On modal μ-calculus with explicit interpolants

Abstract

This paper deals with the extension of Kozen's μ-calculus with the so-called "existential bisimulation quantifier". By using this quantifier one can express the uniform interpolant of any formula of the μ-calculus. In this work we provide an explicit form for the uniform interpolant of a disjunctive formula and see that it belongs to the same level of the fixpoint alternation hierarchy of the μ-calculus than the original formula. We show that this result cannot be generalized to the whole logic, because the closure of the third level of the hierarchy under the existential bisimulation quantifier is the whole μ-calculus. However, we prove that the first two levels of the hierarchy are closed. We also provide the μ-logic extended with the bisimulation quantifier with a complete calculus.

On the size of disjunctive formulas in the μ-calculus

“…While disjunctive formulas correspond to a proper syntactic fragment of the µ-calculus, it is a somewhat surprising fact that each formula of the µ-calculus is semantically equivalent to a disjunctive one. This has many applications, e.g., satisfiability checking of a disjunctive formula can be carried out efficiently [13] (being ExpTime-complete in for arbitrary formulas [7,8]) and disjunctive formulas facilitate the computation of uniform interpolants [4,5]. Furthermore, disjunctive formulas play a pivotal role in the completeness theory of the modal µ-calculus [23,9].…”