On the strictness of the quantifier structure hierarchy in first-order logic

We study a natural hierarchy in first-order logic, namely the quantifier structure hierarchy, which gives a systematic classification of first-order formulas based on structural quantifier resource. We define a variant of Ehrenfeucht-Fraisse games that characterizes quantifier classes and use it to prove that this hierarchy is strict over finite structures, using strategy compositions. Moreover, we prove that this hierarchy is strict even over ordered finite structures, which is interesting in the context of descriptive complexity.Comment: 38 pages, 8 figure

The Power of the Depth of Iteration in Defining Relations by Induction

In this paper we provoke the question of whether the sequence ... is strictly increasing, i.e., the question of whether increasing the depth of iteration increases the expressive power of defining by induction. Solving this question should have a deep impact on computer science as well as on mathematical logic since it is a question in a subject on the crossroads between them, namely, descriptive complexity. We shall mention a potential way of tackling the problem

Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words

We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels

The Arity Hierarchy in the Polyadic μ\mu-Calculus

The polyadic mu-calculus is a modal fixpoint logic whose formulas define relations of nodes rather than just sets in labelled transition systems. It can express exactly the polynomial-time computable and bisimulation-invariant queries on finite graphs. In this paper we show a hierarchy result with respect to expressive power inside the polyadic mu-calculus: for every level of fixpoint alternation, greater arity of relations gives rise to higher expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282