Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy

As is already known from the work of D. Janin & I. Walukiewicz (1996), the mu-calculus is as expressive as the bisimulation-invariant fragment of monadic second-order logic. In this paper, we relate the expressiveness of levels of the fixpoint alternation depth hierarchy of the mu-calculus (the mu-calculus hierarchy) with the expressiveness of the bisimulation-invariant fragment of levels of the monadic quantifiers alternation-depth hierarchy (the monadic hierarchy). From J. van Benthem's (1976) results, we know already that the fixpoint free fragment of the mu-calculus (i.e. polymodal logic) is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 0/ (i.e. first-order logic). We show that the /spl nu/-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 1/ and that the /spl nu//spl mu/-level of the mu-calculus hierarchy is as expressive as the bisimulation-invariant fragment of monadic /spl Sigma//sub 2/, and we show that no other level /spl Sigma//sub k/ (for k>2) of the monadic hierarchy can be related similarly with any other level of the mu-calculus hierarchy. The possible inclusion of all the mu-calculus in some level /spl Sigma//sub k/ of the monadic hierarchy, for some k>2, is also discussed.

[1]  Eric Rosen,et al.  Modal Logic over Finite Structures , 1997, J. Log. Lang. Inf..

[2]  Wolfgang Thomas,et al.  The monadic quantifier alternation hierarchy over graphs is infinite , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[3]  Jerzy Skurczynski.  A characterization of Büchi tree automata , 2002, Inf. Process. Lett..

[4]  Giacomo Lenzi.  A New Logical Characterization of Büchi Automata , 2001, STACS.

[5]  E. Allen Emerson,et al.  Temporal and Modal Logic , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[6]  Julian C. Bradfield.  The Modal µ-Calculus Alternation Hierarchy is Strict , 1998, Theor. Comput. Sci..

[7]  David E. Muller,et al.  Simulating Alternating Tree Automata by Nondeterministic Automata: New Results and New Proofs of the Theorems of Rabin, McNaughton and Safra , 1995, Theor. Comput. Sci..

[8]  Igor Walukiewicz,et al.  Automata for the Modal mu-Calculus and related Results , 1995, MFCS.

[9]  André Arnold,et al.  p329 The µ-calculus alternation-depth hierarchy is strict on binary trees , 1999, RAIRO Theor. Informatics Appl..

[10]  E. Allen Emerson,et al.  An Automata Theoretic Decision Procedure for the Propositional Mu-Calculus , 1989, Inf. Comput..

[11]  Igor Walukiewicz,et al.  Monadic Second Order Logic on Tree-Like Structures , 1996, STACS.

[12]  Damian Niwinski.  On Fixed-Point Clones (Extended Abstract) , 1986, ICALP.

[13]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[14]  Martin Otto,et al.  Back and forth between guarded and modal logics , 2002, TOCL.

[15]  Silvio Ghilardi,et al.  A Sheaf Representation and Duality for Finitely Presenting Heyting Algebras , 1995, J. Symb. Log..

[16]  Wolfgang Thomas,et al.  Computation Tree Logic CTL* and Path Quantifiers in the Monadic Theory of the Binary Tree , 1987, ICALP.

[17]  Marcin Jurdziński,et al.  Deciding the Winner in Parity Games is in UP \cap co-Up , 1998, Inf. Process. Lett..

[18]  D. Kozen.  Results on the Propositional µ-Calculus , 1982 .

[19]  David Janin,et al.  Automata for the mu-calculus and Related Results , 1995 .

[20]  Faron Moller,et al.  On the expressive power of CTL , 1999, Proceedings. 14th Symposium on Logic in Computer Science (Cat. No. PR00158).

[21]  Marcin Jurdzinski,et al.  Small Progress Measures for Solving Parity Games , 2000, STACS.

[22]  Igor Walukiewicz.  Monadic second-order logic on tree-like structures , 2002, Theor. Comput. Sci..

[23]  M. Rabin.  Weakly Definable Relations and Special Automata , 1970 .

[24]  Igor Walukiewicz,et al.  On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic , 1996, CONCUR.

[25]  Martin Otto,et al.  Bisimulation-invariant PTIME and higher-dimensional µ-calculus , 1999, Theor. Comput. Sci..

[26]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[27]  Dexter Kozen,et al.  RESULTS ON THE PROPOSITIONAL’p-CALCULUS , 2001 .

[28]  Andrzej Wlodzimierz Mostowski.  Hierarchies of Weak Automata and Weak Monadic Formulas , 1991, Theor. Comput. Sci..

[29]  Ronald Fagin,et al.  Reachability is harder for directed than for undirected finite graphs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[30]  M. Rabin.  Decidability of second-order theories and automata on infinite trees. , 1969 .

[31]  Bruno Courcelle,et al.  The Monadic Second order Logic of Graphs VI: on Several Representations of Graphs By Relational Structures , 1994, Discret. Appl. Math..

[32]  Ronald Fagin,et al.  Reachability is harder for directed than for undirected finite graphs , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.