Counterexample Guided Abstraction Refinement Algorithm for Propositional Circumscription

Circumscription is a representative example of a nonmonotonic reasoning inference technique. Circumscription has often been studied for first order theories, but its propositional version has also been the subject of extensive research, having been shown equivalent to extended closed world assumption (ECWA). Moreover, entailment in propositional circumscription is a well-known example of a decision problem in the second level of the polynomial hierarchy. This paper proposes a new Boolean Satisfiability (SAT)-based algorithm for entailment in propositional circumscription that explores the relationship of propositional circumscription to minimal models. The new algorithm is inspired by ideas commonly used in SAT-based model checking, namely counterexample guided abstraction refinement. In addition, the new algorithm is refined to compute the theory closure for generalized close world assumption (GCWA). Experimental results show that the new algorithm can solve problem instances that other solutions are unable to solve

Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satis ability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatis able formulas are equivalent to tree-like resolution proofs. Therefore, lower bounds for tree-like resolution (which are known since 1960s) apply to them

Computing Tiny Clause Normal Forms

International audienceAutomated reasoning systems which build on resolution or superposition typically operate on formulas in clause normal form (CNF). It is well-known that standard CNF translation of a first-order formula may result in an exponential number of clauses. In order to prevent this effect, renaming techniques have been introduced that replace subformulas by atoms over fresh predicates and introduce definitions accordingly. This paper presents generalized renaming. Given a formula and a set of subformulas to be renamed, it is suggested to use one atom to replace all instances of a generalization of a given subformula. A generalized renaming algorithm and an implementation as part of the SPASS theorem prover are described. The new renaming algorithm is faster than the previous one implemented in SPASS. Experiments on the TPTP show that generalized renaming significantly reduces the number of clauses and the average time taken to solve the problems afterward

Relaxed Stratification: A New Approach to Practical Complete Predicate Refinement

Abstract. In counterexample-guided abstraction refinement, a predi-cate refinement scheme is said to be complete for a given theory if it is guaranteed to eventually find predicates sufficient to prove the given property, when such exist. However, existing complete methods require deciding if a proof of the counterexample’s spuriousness exists in some fi-nite language of predicates. Such an exact finite-language-restricted pred-icate search is quite hard for many theories used in practice and incurs a heavy overhead. In this paper, we address the issue by showing that the language restriction can be relaxed so that the refinement process is restricted to infer proofs from some finite language Lbase ∪ Lext but is only required to return a proof when the counterexample’s spurious-ness can be proved in Lbase. Then, we show how a proof-based refine-ment algorithm can be made to satisfy the relaxed requirement and be complete by restricting only the theory-level reasoning in SMT to emit Lbase-restricted partial interpolants (while such an approach has been proposed previously, we show for the first time that it can be done for languages that are not closed under conjunctions and disjunctions). We also present a technique that uses a property of counterexample patterns to further improve the efficiency of the refinement algorithm while still satisfying the requirement. We have experimented with a prototype im-plementation of the new refinement algorithm, and show that it is able to achieve complete refinement with only a small overhead.

On the Complexity of Resolution with Bounded Conjunctions

We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Kraj'icek in [21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreove