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：

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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。