证明搜索中对抗不确定性的互动与深度
Interaction and Depth against Nondeterminism in Proof Search
摘要：
深度推理是一种证明理论方法，它概括了逻辑演算的标准推理概念，推理规则可以应用于逻辑表达式的任何深度。深度推理为不同逻辑的演绎系统的设计提供了更多的自由度，并提供了丰富的证明组合分析。特别是，指数较短的分析证明的建设成为可能，但成本更大的非决定性比微积分。在本文中，我们表明，证明搜索的不确定性可以减少而不会失去较短的证明，而不牺牲证明理论的清洁。为此，我们在逻辑表达式中采用了交互和深度方案。我们展示了我们的方法在乘法线性逻辑和经典逻辑的深度推理系统，讨论其证明复杂性及其与聚焦的关系，并提出了实现。
英文摘要：
Deep inference is a proof theoretic methodology that generalizes the standard notion of inference of the sequent calculus, whereby inference rules become applicable at any depth inside logical expressions. Deep inference provides more freedom in the design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, however with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we show that the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing proof theoretic cleanliness. For this, we exploit an interaction and depth scheme in the logical expressions. We demonstrate our method on deep inference systems for multiplicative linear logic and classical logic, discuss its proof complexity and its relation to focusing, and present implementations.
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