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The closure ordinal of a \(\mu \)-calculus formula \(\varphi (x)\) is the least ordinal \(\alpha \), if it exists, such that, in any model, the least fixed point of \(\varphi (x)\) can be computed in at most \(\alpha \) many steps, by iteration of the meaning function associated with \(\varphi (x)\), starting from the empty set. In this paper we focus on closure ordinals of the two-way modal \(\mu \)-calculus. Our main technical contribution is the construction of a two-way formula \(\varphi _n\) with closure ordinal \(\omega ^n\) for an arbitrary \(n\in \omega \). Building on this construction, as our main result we define a two-way formula \(\varphi _\alpha \) with closure ordinal \(\alpha \) for an arbitrary \(\alpha <\omega ^\omega \).

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Ruitenburg’s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ultimately periodic if f fixes all the generators but one. More precisely, there is N ≥ 0 such that fN +2 = fN , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators.

. Monadic second order logic (MSOL) over transition systems is considered. It is shown that every formula of MSOL which does not distinguish between bisimilar models is equivalent to a formula of the propositional -calculus. This expressive completeness result implies that every logic over transition systems invariant under bisimulation and translatable into MSOL can be also translated into the -calculus. This gives a precise meaning to the statement that most propositional logics of programs can be translated into the -calculus. 1 Introduction Transition systems are structures consisting of a nonempty set of states, a set of unary relations describing properties of states and a set of binary relations describing transitions between states. It was advocated by many authors [26, 3] that this kind of structures provide a good framework for describing behaviour of programs (or program schemes), or even more generally, engineering systems, provided their evolution in time is disc...

This paper contributes to the theory of the modal $\mu$-calculus by proving
some model-theoretic results. More in particular, we discuss a number of
semantic properties pertaining to formulas of the modal $\mu$-calculus. For
each of these properties we provide a corresponding syntactic fragment, in the
sense that a $\mu$-formula $\xi$ has the given property iff it is equivalent to
a formula $\xi'$ in the corresponding fragment. Since this formula $\xi'$ will
always be effectively obtainable from $\xi$, as a corollary, for each of the
properties under discussion, we prove that it is decidable in elementary time
whether a given $\mu$-calculus formula has the property or not.
The properties that we study all concern the way in which the meaning of a
formula $\xi$ in a model depends on the meaning of a single, fixed proposition
letter $p$. For example, consider a formula $\xi$ which is monotone in $p$;
such a formula a formula $\xi$ is called continuous (respectively, fully
additive), if in addition it satisfies the property that, if $\xi$ is true at a
state $s$ then there is a finite set (respectively, a singleton set) $U$ such
that $\xi$ remains true at $s$ if we restrict the interpretation of $p$ to the
set $U$. Each of the properties that we consider is, in a similar way,
associated with one of the following special kinds of subset of a tree model:
singletons, finite sets, finitely branching subtrees, noetherian subtrees
(i.e., without infinite paths), and branches.
Our proofs for these characterization results will be automata-theoretic in
nature; we will see that the effectively defined maps on formulas are in fact
induced by rather simple transformations on modal automata. Thus our results
can also be seen as a contribution to the model theory of modal automata.

It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game â with n nodes and m distinct values (aka colours or priorities) â is proven to be in the class of fixed parameter tractable (FPT) problems when parameterised over m. Both results improve known bounds, from runtime nO(ân) to O(nlog(m)+6) and from an XP-algorithm with runtime O(nÎ(m)) for fixed parameter m to an FPT-algorithm with runtime O(n⁵)+g(m), for some function g depending on m only. As an application it is proven that coloured Muller games with n nodes and m colours can be decided in time O((mm Â· n)⁵); it is also shown that this bound cannot be improved to O((2m Â· n)c), for any c, unless FPT = W[1].

