### An improved inductive definition of two restricted classes of triangulations of the plane

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This paper is a survey of results on finite variable logics in finite model theory. It focusses on the common underlying techniques that unite many such results.

We provide an elementary proof of the fixpoint alternation hierarchy in arithmetic, which in turn allows us to simplify the proof of the modal mu-calculus alternation hierarchy. We further show that the alternation hierarchy on the binary tree is strict, resolving a problem of Niwiński.

For every fixed-point expression e of alternation-depth r, we construct a new fixed-point expression e' of alternation-depth 2 and size $\mathcal{O}(r\xb7|e\left|\right)$. Expression e' is equivalent to e whenever operators are distributive and the underlying complete lattice has a co-continuous least upper bound. We alternation-depth but also w.r.t. the increase in size of the resulting expression.

In the field of automatic proving, the study of the sets of prime implicants or implicates of a formula has proven to be very important. If we focus on non-classical logics and, in particular, on temporal logics, such study is useful even if it is restricted to the set of unitary implicants/implicates [P. Cordero, M. Enciso, and I. de Guzmán: Structure theorems for closed sets of implicates/implicants in temporal logic. (Lecture Notes in Artificial Intelligence 1695.) Springer–Verlag, Berlin 1999]....