Set constraints are inclusions between expressions denoting sets of trees. In atomic set constraints, the syntax of set expressions is restricted not to contain any Boolean set operators. Using a reduction from the Hilbert's Tenth Problem we prove the undecidability of the 9 8-fragment of the rst-order theory of atomic set constraints. This is the minimal undecidable fragment of the rst-order t...

In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. Some special classes of Boolean algebras are also concerned with and, in particular, the relations of Boolean algebras to propositional logic and to set theory are studied in more details. We mention here that all what we do for propositional logic can also be extende...

We present several results that show that systematic (complete) judgment aggregators can be viewed as both (2-valued) Boolean homomorphisms and as syntatic versions of reduced (ultra)products. Thereby, Arrovian judgment aggregators link the Boolean algebraic structures of (i) the set of coalitions (ii) the agenda, and (iii) the set of truth values of collective judgments. Since filters arise na...

The Boolean Differential Calculus is a powerful theory that extends the Boolean algebra significantly. Based on a small number of definitions a lot of theorems were proven. The available operations have been efficiently implemented in several software packages. There is a very wide field of applications. In this paper we combine a compact introduction into the Boolean Differential Calculus with...

The objective of this paper is to uncover the structure of the back-andforth equivalence classes at the finite levels for the class of Boolean algebras. As an application, we obtain bounds on the computational complexity of determining the back-and-forth equivalence classes of a Boolean algebra for finite levels. This result has implications for characterizing the relatively intrinsically Σn re...

We show that a theorem of Leonid B. Shapiro which was proved under MA, is actually independent from ZFC. We also give a direct proof of the Boolean algebra version of the theorem under MA(Cohen).

Boolean circuits are used to represent programs on finite data. Reversible Boolean circuits and quantum Boolean circuits have been introduced to modelize some physical aspects of computation. Those notions are essential in complexity theory, but we claim that a deep mathematical theory is needed to make progress in this area. For that purpose, the recent developments of knot theory is a major s...

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 ...

We obtain a simple, purely game-theoretic characterization of Boolean grammars [A. Okhotin, Boolean grammars, Information and Computation, 194(1) (2004) 19–48]. In particular, we propose a two-player infinite game of perfect information for Boolean grammars, which is equivalent to their well-founded semantics. The game is directly applicable to the simpler classes of conjunctive and context-fre...

Molodtsov's soft set theory provides a general mathematical framework for dealing with uncertainty. The concepts of (M, N)-SI implicative (Boolean) filters of BL-algebras are introduced. Some good examples are explored. The relationships between (M, N)-SI filters and (M, N)-SI implicative filters are discussed. Some properties of (M, N)-SI implicative (Boolean) filters are investigated. In part...