A μ-algebra is a model of a first-order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms where μx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications.Standard μ-algebras are complete meaning that their lattice reduct is a complete lattice. We prove that any nontrivial quasivariety of μ-algebras contains a μ-algebra that has no embedding into a complete μ-algebra.We then focus on modal μ-algebras, i.e. algebraic (...) models of the propositional modal μ-calculus. We prove that free modal μ-algebras satisfy a condition–reminiscent of Whitman’s condition for free lattices–which allows us to prove that modal operators are adjoints on free modal μ-algebras, least prefixed points of Σ1-operations satisfy the constructive relation μx.f=logical and operatorn≥0fn. These properties imply the following statement: the MacNeille–Dedekind completion of a free modal μ-algebra is a complete modal μ-algebra and moreover the canonical embedding preserves all the operations in the class image of the fixed point alternation hierarchy. (shrink)

We show that given a finite, transitive and reflexive Kripke model 〈 W , ≼, ⟦ ⋅ ⟧ 〉 and $${w \in W}$$ , the property of being simulated by w (i.e., lying on the image of a literalpreserving relation satisfying the ‘forth’ condition of bisimulation) is modally undefinable within the class of S4 Kripke models. Note the contrast to the fact that lying in the image of w under a bi simulation is definable in the standard modal language even (...) over the class of K4 models, a fairly standard result for which we also provide a proof. We then propose a minor extension of the language adding a sequent operator $${\natural}$$ (‘tangle’) which can be interpreted over Kripke models as well as over topological spaces. Over finite Kripke models it indicates the existence of clusters satisfying a specified set of formulas, very similar to an operator introduced by Dawar and Otto. In the extended language $${{\sf L}^+ = {\sf L}^{\square\natural}}$$ , being simulated by a point on a finite transitive Kripke model becomes definable, both over the class of (arbitrary) Kripke models and over the class of topological S4 models. As a consequence of this we obtain the result that any class of finite, transitive models over finitely many propositional variables which is closed under simulability is also definable in L + , as well as Boolean combinations of these classes. From this it follows that the μ -calculus interpreted over any such class of models is decidable. (shrink)

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

We study the proof-theoretic relationship between two deductive systems for the modal mu-calculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on non-well-founded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along (...) every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as a corollary. (shrink)

A non-effective cut-elimination proof for modal mu-calculus has been given by G. Jäger, M. Kretz and T. Studer. Later an effective proof has been given for a subsystem M 1 with non-iterated fixpoints and positive endsequents. Using a new device we give an effective cut-elimination proof for M 1 without restriction to positive sequents.

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.

Buchholz’s Ω μ+1-rules provide a major tool for the proof-theoretic analysis of arithmetical inductive definitions. The aim of this paper is to put this approach into the new context of modal fixed point logic. We introduce a deductive system based on an Ω-rule tailored for modal fixed point logic and develop the basic techniques for establishing soundness and completeness of the corresponding system. In the concluding section we prove a cut elimination and collapsing result similar to that of Buchholz (Iterated (...) inductive definitions and subsystems of analysis: recent proof theoretic studies. Lecture notes in mathematics, vol. 897, pp. 189–233, Springer, Berlin, 1981). (shrink)

It is a classical result of Mortimer that $L^2$ , first-order logic with two variables, is decidable for satisfiability. We show that going beyond $L^2$ by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Härtig (or equicardinality) quantifier¶– a choice construct known as Hilbert's $\epsilon$ -operator.In fact all these extensions of $L^2$ prove to be undecidable both for satisfiability, (...) and for satisfiability in finite structures. 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. (shrink)

For some modal fixed point logics, there are deductive systems that enjoy syntactic cut-elimination. An early example is the system in Pliuskevicius [15] for LTL. More recent examples are the systems by the authors of this paper for the logic of common knowledge [5] and by Hill and Poggiolesi for PDL[8], which are based on a form of deep inference. These logics can be seen as fragments of the modal mu-calculus. Here we are interested in how far this approach can (...) be pushed in general. To this end, we introduce a nested sequent system with syntactic cut-elimination which is incomplete for the modal mu-calculus, but complete with respect to a restricted language that allows only fixed points of a certain form. This restricted language includes the logic of common knowledge and PDL. There is also a traditional sequent system for the modal mu-calculus by Jäger et al. [9], without a syntactic cut-elimination procedure. We embed that system into ours and vice versa, thus establishing cut-elimination also for the shallow system, when only the restricted language is considered. (shrink)

We study the strictness of the modal μ-calculus hierarchy over some restricted classes of transition systems. First, we prove that over transitive systems the hierarchy collapses to the alternationfree fragment. In order to do this the finite model theorem for transitive transition systems is proved. Further, we verify that if symmetry is added to transitivity the hierarchy collapses to the purely modal fragment. Finally, we show that the hierarchy is strict over reflexive frames. By proving the finite model theorem for (...) reflexive systems the same results holds for finite models. (shrink)