The Bounded Weak Monadic Quantifier Alternation Hierarchy of Equational Graphs Is Infinite

Here we deal with the question of definability of infinite graphs up to isomorphism by weak monadic second-order formulae. In this respect, we prove that the quantifier alternation bounded hierarchy of equational graphs is infinite. Two proofs are given: the first one is based on the Ehrenfeucht-FraissE games; the second one uses the arithmetical hierarchy. Next, we give a new proof of the Thomas's result according to which the bounded hierarchy of the weak monadic second-order logic of the complete binary tree is infinite.

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