Object oriented constraint programs (OOCPs) emerge as a leading evolution of constraint programming and artificial intelligence, first applied to a range of industrial applications called configuration problems. The rich variety of technical approaches to solving configuration problems (CLP(FD), CC(FD), DCSP, Terminological systems, constraint programs with set variables, . . . ) is a source of difficulty. No universally accepted formal language exists for communicating about OOCPs, which makes...

Let Gₙ be the random graph on [n] = 1,...,n with the probability of i,j being an edge decaying as a power of the distance, specifically the probability being $p_{|i-j|}=1/{|i-j|}^{\alpha }$, where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a “finite compactness”. Lastly, as a consequence, we prove that the zero-one law (for first order logic)...

Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability $p_{|i-j|}=1/{|i-j|}^{\alpha }$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..

Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let ${}^{\infty }$ denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits ${}^{\infty }$ cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a ${}^{\infty }$ function f:Rⁿ → R. This implies ${}^{\infty }$ approximation of definable continuous functions and gluing of ${}^{\infty }$ functions defined on closed definable sets.