Variable Independence in Linear Real Arithmetic

Variable independence and decomposability are algorithmic techniques for simplifying logical formulas by tearing apart connections between free variables. These techniques were originally proposed to speed up query evaluation in constraint databases, in particular by representing the query as a Boolean combination of formulas with no interconnected variables. They also have many other applications in SMT, string analysis, databases, automata theory and other areas. However, the precise complexity of variable independence and decomposability has been left open especially for the quantifier-free theory of linear real arithmetic (LRA), which is central in database applications. We introduce a novel characterization of formulas admitting decompositions and use it to show that it is coNP-complete to decide variable decomposability over LRA. As a corollary, we obtain that deciding variable independence is in $ \Sigma_2^p $. These results substantially improve the best known double-exponential time algorithms for variable decomposability and independence. In many practical applications, it is crucial to be able to efficiently eliminate connections between variables whenever possible. We design and implement an algorithm for this problem, which is optimal in theory, exponentially faster compared to the current state-of-the-art algorithm and efficient on various microbenchmarks. In particular, our algorithm is the first one to overcome a fundamental barrier between non-discrete and discrete first-order theories. Formulas arising in practice often have few or even no free variables that are perfectly independent. In this case, our algorithm can compute a best-possible approximation of a decomposition, which can be used to optimize database queries by exploiting partial variable independence, which is present in almost every logical formula or database query constraint.