The complexity of Boolean formula minimization

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be Σ^P_2-complete and indeed appears as an open problem in Garey and Johnson (1979) [5]. The depth-2 variant was only shown to be Σ^P_2-complete in 1998 (Umans (1998) [13], Umans (2001) [15]) and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is Σ^P_2-complete under Turing reductions for all k ≥ 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ^P_2-complete under Turing reductions

Improved inapproximability factors for some Σ^p₂ minimization problems

We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are Σ^p₂-hard to approximate to within factors of n^(1/3−ϵ) and ^(n1/2−ϵ) (where the previous results achieved n^(1/4−ϵ)), for arbitrarily small constant ϵ > 0. For one problem shown to be inapproximable to within n^(1/2−ϵ), we give a matching O(n^(1/2))-approximation algorithm, running in randomized polynomial time with access to an NP oracle, which shows that this result is tight assuming the PH doesn't collapse

Redundancy in Logic I: CNF Propositional Formulae

A knowledge base is redundant if it contains parts that can be inferred from the rest of it. We study the problem of checking whether a CNF formula (a set of clauses) is redundant, that is, it contains clauses that can be derived from the other ones. Any CNF formula can be made irredundant by deleting some of its clauses: what results is an irredundant equivalent subset (I.E.S.) We study the complexity of some related problems: verification, checking existence of a I.E.S. with a given size, checking necessary and possible presence of clauses in I.E.S.'s, and uniqueness. We also consider the problem of redundancy with different definitions of equivalence.Comment: Extended and revised version of a paper that has been presented at ECAI 200

Explainability via Short Formulas: the Case of Propositional Logic with Implementation

We conceptualize explainability in terms of logic and formula size, giving a number of related definitions of explainability in a very general setting. Our main interest is the so-called special explanation problem which aims to explain the truth value of an input formula in an input model. The explanation is a formula of minimal size that (1) agrees with the input formula on the input model and (2) transmits the involved truth value to the input formula globally, i.e., on every model. As an important example case, we study propositional logic in this setting and show that the special explainability problem is complete for the second level of the polynomial hierarchy. We also provide an implementation of this problem in answer set programming and investigate its capacity in relation to explaining answers to the n-queens and dominating set problems.publishedVersionPeer reviewe