Resolución Eficiente de Horn-SAT mediante Algoritmos Bottom-Up y Top-Down

Abstract

Propositional Horn satisfiability (Horn-SAT) is fundamental in artificial intelligence, representing a pillar in main fields: (i) query/answering propositional logic programs, (ii) unit propagation in modern SAT solvers, and (iii) modeling through directed hypergraphs (And/Or-graphs). Horn-SAT constitutes the most prominent tractable problem characterized by formulas with Horn clauses containing at most one positive literal, reducing complexity from exponential to linear. Although W.F. Dowling and J.H. Gallier (1984) developed a pioneering Linear Bottom-Up (LBU) algorithm, it presents the fundamental limitation of inferring almost all logical consequences regardless of the specific query, being impractical for expressive logics where it generates potentially infinite models. Previous efforts on linear top-down algorithms have resulted incomplete or technically insufficient, leaving a critical gap in the literature. This work formulates and implements a basic version of the Top-Down (LTD) algorithm for Horn-SAT without considering cycles, solving fundamental deficiencies of previous approaches through query-driven marking strategies. The main contribution consists in offering a correctly specified implementation of the basic top-down approach along with an exhaustive comparative analysis that demonstrates when each strategy is superior. Experimental evaluation uses parametrized Horn formulas as benchmarks (complete, incremental, and matrix) that expose challenges inherent to the Horn class, demonstrating that relative efficiency depends fundamentally on topological characteristics: LTD outperforms LBU in incremental structures by effectively restricting the query-driven search space, while LBU retains advantages in configurations where recursion overhead impacts LTD's practical performance.