 的众所周知的嵌入组合到 $$\mathsf {MALL} $$ MALL 中,我们获得了 $$\mathsf { MALL} $$ MALL 公式,再次为多项式层次结构的每个级别雕刻出完整的片段。这改进了众所周知的结果 $$\mathsf {MALL} {\mathsf {w} }$$ MALL w 和 $$\mathsf {MALL} $$ MALL 都是 $$\mathbf {PSPACE}$$ PSPACE -完全的。一个关键的见解是,我们必须改进聚焦的通常表示,以解释证明搜索中的确定性计算,这对应于不分支的可逆规则。这样我们就可以更忠实地将重点证明搜索的各个阶段与其交替时间复杂度联系起来。与通常的演示相比,这种演示似乎在证明搜索的层面上揭示了更多的二元性,因此就其本身而言可能具有证明理论的兴趣。对应于不分支的可逆规则。这样我们就可以更忠实地将重点证明搜索的各个阶段与其交替时间复杂度联系起来。与通常的演示相比,这种演示似乎在证明搜索的层面上揭示了更多的二元性,因此就其本身而言可能具有证明理论的兴趣。对应于不分支的可逆规则。这样我们就可以更忠实地将重点证明搜索的各个阶段与其交替时间复杂度联系起来。与通常的演示相比,这种演示似乎在证明搜索的层面上揭示了更多的二元性,因此就其本身而言可能具有证明理论的兴趣。
更新日期:2020-05-22
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