New Models for Generating Hard Random Boolean Formulas and Disjunctive Logic Programs

We propose two models of random quantified boolean formulas and their natural random disjunctive logic program counterparts. The models extend the standard models of random k-CNF formulas and the Chen-Interian model of random 2QBFs. The first model controls the generation of programs and QSAT formulas by imposing a specific structure on rules and clauses, respectively. The second model is based on a family of QSAT formulas in a non-clausal form. We provide theoretical bounds for the phase transition region in our models, and show experimentally the presence of the easy-hard-easy pattern and its alignment with the location of the phase transition. We show that boolean formulas and logic programs from our models are significantly harder than those obtained from the standard k-CNF and Chen-Interian models, and that their combination yields formulas and programs that are “super-hard” to evaluate. We also provide evidence suggesting that formulas from one of our models are well suited for assessing solvers tuned to real-world instances. Finally, it is noteworthy that, to the best of our knowledge, our models and results on random disjunctive logic programs are the first of their kind.