The (modal) μ-calculus ([14]) is a very powerful extension of modal logic with least and greatest fixed point operators. It is of great interest to computer science for expressing properties of processes such as termination (every run is finite) and fairness (on every infinite run, no action is repeated infinitely often to the exclusion of all others).
The power of the μ-calculus is also evident from a more theoretical perspective. The μ-calculus is a fragment of monadic second-order logic (MSO) containing only formulae that are invariant for bisimulation , in the sense that they cannot distinguish between bisimilar states. Janin and Walukiewicz prove the converse: any property which is invariant for bisimulation and MSO-expressible is already expressible in the μ-calculus ([13]). Yet the μ-calculus enjoys many desirable properties which MSO lacks, like a complete sequent-calculus ([29]), an exponential-time decision procedure, and the finite model property ([25]). Switching from MSO to its bisimulation-invariant fragment gives us these desirable properties.
In this paper we take a classical logician's view of the μ-calculus. As far as we are concerned a new logic should not be allowed into the community of logics without at least considering the standard questions that any logic is bothered with. In this paper we perform this rite of passage for the μ-calculus. The questions we will be concerned with are the following.

The closure ordinal of a formula of modal μ-calculus μXℓ is the least ordinal κ, if it exists, such that the denotation of the formula and the κ-th iteration of the monotone operator induced by ℓ coincide across all transition systems (finite and infinite). It is known that for every α < ω2 there is a formula ℓ of modal logic such that μXℓ has closure ordinal α [3]. We prove that the closure ordinals arising from the alternation-free fragment of modal μ-calculus (the syntactic class capturing 2 ∩ ∏2) are bounded by ω2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulæ into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulæ and embedded order-types of the corresponding tableaux.

In this paper we define and study a propositional μ-calculus Lμ, which consists essentially of propositional modal logic with a least fixpoint operator. Lμ is syntactically simpler yet strictly more expressive than Propositional Dynamic Logic (PDL). For a restricted version we give an exponential-time decision procedure, small model property, and complete deductive system, theory subsuming the corresponding results for PDL.

We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.

In this paper we investigate the Scott continuous fragment of the modal μ-calculus. We discuss its relation with constructivity, where we call a formula constructive if its least fixpoint is always
reached in at most ω steps. Our main result is a syntactic characterization of this continuous fragment. We also show that it is decidable whether
a formula is continuous.

Propositional -calculus is an extension of the propositional modal logic with the least xp oint 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.

Consider the following problem: given a formula of the modal μ-calculus, decide whether this formula is equivalently expressible
in basic modal logic. It is shown that this problem is decidable, in fact in deterministic exponential time. The decidability
result can be obtained through a model theoretic reduction to the monadic second-order theory of the complete binary tree,
which by Rabin’s classical result is decidable, albeit of non-elementary complexity. An improved analysis based on tree automata
yields an exponential time decision procedure.

The propositional |ì-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 consider the propositional μ-calculus as introduced by D.
Kozen (1983). In that paper a natural proof system was proposed and its
completeness stated as an open problem. We show that the system is
complete

The computational complexity of testing nonemptiness of finite-state automata on infinite trees is investigated. It is shown that for tree automata with m states and n pairs nonemptiness can be tested in time O(( mn )<sup>3n</sup>), even though the problem is in general NP-complete. The nonemptiness algorithm is used to obtain exponentially improved, essentially tight upper bounds for numerous important modal logics of programs, interpreted with the usual semantics over structures generated by binary relations. For example, it is shown that satisfiability for the full branching time logic CTL* can be tested in deterministic double exponential time. It also follows that satisfiability for propositional dynamic logic with a repetition construct (PDL-delta) and for the propositional mu-calculus ( L μ) can be tested in deterministic single exponential time

An exploration of closure ordinals in the modal $$mu $$-calculus. Master’s thesis, Institute for Logic, Language and Computation

- G Milanese

How fast can the fixpoints in modal $$mu $$-calculus be reached?

- M Czarnecki

- M J Gouveia
- L Santocanale