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 shape 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 propositional variables as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each propositional variable into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of all the constructions in this paper. To accommodate self-dual non-commutativity, we compose proofs by deep inference.

中文翻译：

---

决策树的亚原子证明系统

我们为命题经典逻辑设计了一个证明系统，该系统集成了两种用于布尔函数的语言：标准合取与非运算和二进制决策树。我们给出两个理由。首先是证明理论的自然性：该系统仅由最近引入的亚原子逻辑的单一，简单，线性形状所产生的所有推理规则组成。由于这种规律性，可以通过自然的结构消除割伤。第二个原因是系统生成有效的证明。确实，我们证明了由于Statman而导致的某些重言式，在随后的演算中，没有比指数免割证明更好的了，在我们的系统中具有多项式免割证明。我们使用与消除切割相同的构造来实现这一目标。总之，通过扩展命题逻辑的语言，我们使它的证明理论更加规范，并产生了更多的证明，其中有些非常有效。通过将命题变量视为其真值的叠加，使这种设计成为可能，这些命题变量通过自对偶，非交换性连接词进行连接。然后可以通过每个命题变量将一个证明投影到两个证明中，每个真值都需要一个证明，而无需削减。这些预测在语义上是自然的，并且是本文所有结构的核心。为了适应自对偶的不可交换性，我们通过深入的推理来构成证明。通过将命题变量视为其真值的叠加，使这种设计成为可能，这些命题变量通过自对偶，非交换性连接词进行连接。然后可以通过每个命题变量将一个证明投影到两个证明中，每个真值都需要一个证明，而无需削减。这些预测在语义上是自然的，并且是本文所有结构的核心。为了适应自对偶的不可交换性，我们通过深入的推理来构成证明。通过将命题变量视为其真值的叠加，使这种设计成为可能，这些命题变量通过自对偶，非交换性连接词进行连接。然后可以通过每个命题变量将一个证明投影到两个证明中，每个真值都需要一个证明，而无需削减。这些预测在语义上是自然的，并且是本文所有结构的核心。为了适应自对偶的不可交换性，我们通过深入的推理来构成证明。