AND decomposition of a boolean formula means finding two (or several) formulas such that their conjunction is equivalent to the given one. Decomposition is called disjoint if the component formulas do not have variables in common. In the paper, we show that deciding AND decomposability is intractable for bool ean formulas given in CNF or DNF and prove tractability of computing disjoint AND decomposition com ponents of boolean formulas given in positive DNF, Full DNF, and ANF. The latter result follows from trac tability of multilinear polynomial factorization over the finite field of order 2, for which we provide a polytime factorization algorithm based on identity testing for partial derivatives of multilinear polynomials.

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In many tasks related to reasoning about consequences of a logical theory, it is desirable to decompose the theory into a number of weakly-related or independent components. However, a theory may represent knowledge that is subject to change due to execution of actions that have effects on some properties mentioned in the theory. Having once computed a decomposition of a theory, one would like to know whether a decomposition has to be computed again in the theory obtained from taking into account changes resulting from execution of an action. In the paper, we address this problem in the scope of the situation calculus, where a change of an initial theory is related to the notion of progression. We undertake a study of the decomposability and inseparability properties known from the literature. We contribute by studying these properties wrt progression and the related notion of forgetting. We provide negative examples and identify cases when these properties are preserved under progression of initial theories and under forgetting in local-effect basic action theories of the situation calculus.

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