The theory of NP-completeness has its roots in a foundational result by Cook, who showed that Boolean satisfiability (SAT) is NP-complete and thus unlikely to admit an efficient solution. We prove an analogous result using binary integer programming in the place of SAT. The proof gives deeper insight into the theory of NP-completeness to operations researchers for whom the language of integer p...

Hazardous wastes include numerous kinds of discarded chemicals and other wastes generated from industrial, commercial, and institutional activities. These types of waste present immediate or long-term risks to humans, animals, plants, or the environment and therefore require special handling for safe disposal. Landfills that can accept hazardous wastes are excavated or engineered sites where th...

UNLABELLED A theory is presented that attempts to answer two questions. What visual contents can an observer consciously access at one moment? ANSWER only one feature value (e.g., green) per dimension, but those feature values can be associated (as a group) with multiple spatially precise locations (comprising a single labeled Boolean map). How can an observer voluntarily select what to acces...

Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methodsf and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington.J Since they embody in abstract form the principal algebraic rules governing the manipulation of classes or aggregates, these systems are of ...

This theory defines a type constructor representing the free Boolean algebra over a set of generators. Values of type (α)formula represent propositional formulas with uninterpreted variables from type α, ordered by implication. In addition to all the standard Boolean algebra operations, the library also provides a function for building homomorphisms to any other Boolean algebra type. 1 Free Boo...

Probability is usually closely related to Boolean structures, i.e. Boolean algebras or propositional logic. Here we show, how probability can be combined with Non-Boolean structures, and in particular Non-Boolean logics. The basic idea is to describe uncertainty by (Boolean) assumptions, which may or may not be valid. The uncertain information depends then on these uncertain assumptions, scenar...

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...

We investigate in depth the properties of a natural generalisation, called “sets of no-goods” in the AI literature, of boolean formulas in conjunctive normal forms such that non-boolean variables can be used. After building up a solid foundation, we generalise the notion of deficiency, and we obtain polynomial time satisfiability decision for generalised clause-sets with fixed maximal deficienc...

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a | M |= φ(a)} defined by some formula φ) is invariant under all automorphisms of ...

In this paper we apply information theory concepts to evolutionary Boolean circuit synthesis. We discuss the schema destruction problem when simple conditional entropy is used as fitness function. The design problem is the synthesis of Boolean functions by using the minimum number of binary multiplexers. We show that the fitness landscape of normalized mutual information exhibits better charact...