A Toolkit for First Order Extensions of Monadic Games

In 1974 R. Fagin proved that properties of structures which are in NP are exactly the same as those expressible by existential second order sentences, that is sentences of the form: there exist such that , where is a tuple of relation symbols. and is a first order formula. Fagin was also the first to study monadic NP: the class of properties expressible by existential second order sentences where all the quantified relations are unary. In [AFS00] Ajtai, Fagin and Stockmeyer introduce closed monadic NP: the class of properties which can be expressed by a kind of monadic second order existential formula, where the second order quantifiers can interleave with first order quantifiers. In order to prove that such alternation of quantifiers gives substantial additional expressive power they construct graph properties and : is expressible by a sentence with the quantifier prefix in the class 1 but not by a boolean combination of sentences from monadic NP (i.e with the prefix of the form and is expressible by a sentence but not by a Boolean combination of sentences of the form . A natural question arises here whether the hierarchy inside closed monadic NP, defined by the number of blocks of second order existential quantifiers, is strict. In this paper we present a technology for proving some non expressibility results for monadic second order logic. As a corollary we get a new, easy, proof of the two results from [AFS00] mentioned above. With our technology we can also make a first small step towards an answer to the hierarchy question by showing that the hierarchy inside closed monadic NP does not collapse on a first order level. The monadic complexity of properties definable in Kozen’s mu-calculus is also considered as our technology also applies to the mu-calculus itself.