A zero-one law for first-order logic on random images
Coupier, David ; Desolneux, Agnès ; Ycart, Bernard
HAL, hal-00020672 / Harvested from HAL
Text on HAL
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.
Publié le : 2004-07-05
Classification:  threshold function,  zero-one law,  first-order logic,  random image,  threshold function.,  60 F 20,  [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] 
@article{hal-00020672,
     author = {Coupier, David and Desolneux, Agn\`es and Ycart, Bernard},
     title = {A zero-one law for first-order logic on random images},
     journal = {HAL},
     volume = {2004},
     number = {0},
     year = {2004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/hal-00020672}
}

Coupier, David; Desolneux, Agnès; Ycart, Bernard. A zero-one law for first-order logic on random images. HAL, Tome 2004 (2004) no. 0, . http://gdmltest.u-ga.fr/item/hal-00020672/