Abstract

It has been conjectured [FSV93] that an existential secondoder formula, in which the second-order quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary built-in relations. In this paper it is shown that Graph Connectivity cannot be expressed by Monadic NP formulas in the presence of arbitrary built-in relations of degree n^o(1). The result is obtained by using a simplified version of a method introduced in [Sch94] that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy.

Citations

242	Generalized first-order spectra and polynomial-time recognizable sets - Fagin - 1974
146	Applications of games to the completeness problem for formalized theories - Ehrenfeucht - 1961
88	Sigma 1 formulae on finite structures - Ajtai
79	On monadic NP vs. monadic co-NP - Fagin, Stockmeyer, et al. - 1995
69	Reachability is harder for directed than for undirected finite graphs - Ajtai, Fagin - 1990
69	Sur quelques classifications des systemes de relations - Frai’sse - 1954
57	Monadic generalized spectra, Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik - Fagin - 1975
21	Second-order and inductive definability on finite structures. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik - Rougemont - 1987
18	2007): Einführung in die mathematische Logik, 5 edition. Spektrum Akademischer - Ebbinghaus, Flum, et al.
18	Graph connectivity and monadic NP - Schwentick - 1994
11	Logical reducibility and monadic NP, in - Cosmadakis - 1993