Modal Fixpoint Logics

Context. Modal fixpoint logics constitute a research field of considerable interest, not only because of its many applications in computer science, but also because of its rich mathematical theory, which features deep connections with fields as diverse as lattice theory, automata theory, and universal coalgebra. In the 1970s computer scientists started to use modal logics as specification languages. It was soon realized that the basic modal languages lack expressive power to formalize properties related to the ongoing behavior of processes. As a solution to such a problem it was devised to add a limited possibility of inductive definitions to modal languages, in the form of fixpoint operators. Such enrichment of the logical language may achieve a great increase in expressive power of a modal language, while keeping the formalism computationally manageable. Therefore fixpoint operators provide logical formalisms of considerable practical interest. Many modal fixpoint logics, such as Propositional Dynamic Logic (pdl) [20] or Computational Tree Logic (ctl or ctl∗) [13] involve implicit fixpoints only. Almost all these logics can be seen as fragments of the modal μ-calculus Lμ. This is a formalism introduced by Kozen [28] by extending the basic (poly-)modal language with explicit least and greatest fixpoint operators. Due to a very attractive combination of properties, computational, expressive and other, this formalism has become the most widely studied of all program logics. Even if some monographs dedicated to μ-calculi have already appeared [11, 4], the number of recent contributions [12, 10, 52, 36, 21] indicates this is still a very active research subject moving onto many diversified directions. Over the years, a wealth of information has been obtained about fixpoint logics in general, and the modal μ-calculus in particular, mostly by automata and game-theoretic means [4, 55, 19]. Since Rabin’s result [40] the relations between logic and automata theory have been deeply investigated. By now, it is common knowledge [23] that alternating automata are combinatorial representations of μ-calculi propositional formulas. The model checking problem for the modal μ-calculus also has an equivalent formulation in terms of solving parity games [14]. Through this equivalence many efforts have been directed towards an exact understanding of the complexity of the model checking problem [26]. On the side of axiomatic, the main result is Walukiewicz’ theorem [59] asserting the completeness of Kozen’s axiomatization [28] of the modal μ-calculus. Despite the richness and difficulty of its proof, this theorem did not have until now many consequences as it could be expected and, for what concerns the axiomatic of fixpoint logics, this theorem stands surprisingly isolated. The previous discussion lead us to the main remark motivating our research proposal: compared to other formalisms that enhance the expressive power of the basic modal language, such as hybrid logic [3], there are surprisingly large gaps in our theoretical knowledge of modal fixpoint logics, and some results of considerable interest exist in isolation of the general theory of modal logic [5]. This applies in particular to model-theoretic and algebraic perspectives on modal fixpoint logics. It is in this direction that we intend to aim our collaborative research efforts. ∗Laboratoire d’Informatique Fondamentale, Université de Provence. †Institute for Logic, Language and Computation and Faculty of Science, Universiteit van Amsterdam.