Toolkit for First Order Extensions of Monadi Games

Games David Janin1 and Jerzy Mar inkowski2 1 Laboratoire Bordelais de Re her he en Informatique, Universit e de Bordeaux I, 351, ours de la Lib eration, 33 405 Talen e edex, janin labri.u-bordeaux.fr 2 Institute of Computer S ien e, University of Wro law, Przesmy kiego 20, 51151 Wro law, Poland, jma t s.uni.wro .pl Abstra t. In 1974 R. Fagin proved that properties of stru tures whi h are in NP are exa tly the same as those expressible by existential se ond order senten es, that is senten es of the form: there exist ~ P su h that ', where ~ P is a tuple of relation symbols. and ' is a rst order formula. Fagin was also the rst to study monadi NP: the lass of properties expressible by existential se ond order senten es where all the quanti ed relations are unary. In [?℄ Ajtai, Fagin and Sto kmeyer introdu e losed monadi NP: the lass of properties whi h an be expressed by a kind of monadi se ond order existential formula, where the se ond order quanti ers an interleave with rst order quanti ers. In order to prove that su h alternation of quanti ers gives a lot of additional expressive power they onstru t graph properties P1 and P2: P1 is expressible by a senten e with the quanti er pre x in the lass (98) E (98) 1 but not by a boolean ombination of senten es from monadi NP (i.e with the pre x of the form