Finding solution values for unknowns in Boolean equations was, along with second-order quantifier elimination, a principal reasoning mode in the Algebra of Logic of the 19th century. Schröder [19] investigated it as Auflösungsproblem (solution problem). It is closely related to the modern notion of Boolean unification. For a given formula that contains unknowns formulas are sought such that aft...

The first hard problem we will examine is what is known as Satisfiability or SAT. As input, we are given a set of n boolean variables X = {x1, x2, . . . , xn} (i.e., each variable can be set to either true or false). We are then given a boolean formula over these variables of the following form (noting that this is just a specific example where X = {x1, x2, x3, x4}): (x1 ∨ x2 ∨ x3) ∧ (x2 ∨ x3 ∨...

We show that Nechiporuk’s method [26] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an Ω(n/ log n) lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [27] states that the majority function does not have a linear–size quantum formula. We also show that quantum formul...

It is difficult to find Boolean functions achieving many good cryptographic properties. Recently, Tu and Deng obtained two classes of Boolean functions with good properties based on a combinatorial conjecture about binary strings. In this paper, using different approaches, we prove this conjecture is true in some cases. This conjecture has resisted different attempts of proof since it is hard t...

Boolean networks (BNs) are discrete-time dynamical systems with Boolean state-variables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks with Boolean inputs. We consider the problem of steering a BCN from a given state to a desired stat...

In this paper we provide an explicit formula for calculating the boolean number of a Ferrers graph. By previous work of the last two authors, this determines the homotopy type of the boolean complex of the graph. Specializing to staircase shapes, we show that the boolean numbers of the associated Ferrers graphs are the Genocchi numbers of the second kind, and obtain a relation between the Legen...

Symbolic decision trees are not the only way to correlate the relationship between flags and numeric variables. Boolean formulae can also represent such relationships where the integer variables are modelled with bit-vectors of propositional variables. Boolean formula can be composed to express the semantics of a block and program state, but they are hardly tractable, hence the need to compute ...

Recently, the performance of planners has been increased significantly. Many of the new methods responsible for the speed-up have used SAT-compilation techniques. These methods convert the planning problem specification into a Boolean CNF formula, which is then solved by a range of fast SAT solvers. However, some of the properties of the planning problems can be more concisely expressed using a...

Consider the problem of deciding whether a Boolean formula f is self-dual, i.e. f is logically equivalent to its dual formula f d , deened by f d (x) = f (x). This problem is a well-studied problem in several areas like theory of coteries, database theory, hypergraph theory or computational learning theory. In this paper we exhibit polynomial time algorithms for testing self-duality for several...

While it is empirically confirmed folklore that conjunctive normal form (CNF) is not the ideal input format for QBF solvers, most tool developers and therefore also the users focus on formulas in this restricted structure. One important factor for establishing non-CNF solving is the input format. To overcome drawbacks of available formats, the QCIR format has recently been presented. The QCIR f...