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Fixpoint Alternation: Arithmetic, Transition Systems, and the Binary Tree

1998
*
BRICS Report Series
*

We provide an elementary proof

doi:10.7146/brics.v5i53.19499
fatcat:t3btzhwmgvcsreaectbnk36a2q
*of**the*fixpoint alternation<br />*hierarchy*in arithmetic, which in turn allows us to simplify*the*proof*of**the**modal*mu-*calculus*alternation*hierarchy*. ... We further show that*the*alternation*hierarchy**on**the*binary tree is strict, resolving a problem*of*Niwinski. ...*One*can define*the*syntactic alternation classes for arithmetic just as for*the**modal*mu-*calculus*: First-order formulae are Σ*µ*0 and Π*µ*0 , as are set*variables*. ...##
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The Arity Hierarchy in the Polyadic μ-Calculus

2015
*
Electronic Proceedings in Theoretical Computer Science
*

*The*polyadic mu-

*calculus*is a

*modal*fixpoint logic whose formulas define relations

*of*nodes rather than just sets in labelled transition systems. ... In this paper we show a

*hierarchy*result with respect to expressive power inside

*the*polyadic mu-

*calculus*: for every level

*of*fixpoint alternation, greater arity

*of*relations gives rise to higher expressive ...

*The*syntax

*of*

*the*polyadic

*modal*

*µ*-

*calculus*L ω

*µ*is similar to that

*of*

*the*ordinary

*modal*

*µ*-

*calculus*. ...

##
###
The mu-calculus and Model Checking
[chapter]

2018
*
Handbook of Model Checking
*

It gives a quite detailed explanation

doi:10.1007/978-3-319-10575-8_26
fatcat:7adhegoggvcjthaa5x5uhfpt24
*of**the*satisfiability algorithm, followed by*the*results*on*alternation*hierarchy*, proof systems, and interpolation. ... Then it develops*the*notion*of**modal*automaton: an automaton-based model behind*the*mu-*calculus*. ...*The*fixpoint operators*µ*X and νX bind occurrences*of**the**variable*X, in*the*sense that*the*meaning*of**µ*X.α does not depend*on**the*valuation*of*X. ...##
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The modal μ-calculus hierarchy over restricted classes of transition systems

2009
*
Journal of Symbolic Logic (JSL)
*

We study

doi:10.2178/jsl/1254748696
fatcat:vlgtshyd7vc3nkssw2cb7lpo6e
*the*strictness*of**the**modal**μ*-*calculus**hierarchy*over some restricted classes*of*transition systems. ... 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. ... We thank Igor Walukiewicz, and especially Giovanna D'Agostino and Giacomo Lenzi for various comments*on*a preliminary version*of*this paper. We also thank an anonymous referee for valuable comments. ...##
###
On modal μ-calculus with explicit interpolants

2006
*
Journal of Applied Logic
*

By using this quantifier

doi:10.1016/j.jal.2005.06.008
fatcat:4drsjumrunhyhp5h4srhgxy4oy
*one*can express*the*uniform interpolant*of*any formula*of**the**μ*-*calculus*. ... In this work we provide an explicit form for*the*uniform interpolant*of*a disjunctive formula and see that it belongs to*the*same level*of**the*fixpoint alternation*hierarchy**of**the**μ*-*calculus*than*the*...*The*same result is known for*modal*logic, i.e., for level N 0*of**the**μ*-*calculus*. Here we prove that*the*same holds for levels N 1 and N 2*of**the**μ*-*calculus**hierarchy*. Proof. ...##
###
THE MODAL µ-CALCULUS: A SURVEY

2005
*
TASK Quarterly
*

*The*

*modal*

*µ*-

*calculus*is an extension

*of*

*modal*logic with two operators

*µ*and ν, which give

*the*least and greatest fixpoints

*of*monotone operators

*on*powersets. ... In this survey we review both

*the*theoretical aspects

*of*

*the*

*modal*

*µ*-

*calculus*and its applications to computer science. ... As

*the*name suggests,

*modal*

*µ*-

*calculus*is built

*on*top

*of*

*modal*logic; for many other

*µ*-calculi see [2] . ...

##
###
On P-transitive graphs and applications

2011
*
Electronic Proceedings in Theoretical Computer Science
*

First we show that

doi:10.4204/eptcs.54.16
fatcat:dhrkttwzonacro2sqwwq6dhxvq
*the*analogue*of*de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we show that*the*mu-*calculus*fixpoint*hierarchy*is infinite for P-transitive graphs. ... Both results contrast with*the*case*of*transitive graphs. We give also an undecidability result for an enriched mu-*calculus**on*P-transitive graphs. ... It is known that there is a collapse*of**the**µ*-*calculus**on*transitive graphs, not to*modal*logic, but to*the*second level*of**the*fixpoint*hierarchy*, see [2] , [10] and [11] ; moreover,*on*arbitrary ...##
###
On Modal μ-Calculus over Finite Graphs with Bounded Strongly Connected Components

