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Provability in BI's Sequent Calculus is Decidable
arXiv - CS - Symbolic Computation Pub Date : 2021-03-03 , DOI: arxiv-2103.02343 Alexander Gheorghiu, Simon Docherty, David Pym
arXiv - CS - Symbolic Computation Pub Date : 2021-03-03 , DOI: arxiv-2103.02343 Alexander Gheorghiu, Simon Docherty, David Pym
The logic of Bunched Implications (BI) combines both additive and
multiplicative connectives, which include two primitive intuitionistic
implications. As a consequence, contexts in the sequent presentation are not
lists, nor multisets, but rather tree-like structures called bunches. This
additional complexity notwithstanding, the logic has a well-behaved metatheory
admitting all the familiar forms of semantics and proof systems. However, the
presentation of an effective proof-search procedure has been elusive since the
logic's debut. We show that one can reduce the proof-search space for any given
sequent to a primitive recursive set, the argument generalizing Gentzen's
decidability argument for classical propositional logic and combining key
features of Dyckhoff's contraction-elimination argument for intuitionistic
logic. An effective proof-search procedure, and hence decidability of
provability, follows as a corollary.
中文翻译:
BI的后续演算中的可证明性是可以确定的
束缚蕴涵(BI)的逻辑结合了加性和乘积性连词,其中包括两个原始的直觉性蕴涵。因此,后续演示中的上下文不是列表,也不是多集,而是树状结构,称为束。尽管存在这种额外的复杂性,但该逻辑具有一个行为良好的元理论,可以接受所有熟悉形式的语义和证明系统。然而,自逻辑学问世以来,有效的证明搜索程序的介绍一直难以捉摸。我们表明,对于原始递归集而言,对于给定的任何给定序列,可以减少证明搜索空间,该论点推广了经典命题逻辑的Gentzen可判定性论点,并结合了直觉逻辑的Dyckhoff消除收缩论点的关键特征。
更新日期:2021-03-04
中文翻译:
BI的后续演算中的可证明性是可以确定的
束缚蕴涵(BI)的逻辑结合了加性和乘积性连词,其中包括两个原始的直觉性蕴涵。因此,后续演示中的上下文不是列表,也不是多集,而是树状结构,称为束。尽管存在这种额外的复杂性,但该逻辑具有一个行为良好的元理论,可以接受所有熟悉形式的语义和证明系统。然而,自逻辑学问世以来,有效的证明搜索程序的介绍一直难以捉摸。我们表明,对于原始递归集而言,对于给定的任何给定序列,可以减少证明搜索空间,该论点推广了经典命题逻辑的Gentzen可判定性论点,并结合了直觉逻辑的Dyckhoff消除收缩论点的关键特征。