On the Succinctness of Modal μ-Calculus Based on Covariant–Contravariant Refinement

Abstract

The expressive power of logics is one of the major research topics in mathematical logic and computer science. One way of comparing the complexities of different formalisms (e.g. knowledge representation formalisms) stems from the perspective of representational succinctness. The concept of covariant–contravariant refinement (CC-refinement, for short) generalizes the concepts of refinement, simulation and bisimulation. We introduce an extension of the standard multi-agent modal [Formula: see text]-calculus system [Formula: see text] with CC-refinement operators ([Formula: see text]) and show that [Formula: see text] is equivalently expressive to [Formula: see text]. This paper presents the succinctness results of [Formula: see text] compared with [Formula: see text] from two viewpoints based on the sets of covariant and contravariant actions. We establish a family of CC-refinement formulas for comparing the succinctness based on the well-known parity symmetric alternating automata, which is often used to describe modal [Formula: see text]-calculus formulas.