2010
*
Electronic Proceedings in Theoretical Computer Science
*

We show that for every k,

doi:10.4204/eptcs.25.9
fatcat:mfzghgt6inab3is4nar3uv3oue
*the**Modal*mu-*Calculus*fixpoint*hierarchy**on*SCCk collapses to*the*level Delta2, but not to Comp(Sigma1,Pi1) (compositions*of*formulas*of*level Sigma1 and Pi1). ... This contrasts with*the*class*of*all graphs, where Delta2=Comp(Sigma1,Pi1). ... games for*the*formal verification*of*complex systems. ...##
###
Methods for modalities 3

2006
*
Journal of Applied Logic
*

"

doi:10.1016/j.jal.2005.06.006
fatcat:voymp7oetvbrrid7zwsr2embte
*On**modal**μ*-*calculus*with explicit interpolants" by D'Agostino and Lenzi. ... third level has already*the*expressivity*of**the*whole*μ*-*calculus*. ...##
###
The Variable Hierarchy of the μ-Calculus Is Strict
[chapter]

2005
*
Lecture Notes in Computer Science
*

Most

doi:10.1007/978-3-540-31856-9_8
fatcat:muut5tpmojc7hkobd6njjurhgm
*of**the*logics commonly used in verification, such as LTL, CTL, CTL * , and PDL can be embedded into*the*two-*variable*fragment*of**the**µ*-*calculus*. ... We answer this question negatively, and prove that*the*variablehierarchy*of**the**µ*-*calculus*is semantically strict. ...*The*authors wish to thank Thomas Colcombet for his valuable feedback*on*this article.*The*third author expresses his gratitute to André Arnold for his continuing support and advice. ...##
###
The Variable Hierarchy of the μ-Calculus Is Strict

2007
*
Theory of Computing Systems
*

Most

doi:10.1007/s00224-006-1317-8
fatcat:osnj2njlzrhhpcbkwkzwcaoyaq
*of**the*logics commonly used in verification, such as LTL, CTL, CTL * , and PDL can be embedded into*the*two-*variable*fragment*of**the**µ*-*calculus*. ... We answer this question negatively, and prove that*the*variablehierarchy*of**the**µ*-*calculus*is semantically strict. ...*The*authors wish to thank Thomas Colcombet for his valuable feedback*on*this article.*The*third author expresses his gratitute to André Arnold for his continuing support and advice. ...##
###
Fixpoints, games and the difference hierarchy

2003
*
RAIRO - Theoretical Informatics and Applications
*

This both provides a simple characterization

doi:10.1051/ita:2003011
fatcat:4hwj5rljynabbm7vc7af2y4kim
*of**the*fixpoint*hierarchy*, and refines existing results*on**the*power*of**the*game quantifier in descriptive set theory. ... We raise*the*problem*of*transfinite fixpoint*hierarchies*. Mathematics Subject Classification. 03E15, 68Q45. ... These alternating Rabin automata are important in temporal logic, as they are*one*characterization*of**modal*mu-*calculus*. ...##
###
Characterizing polynomial and exponential complexity classes in elementary lambda-calculus

2018
*
Information and Computation
*

Types are formulae

doi:10.1016/j.ic.2018.05.005
fatcat:ycucrc732jfhpbwsjeau6kxlfm
*of*Elementary Linear Logic (ELL), and*the**hierarchy**of*complexity classes k-EXP is characterized by a*hierarchy**of*types. ... In this paper an implicit characterization*of**the*complexity classes k-EXP and k-FEXP, for k ≥ 0, is given, by a type assignment system for a stratified λ-*calculus*, where types for programs are witnesses ... Note also that there is no axiom rule for*variables*in*the**modal*basis, so*the*only way to introduce a*variable*in this basis is*the*(!) rule, moving*variables*from*the*parking to*the**modal*basis. ...##
###
First-Order Modal ξ-Calculus: Application and Bisimulation
[article]

2022
*
arXiv
*
pre-print

Inspired by

arXiv:2110.07689v3
fatcat:n3u6rhce3fehxip7w7ceaipqcq
*modal**μ*-*calculus*, first-order*modal*ξ-*calculus*takes a quite similar form and extends its inductive expressivity onto a different dimension. ... We elaborate*on*several vivid examples that demonstrate this logic's profound utility, especially for depicting genealogy*of*concurrent computer processes. ... A handful*of*anonymous reviewers*on*previous versions*of**the*manuscript have also provided tremendous valuable suggestions to help*the*author improve this paper. ...##
###
On the μ-calculus over transitive and finite transitive frames

2010
*
Theoretical Computer Science
*

We prove that

doi:10.1016/j.tcs.2010.09.002
fatcat:jopolvudajdqxgtj4ktr7pl5dm
*the**modal**µ*-*calculus*collapses to first order logic over*the*class*of*finite transitive frames. ... Moreover, we prove that*the**modal**µ*-*calculus*is Büchi and co-Büchi definable over*the*class*of*all models where, in a strongly connected component, vertexes are distinguishable by means*of**the*propositions ... Acknowledgements We would like to thank Luca Alberucci and Alessandro Facchini for their careful reading*of*Section 3*of*this paper. We also thank*the*anonymous referee for his suggestions. ...
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