DICHOTOMY RESULT FOR INDEPENDENCE-FRIENDLY PREFIXES OF GENERALIZED QUANTIFIERS

We study the expressive power of independence-friendly quantifier prefixes composed of universal $\left( {\forall x/X} \right)$ , existential $\left( {\exists x/X} \right)$ , and majority quantifiers $\left( {Mx/X} \right)$ . We provide four quantifier prefixes that can express NP hard properties and show that all quantifier prefixes capable of expressing NP-hard properties embed at least one of these four quantifier prefixes. As for the quantifier prefixes that do not embed any of these four quantifier prefixes, we show that they are equivalent to a first-order quantifier prefix composed of $\forall x$ , $\exists x$ , and M x . In unison, our results imply a dichotomy result: every independence-friendly quantifier prefix is either decidable in LOGSPACE or NP hard.

[1]  K. Jon Barwise,et al.  On branching quantifiers in English , 1979, J. Philos. Log..

[2]  Fausto Barbero,et al.  ON EXISTENTIAL DECLARATIONS OF INDEPENDENCE IN IF LOGIC , 2012, The Review of Symbolic Logic.

[3]  Thomas Schwentick,et al.  Existential second-order logic over graphs: Charting the tractability frontier , 2004, JACM.

[4]  Xavier Caicedo,et al.  Equivalence and quantifier rules for logic with imperfect information , 2009, Log. J. IGPL.

[5]  H. Enderton.  Finite Partially-Ordered Quantifiers , 1970 .

[6]  Merlijn Sevenster,et al.  Branches of imperfect information : logic, games, and computation , 2002 .

[7]  Fredrik Engström,et al.  Generalized Quantifiers in Dependence Logic , 2012, J. Log. Lang. Inf..

[8]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[9]  Gabriel Sandu,et al.  On the logic of informational independence and its applications , 1993, J. Philos. Log..

[10]  Wilfrid Hodges,et al.  Compositional Semantics for a Language of Imperfect Information , 1997, Log. J. IGPL.

[11]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[12]  Wilbur John Walkoe,et al.  Finite Partially-Ordered Quantification , 1970, J. Symb. Log..

[13]  Merlijn Sevenster,et al.  Independence-Friendly Logic - a Game-Theoretic Approach , 2011, London Mathematical Society lecture note series.

[14]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[15]  Andreas Blass,et al.  Henkin quantifiers and complete problems , 1986, Ann. Pure Appl. Log..