The µ-calculus can be viewed as essentially the "ultimate" program logic, as it expressively subsumes all propositional program logics, including dynamic logics, process logics, and temporal logics. It is known that the satisfiability problem for the µ-calculus is EXPTIME-complete. This upper bound, however, is known for a version of the logic that has only forward modalities, which express weakest preconditions, but not backward modalities, which express strongest postconditions. Our main result in this paper is an exponential time upper bound for the satisfiability problem of the µ-calculus with both forward and backward modalities. To get this result we develop a theory of two-way alternating automata on infinite trees.

This paper is a survey and systematic presentation of decidability and complexity issues for modal and non-modal two-variable logics. A classical result due to Mortimer says that the two-variable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2 . Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points etc. Examples of such logics are computation tree logic CTL, the modal ¯-calculus L¯ , or popular description logics used in artificial intelligence. Although the additional features are usually not first-order...

It is a classical result of Mortimer that L², first-order logic with two variables, is decidable for satisfiability. We show that going beyond L² by adding any one of the following leads to an undecidable logic: ffl very weak forms of recursion, viz. (i) transitive closure operations (ii) (restricted) monadic fixed-point operations ffl weak access to cardinalities, through the Hartig (or equicardinality) quantifier ffl a choice construct known as Hilbert's "-operator. In fact all these extensions of L² prove to be undecidable both for satisfiability, and for satisfiability in finite models. Moreover most of them are hard for \Sigma 1 1 , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic.

Propositional -calculus is an extension of the propositional modal logic with the least fixpoint operator. In the paper introducing the logic Kozen posed a question about completeness of the axiomatisation which is a small extension of the axiomatisation of the modal system K. It is shown that this axiomatisation is complete.

Modal logics are widely used in a number of areas in computer science, in particular

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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are ffl recognizable in polynomial time, and ffl closed under bisimulation equivalence. It is shown that the class of these bisimulation-invariant Ptime queries has a natural logical characterization. It is captured by the straightforward extension of propositional µ-calculus to arbitrary finite dimension. Bisimulation-invariant Ptime, or the modal fragment of Ptime, thus proves to be one of the very rare cases in which a logical characterization is known in a setting of unordered structures. It is also shown that higher-dimensional µ-calculus is undecidable for satisfiability in finite structures, and even \Sigma 1 1 -hard over general structures.

The propositional µ-calculus as introduced by Kozen in [12] is considered. In that paper

This paper shows how modal mu-calculus formulae characterizing finite-state processes up to strong or weak bisimulation can be derived directly from the well-known greatest fixpoint characterizations of the bisimulation relations. Our derivation simplifies earlier proofs for the strong bisimulation case and, by virtue of derivation, immediately generalizes to various other bisimulation-like relations, in particular weak bisimulation.