The Horn Non-Clausal Class and its Polynomiality

Abstract

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 H_NC of Horn Non-Clausal (Horn-NC) formulas, by adequately lifting the Horn pattern to NC form, and argue that 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 H_NC; (ii) prove that syntactically H_NC subsumes the Horn class but semantically both classes are equivalent, and (iii) characterize the non-clausal formulas belonging to H_NC. Thirdly, we define the Non-Clausal Unit-Resolution calculus, UR_NC, and prove that it checks the satisfiability of H_NC in polynomial time. This fact, to our knowledge, makes H_NC the first characterized polynomial class in NC reasoning. Finally, we prove that 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 H_NC and UR_NC and that, as a by-product of its proved properties, H_NC arises as a new alternative to analyze Horn functions and implication systems.

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