The Horn Non-Clausal Class and its Polynomiality

The expressiveness of propositional non-clausal (NC) formulas is exponentially richer than that of clausal formulas. Yet, clausal efficiency outperforms non-clausal one. Indeed, a major weakness of the latter is that, while Horn clausal formulas, along with Horn algorithms, are crucial for the high efficiency of clausal reasoning, no Horn-like formulas in non-clausal form had been proposed. To overcome such weakness, we define the hybrid class $\mathbb{H_{NC}}$ of Horn Non-Clausal (Horn-NC) formulas, by adequately lifting the Horn pattern to NC form, and argue that $\mathbb{H_{NC}}$, along with future Horn-NC algorithms, shall increase non-clausal efficiency just as the Horn class has increased clausal efficiency. Secondly, we: (i) give the compact, inductive definition of $\mathbb{H_{NC}}$; (ii) prove that syntactically $\mathbb{H_{NC}}$ subsumes the Horn class but semantically both classes are equivalent, and (iii) characterize the non-clausal formulas belonging to $\mathbb{H_{NC}}$. Thirdly, we define the Non-Clausal Unit-Resolution calculus, $UR_{NC}$, and prove that it checks the satisfiability of $\mathbb{H_{NC}}$ in polynomial time. This fact, to our knowledge, makes $\mathbb{H_{NC}}$ the first characterized polynomial class in NC reasoning. Finally, we prove that $\mathbb{H_{NC}}$ is linearly recognizable, and also that it is both strictly succincter and exponentially richer than the Horn class. We discuss that in NC automated reasoning, e.g. satisfiability solving, theorem proving, logic programming, etc., can directly benefit from $\mathbb{H_{NC}}$ and $UR_{NC}$ and that, as a by-product of its proved properties, $\mathbb{H_{NC}}$ arises as a new alternative to analyze Horn functions and implication systems.