The Possibilistic Horn Non-Clausal Knowledge Bases

Posibilistic logic is the most extended approach to handle uncertain and partially inconsistent information. Regarding normal forms, advances in possibilistic reasoning are mostly focused on clausal form. Yet, the encoding of real-world problems usually results in a non-clausal (NC) formula and NC-to-clausal translators produce severe drawbacks that heavily limit the practical performance of clausal reasoning. Thus, by computing formulas in its original NC form, we propose several contributions showing that notable advances are also possible in possibilistic non-clausal reasoning. {\em Firstly,} we define the class of {\em Possibilistic Horn Non-Clausal Knowledge Bases,} or $\mathcal{\overline{H}}_\Sigma$, which subsumes the classes: possibilistic Horn and propositional Horn-NC. $\mathcal{\overline{H}}_\Sigma $ is shown to be a kind of NC analogous of the standard Horn class. {\em Secondly}, we define {\em Possibilistic Non-Clausal Unit-Resolution,} or $ \mathcal{UR}_\Sigma $, and prove that $ \mathcal{UR}_\Sigma $ correctly computes the inconsistency degree of $\mathcal{\overline{H}}_\Sigma $members. $\mathcal{UR}_\Sigma $ had not been proposed before and is formulated in a clausal-like manner, which eases its understanding, formal proofs and future extension towards non-clausal resolution. {\em Thirdly}, we prove that computing the inconsistency degree of $\mathcal{\overline{H}}_\Sigma $ members takes polynomial time. Although there already exist tractable classes in possibilistic logic, all of them are clausal, and thus, $\mathcal{\overline{H}}_\Sigma $ turns out to be the first characterized polynomial non-clausal class within possibilistic reasoning.

A First Polynomial Non-Clausal Class in Many-Valued Logic

The relevance of polynomial formula classes to deductive efficiency motivated their search, and currently, a great number of such classes is known. Nonetheless, they have been exclusively sought in the setting of clausal form and propositional logic, which is of course expressively limiting for real applications. As a consequence, a first polynomial propositional class in non-clausal (NC) form has recently been proposed. Along these lines and towards making NC tractability applicable beyond propositional logic, firstly, we define the Regular many-valued Horn Non-Clausal class, or RH, obtained by suitably amalgamating both regular classes: Horn and NC. Secondly, we demonstrate that the relationship between (1) RH and the regular Horn class is that syntactically RH subsumes the Horn class but that both classes are equivalent semantically; and between (2) RH and the regular non-clausal class is that RH contains all NC formulas whose clausal form is Horn. Thirdly, we define Regular Non-Clausal Unit-Resolution, or RUR-NC , and prove both that it is complete for RH and that checks its satisfiability in polynomial time. The latter fact shows that our intended goal is reached since RH is many-valued, non-clausal and tractable. As RH and RUR-NC are, both, basic in the DPLL scheme, the most efficient in propositional logic, and can be extended to some other non-classical logics, we argue that they pave the way for efficient non-clausal DPLL-based approximate reasoning.

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.