A zero-one law for first-order logic on random images

Details A zero-one law for first-order logic on random images A zero-one law for first-order logic on random images

2006-03-14

arxiv.org/abs/math/0603333v1

Trends in Mathematics, 2004 Birkh\"{a}user Verlag Basel / Switzerland (2004) 495-505

Mathematics

Probability

Scientific paper

For an $n\times n$ random image with independent pixels, black with probability $p(n)$ and white with probability $1-p(n)$, the probability of satisfying any given first-order sentence tends to 0 or 1, provided both $p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for any integer $k$. The result is proved by computing the threshold function for basic local sentences, and applying Gaifman's theorem.

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