Axiomatizing rectangular grids with no extra non-unary relations

12/20/2019

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by   Eryk Kopczyński, et al.

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We construct a formula ϕ which axiomatizes non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆N is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) ≤ n and t(n),s(n) = Ω(log(n)).