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 n0(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.
In fact, Ajtai's result is far more general and not restricted to the monadic case.
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References
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Schwentick, T. (1995). Graph Connectivity, Monadic NP and built-in relations of moderate degree. In: Fülöp, Z., Gécseg, F. (eds) Automata, Languages and Programming. ICALP 1995. Lecture Notes in Computer Science, vol 944. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60084-1_92
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DOI: https://doi.org/10.1007/3-540-60084-1_92
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