A solvable class of quadratic 0–1 programming

AbstractWe show that the minimum of the pseudo-Boolean quadratic function ƒ(x) = xTQx + cTx can be found in linear time when the graph defined by Q is transformable into a combinatorial circuit of AND, OR, NAND, NOR or NOT logic gates. A novel modeling technique is used to transform the graph defined by Q into a logic circuit. A consistent labeling of the signals in the logic circuit from the set {0, 1} corresponds to the global minimum of ƒ and the labeling is determined through logic simulation of the circuit. Our approach establishes a direct and constructive relationship between pseudo-Boolean functions and logic circuits.In the restricted case when all the elements of Q are nonpositive, the minimum of ƒ can be obtained in polynomial time [15]. We show that the problem of finding the minimum of ƒ, even in the special case when all the elements of Q are positive, is NP-complete

Resolving Distributed Knowledge

Distributed knowledge is the sum of the knowledge in a group; what someone who is able to discern between two possible worlds whenever any member of the group can discern between them, would know. Sometimes distributed knowledge is referred to as the potential knowledge of a group, or the joint knowledge they could obtain if they had unlimited means of communication. In epistemic logic, the formula D_G{\phi} is intended to express the fact that group G has distributed knowledge of {\phi}, that there is enough information in the group to infer {\phi}. But this is not the same as reasoning about what happens if the members of the group share their information. In this paper we introduce an operator R_G, such that R_G{\phi} means that {\phi} is true after G have shared all their information with each other - after G's distributed knowledge has been resolved. The R_G operators are called resolution operators. Semantically, we say that an expression R_G{\phi} is true iff {\phi} is true in what van Benthem [11, p. 249] calls (G's) communication core; the model update obtained by removing links to states for members of G that are not linked by all members of G. We study logics with different combinations of resolution operators and operators for common and distributed knowledge. Of particular interest is the relationship between distributed and common knowledge. The main results are sound and complete axiomatizations.Comment: In Proceedings TARK 2015, arXiv:1606.0729

Semantic A-translation and Super-consistency entail Classical Cut Elimination

We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a semantic version of Friedman's A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated algebra and its A-translation in the original algebra. This allows to show the stability of the super-consistency criterion and the cut elimination theorem