We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, st-connectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic st-connectivity for constant width grid graphs with time bound O (log log n) per operation on a random access machine. The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools.

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In this paper we study the logic L !! , which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of L !! with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic L !! is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ae M (i.e., a subset S = fa j M j= '(a)g defined by some formula ') is invariant under all automorphisms of M . The same is of course true for subsets of M