The Horn Non-Clausal Class and its Polynomiality
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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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