We prove a regularity lemma with respect to arbitrary Keisler measures μ on V , ν on W where the bipartite graph (V,W,R) is definable in a saturated structure M̄ and the formula R(x, y) is stable. The proof is rather quick, making use of local stability theory. The special case where (V,W,R) is pseudofinite, μ, ν are the counting measures, and M̄ is suitably chosen (for example a nonstandard mode...

In this note we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on recent results of Nagle, Schacht and the authors, we give here solutions to these problems. In particular, we prove the following: Let K (k) t be the complete kuniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(n) copies of ...

We address a problem connected to the unfolding semantics of functional programming languages: give a useful characterization of those infinite λ-terms that are λletrec-expressible in the sense that they arise as infinite unfoldings of terms in λletrec, the λ-calculus with letrec. We provide two characterizations, using concepts we introduce for infinite λ-terms: regularity, strong regularity, ...

Understanding how humans code for and respond to environmental uncertainty/regularity is a question shared by current computational and neurobiological approaches to human cognition. To date, studies investigating neurobiological systems that track input uncertainty have examined responses to uni-sensory streams. It is not known, however, whether there exist brain systems that combine informati...

Black-box optimization (BBO) algorithms are concerned with finding the best solutions for problems missing analytical details. Most classical methods such based on strong and fixed a priori assumptions, as Gaussianity. However, complex real-world problems, especially when global optimum is desired, could be very far from assumptions because of their diversities, causing unexpected obstacles. In...

We analyze the consistency strength of regularity on the basis of Feferman’s operational set theory OST. Our main result shows that regularity over OST for operations corresponds to regularity with respect to set-theoretic functions in frameworks like Kripke-Platek set theory. As a consequence, we obtain that OST plus the operational axiom (Inac) is consistency-equivalent to Kripke-Platek set t...

We investigate decompositions of a graph into a small number of low diameter subgraphs. Let P (n, 2, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ . . . ∪ Ek such that |E0| ≤ 2n and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi’s regularity lemma, Polcyn and Ruciński showed that P (n, 2, 4) is...

We prove a variant of the abstract probabilistic version of Szemerédi’s regularity lemma, due to Tao, which applies to a number of structures (including graphs, hypergraphs, hypercubes, graphons, and many more) and works for random variables in Lp for any p > 1. Our approach is based on martingale difference sequences.