Intebooleana

Intebooleana is a theoretical framework that combines elements of integer arithmetic with Boolean algebra to model logical constraints that include quantitative dimensions. The term is used in discussions of algebraic logic and constraint-based reasoning. In this framework, truth values are represented by integers and logical connectives are realized through arithmetic operations on these integers, enabling a uniform treatment of logical constraints alongside numeric constraints.

Formal definition: A typical intebooleana structure consists of a finite set of integer-valued variables; a binary predicate for equality; and the three standard logical connectives defined on the 0/1 Boolean subset by NOT x = 1 - x; x AND y = x*y; x OR y = x + y - x*y. Interpreted variables take values in the set {0,1} for propositional reasoning, or other integers with projection to {0,1} for mixed reasoning.

Related models: It shares ideas with SAT modulo theories (SMT), integer linear programming, and Boolean algebra. In practice, intebooleana-like formalisms support encoding of logical constraints within integer programming problems, enabling the use of MILP solvers for logic problems.

Applications: digital circuit verification, constraint satisfaction problems, software model checking, and optimization where logical and numeric constraints co-exist.

Limitations and variants: The approach is largely a modeling convenience; not all logical theories translate directly into arithmetic without loss of expressive power. Variants may replace the min/max with other t-norm based definitions for multi-valued logics.

See also: Boolean algebra, propositional logic, SAT, SMT, integer programming.

References: This article summarizes a usage of the term; for detailed formal treatments, consult works on SAT/SMT and algebraic logic.