Abstract In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) for obtaining a propositional abstraction of the QBF. If this formula is false, the truth value of the QBF is decided, otherwise further refinement steps are necessary. Classically, expansion-based solvers process the given formula quantifier-block wise and use one SAT solver per quantifier block. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided and only two incremental SAT solvers are required. While our algorithm is naturally based on the $$\forall $$ ∀ Exp+Res calculus that is the formal foundation of expansion-based solving, it is conceptually simpler than present recursive approaches. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques.

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) and pass the obtained formula to a SAT solver for deciding the QBF. State-of-the-art expansion-based solvers process the given formula quantifier-block wise and recursively apply expansion until a solution is found. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques.

Reactive synthesis supports designers by automatically constructing correct hardware from declarative specifications. Synthesis algorithms usually compute a strategy, and then construct a circuit that implements it. In this work, we study SAT- and QBF-based methods for the second step, i.e., computing circuits from strategies. This includes methods based on QBF-certification, interpolation, and computational learning. We present optimizations, efficient implementations, and experimental results for synthesis from safety specifications, where we outperform BDDs both regarding execution time and circuit size.

We consider logic-based argumentation in which an argument is a pair (Φ, α), where the support Φ is a minimal consistent set of formulae taken from a given knowledge base (usually denoted by Δ) that entails the claim α (a formula). We study the complexity of three central problems in argumentation: the existence of a support Φ⊆Δ, the verification of a support, and the relevance problem (given ψ, is there a support Φ such that ψ ∈ Φ?). When arguments are given in the full language of propositional logic, these problems are computationally costly tasks: the verification problem is DP-complete; the others are Σ p 2 -complete. We study these problems in Schaefer's famous framework where the considered propositional formulae are in generalized conjunctive normal form. This means that formulae are conjunctions of constraints built upon a fixed finite set of Boolean relations Γ (the constraint language). We show that according to the properties of this language Γ, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete, or Σ p 2 -complete. We present a dichotomous classification, P or DP-complete, for the verification problem and a trichotomous classification for the relevance problem into either polynomial, NP-complete, or Σ p 2 -complete. These last two classifications are obtained by means of algebraic tools.

We consider logic-based argumentation in which an argument is a pair (Φ, α), where the support Φ is a minimal consistent set of formulæof a given knowledge base that entails the formula α. We study the complexity of two different problems: the existence of a support and the verification of the validity of an argument. When arguments are given in the full language of propositional logic these problems are computationally costly tasks, they are respectively ΣP2- and DP-complete. We study these problems in Schaefer's famous framework. We consider the case where formulæare taken from a class of formulæin generalized conjunctive normal form. This means that the propositional formulæ considered are conjunctions of constraints taken from a fixed finite language Γ. We show that according to the properties of this language Γ, deciding whether there exists a support for a claim in a given knowledge base is either polynomial, NP-complete, coNP-complete or ΣP2

The QXORSAT problem is the quantified version of the satisfiability problem XORSAT in which the connective exclusive-or is used instead of the usual or. We study the phase transition associated with random QXORSAT instances. We give a description of this phase transition in the case of one alternation of quantifiers, thus performing an advanced practical and theoretical study on the phase transition of a quantified roblem.

It is well-known that almost all nonmonotonic formalisms have a higher worst-case complexity than classical reasoning. In some sense, this observation denies one of the original motivations of nonmonotonic systems, which was the expectation taht nonmonotonic rules should help to speed-up the reasoning process, and not make it more difficult. In this paper, we look at this issue from a proof-theoretical perspective. We consider analytic calculi for certain nonmonotonic logis and analyze to what extent the presence of nonmonotonic rules can simplify the search for proofs. In particular, we show that there are classes of first-order formulae which have only extremely long “classical” proofs, i.e., proofs without applications of nonmonotonic rules, but there are short proofs using nonmonotonic inferences. Hence,despite the increase of complexity in the worst case, there are instances where nonmonotonic reasoning can be much simpler than classical (cut-free) reasoning.

We present a translation of intuitionistic sequent proofs from a multi-succedent calculus ℒ𝒥mc into a single-succedent calculus ℒ𝒥. The former gives a basis for automated proof search whereas the latter is better suited for proof presentation and p

In this paper, we examine different definitional transformations into normal form for intuitionistic logic. In contrast to the classical case, “intuitionistic clauses” may contain implications and quantifiers. Usually, such definitional transformations introduce labels defining subfor-mulae. An obvious optimization is the use of implications instead of equivalences whenever possible, which can reduce the size of the resulting normal form. We compare the optimized transformation with the unoptimized transformation with respect to the shortest cut-free LJ-derivation of the resulting normal forms. The comparison is based on a sequence (H k ) k∈N of formulae for which the following hold: (i) there exist cut-free LJ-proofs of the unoptimized normal form of H k with length double-exponential in k, and (ii) any cut-free LJ-proof of the optimized normal form of H k has length non-elementary in k. The reason for the different behaviour is the simulation of analytic cuts by the unoptimized translation, which is not possible when the optimization is applied.