We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: the system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this paper. To accommodate self-dual non-commutativity, we compose proofs in deep inference.
翻译:我们设计了一种理论经典逻辑的验证系统, 将布林函数的两种语言融合在一起: 标准交错否定和二进制决定树。 我们给出了两种理由。 首先, 证明- 理论性自然性: 系统由最近引入的亚原子逻辑的单一、 简单、 线性计划产生的所有推论规则组成, 而由于这种规律性, 削减会通过自然构造而消除。 第二个理由是, 系统产生有效的证明。 事实上, 我们显示, 由斯塔曼 带来的某种深层次调调调调调, 它不能比序列计算中的指数性断线性证明更好。 我们这样做有两个理由。 首先是证明- 证明- 理论性断线性证明: 系统是由所有单一、 简单、 线性推导法逻辑产生的。 简而言之, 我们通过扩展推理性逻辑语言, 使证据理论理论更经常化, 并产生更多证据, 其中有一些非常有效的。 设计之所以成为可能, 是因为将数据视为其真实价值的超置位置, 由非自我、 非直接的校准的校准的计算, 可以通过两种推论来推论来推算。