In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function f is called the inversion complexity of f . In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that ⌈log 2 (n + 1)⌉ NO...

We present a simplified proof that Brent/Spira restructuring of Boolean formulas can be improved to allow a Boolean formula of size n to be transformed into an equivalent log depth formula of size O(nα) for arbitrary α > 1.

theory atoms LTL(Th) formula φ φabs pseudo-Boolean input variables Xabs Input variables X Output variables Y pseudo-Boolean LTL LTL controller check whether the input formula is realizable synthesis

We formulate and prove a resolution-like theorem that reduces the unnegated variables of a Boolean formula in Conjunctive Normal Form by one, while preserving the maximal models of the formula. This has an impact to the generation of all maximal models of a Horn formula as repeated applications of the theorem lead to a Boolean formula in which each clause has only negated variables and thus red...

We define formalisms to reason about Coalitional Games (CGs), in which one can express what coalitions of agents can achieve. We start with Quantified CGs (QCGs), in which each agent has some goals he wants to satisfy, which may change over time. Then we focus on CGs themselves. Although CGs can be well analysed in a formalism close to Pauly’s Coalition Logic, in QCGs, when having preferences, ...

In this paper we describe the QBFEVAL web portal, an online resource supporting the participants and the organizers of the yearly evaluation of QBF solvers and instances.

In this paper we report about QBFEVAL’10, the seventh in a series of events established with the aim of assessing the advancements in reasoning about quantified Boolean formulas (QBFs). The paper discusses the results obtained and the evaluation setup, from the criteria used to select QBF instances down to the hardware infrastructure. We also discuss the current state-of-the-art in light of pas...

In this paper we approach the problem of reasoning with quantified Boolean formulas (QBFs) by combining search and resolution, and by switching between them according to structural properties of QBFs. We provide empirical evidence that QBFs which cannot be solved by search or resolution alone, can be solved by combining them, and that our approach makes a proof-of-concept implementation competi...

A new method for analyzing relational specifications is described. A property to be checked is cast as a relational formula, which, if the property holds, has no finite models. The relational formula is translated into a boolean formula that has a model for every model of the relational formula within some finite scope. Errors in specifications can usually be demonstrated with small counterexam...

In this paper we compare the performance of all the currently available suites to evaluate and certify QBFs. Our aim is to assess the current state of the art, and also to understand to which extent QBF encodings can be evaluated producing certificates that can be checked in a reliable and efficient way. We conclude that, while the evaluation of some QBFs is still an open challenge, producing a...