Conditionsat

Conditionsat is a theoretical and applied problem in logic and computer science that concerns determining whether a set of conditions can be simultaneously satisfied under a given collection of logical constraints. It is often described as a generalization of the Boolean satisfiability problem (SAT) that supports conditional rules.

An instance of conditionsat consists of a finite set of boolean variables X = {x1, ..., xn}, a set H of hard constraints expressed in propositional logic (typically in conjunctive normal form), and a set C of conditional constraints of the form A => B, where A and B are propositional formulas over X. An assignment to the variables satisfies the instance if it satisfies all hard constraints in H and makes each A => B true. Since A => B is equivalent to ¬A ∨ B, the problem can be reduced to a SAT instance with additional structure.

In terms of complexity, conditionsat is NP-complete in general, as it subsumes SAT. Special cases with restricted forms of A and B (for example, when C consists of Horn clauses or when all A and B are literals) may admit polynomial-time algorithms. Practical approaches often translate conditionsat instances into SAT or employ constraint programming techniques, leveraging backtracking, unit propagation, and learning.

Applications include software product lines, feature configuration, policy compliance, decision automation, and AI planning, where conditional rules govern when certain propositions become active or required. The concept serves as a modeling framework rather than a standalone standard, with research focusing on expressive power, solving efficiency, and identifying tractable fragments.

See also: SAT, CSP, constraint programming, conditional logic, boolean formula, product configuration.