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

Julian C. Bradfield
1998 BRICS Report Series  
We provide an elementary proof 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.  ... 
doi:10.7146/brics.v5i53.19499 fatcat:t3btzhwmgvcsreaectbnk36a2q

The Arity Hierarchy in the Polyadic μ-Calculus

Martin Lange
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.  ... 
doi:10.4204/eptcs.191.10 fatcat:oi5irp6ef5fd3getjjvgiiit2m

The mu-calculus and Model Checking [chapter]

Julian Bradfield, Igor Walukiewicz
2018 Handbook of Model Checking  
It gives a quite detailed explanation 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 µX and νX bind occurrences of the variable X, in the sense that the meaning of µX.α does not depend on the valuation of X.  ... 
doi:10.1007/978-3-319-10575-8_26 fatcat:7adhegoggvcjthaa5x5uhfpt24

The modal μ-calculus hierarchy over restricted classes of transition systems

Luca Alberucci, Alessandro Facchini
2009 Journal of Symbolic Logic (JSL)  
We study 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.  ... 
doi:10.2178/jsl/1254748696 fatcat:vlgtshyd7vc3nkssw2cb7lpo6e

On modal μ-calculus with explicit interpolants

G. D'Agostino, G. Lenzi
2006 Journal of Applied Logic  
By using this quantifier 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.  ... 
doi:10.1016/j.jal.2005.06.008 fatcat:4drsjumrunhyhp5h4srhgxy4oy

THE MODAL µ-CALCULUS: A SURVEY

GIACOMO LENZI
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] .  ... 
doaj:3c3858f737b04247b9bda1a46646b7c7 fatcat:gnx6n4z5srg4tb354bp6ngn2ya

On P-transitive graphs and applications

Giacomo Lenzi
2011 Electronic Proceedings in Theoretical Computer Science  
First we show that 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  ... 
doi:10.4204/eptcs.54.16 fatcat:dhrkttwzonacro2sqwwq6dhxvq

On Modal μ-Calculus over Finite Graphs with Bounded Strongly Connected Components

Giovanna D'Agostino, Giacomo Lenzi
2010 Electronic Proceedings in Theoretical Computer Science  
We show that for every k, 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.  ... 
doi:10.4204/eptcs.25.9 fatcat:mfzghgt6inab3is4nar3uv3oue

Methods for modalities 3

Carlos Areces
2006 Journal of Applied Logic  
"On modal μ-calculus with explicit interpolants" by D'Agostino and Lenzi.  ...  third level has already the expressivity of the whole μ-calculus.  ... 
doi:10.1016/j.jal.2005.06.006 fatcat:voymp7oetvbrrid7zwsr2embte

The Variable Hierarchy of the μ-Calculus Is Strict [chapter]

Dietmar Berwanger, Giacomo Lenzi
2005 Lecture Notes in Computer Science  
Most 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.  ... 
doi:10.1007/978-3-540-31856-9_8 fatcat:muut5tpmojc7hkobd6njjurhgm

The Variable Hierarchy of the μ-Calculus Is Strict

Dietmar Berwanger, Erich Gradel, Giacomo Lenzi
2007 Theory of Computing Systems  
Most 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.  ... 
doi:10.1007/s00224-006-1317-8 fatcat:osnj2njlzrhhpcbkwkzwcaoyaq

Fixpoints, games and the difference hierarchy

Julian C. Bradfield
2003 RAIRO - Theoretical Informatics and Applications  
This both provides a simple characterization 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.  ... 
doi:10.1051/ita:2003011 fatcat:4hwj5rljynabbm7vc7af2y4kim

Characterizing polynomial and exponential complexity classes in elementary lambda-calculus

Patrick Baillot, Erika De Benedetti, Simona Ronchi Della Rocca
2018 Information and Computation  
Types are formulae 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.  ... 
doi:10.1016/j.ic.2018.05.005 fatcat:ycucrc732jfhpbwsjeau6kxlfm

First-Order Modal ξ-Calculus: Application and Bisimulation [article]

Xinyu Wang
2022 arXiv   pre-print
Inspired by 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.  ... 
arXiv:2110.07689v3 fatcat:n3u6rhce3fehxip7w7ceaipqcq

On the μ-calculus over transitive and finite transitive frames

Giovanna D'Agostino, Giacomo Lenzi
2010 Theoretical Computer Science  
We prove that 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.  ... 
doi:10.1016/j.tcs.2010.09.002 fatcat:jopolvudajdqxgtj4ktr7pl5dm
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