For every positive integer k we consider the class SCCk of all finite graphs
whose strongly connected components have size at most k. 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)

Angelo Montanari

Giacomo Lenzi

Giovanna D'Agostino

Margherita Napoli

Mimmo Parente

Electronic Proceedings in Theoretical Computer Science

English

arXiv

For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. 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)

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

There is a problem with corollaries 6.2 and 7.2. Their proofs
rely on the wrong assumption that the parity automata defined here
and the standard parity alternating automata coincide level by level.
So, the proofs of corollaries 6.2 and 7.2 are wrong and we do not know
whether the statements are true. The summary of the paper has to be
changed accordingly as follows.

Summary: For every positive integer k we consider the class SCCk of
all finite graphs whose strongly connected components have size at most
k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy
on SCCk collapses to the level Delta_2; moreover, we introduce a
kind of automata analogous to Janin-Walukiewicz disjunctive formulas,
and we show that the mu-calculus in SCC1 does not collapse to weak
automata (as they are defined here)

Lenzi, G.

D'Agostino, G.

Archivio della Ricerca - Università di Salerno

On modal mu-calculus over finite graphs with bounded strongly 
connected components.

Crossref

On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly
  Connected Components

On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components

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Directory of Open Access Journals

Archivio della Ricerca - Università di Salerno

Crossref