"Weak yet strong'' restrictions of Hindman's Finite Sums Theorem

We present a natural restriction of Hindman’s Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman’s Theorem with analogous propertie

On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝ We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles

Unprovability results involving braids

We construct long sequences of braids that are descending with respect to the standard order of braids (``Dehornoy order''), and we deduce that, contrary to all usual algebraic properties of braids, certain simple combinatorial statements involving the braid order are true, but not provable in the subsystems ISigma1 or ISigma2 of the standard Peano system.Comment: 32 page

An optimization procedure based on thermal discomfort minimization to support the design of comfortable Net Zero Energy Buildings

The European standard EN 15251 specifies design criteria for dimensioning of building systems. In detail, it proposes that the adaptive comfort model is used, at first, for dimensioning passive means; but, if indoor operative temperature does not meet the chosen long-term adaptive comfort criterion in the “cooling season”, the design would include a mechanical cooling system. In this case, the reference design criteria are provided accordingly the Fanger comfort model. However, there is a discontinuity by switching from the adaptive to the Fanger model, since the best building variant, according to the former, may not coincide with the optimal according to the latter. In this paper, an optimization procedure to support the design of a comfort-optimized net zero energy building is proposed. It uses an optimization engine (GenOpt) for driving a dynamic simulation engine (EnergyPlus) towards those building variants that minimize, at first, two seasonal long-term discomfort indices based on an adaptive model; and if indoor conditions do not meet the adaptive comfort limits or analyst’s expectations, it minimizes two seasonal long-term discomfort indices based on the Fanger model. The calculation of such indices has been introduced in EnergyPlus via the Energy Management System module, by writing computer codes in the EnergyPlus Reference Language. The used long-term discomfort indices proved to provide similar ranking capabilities of building variants, even if they are based on different comfort models, and the proposed procedure meets the two- step procedure suggested by EN 15251 without generating significant discontinuities

Hindman's theorem for sums along the full binary tree, Sigma02-induction and the Pigeonhole principle for trees

We formulate a restriction of Hindman's Finite Sums Theorem in which monochromaticity is required only for sums corresponding to rooted finite paths in the full binary tree. We show that the resulting principle is equivalent to Sigma(0)(2) -induction over RCA(0). The proof uses the equivalence of this Hindman-type theorem with the Pigeonhole Principle for trees T T-1 with an extra condition on the solution tree

AutoDIAL: Automatic DomaIn Alignment Layers

Classifiers trained on given databases perform poorly when tested on data acquired in different settings. This is explained in domain adaptation through a shift among distributions of the source and target domains. Attempts to align them have traditionally resulted in works reducing the domain shift by introducing appropriate loss terms, measuring the discrepancies between source and target distributions, in the objective function. Here we take a different route, proposing to align the learned representations by embedding in any given network specific Domain Alignment Layers, designed to match the source and target feature distributions to a reference one. Opposite to previous works which define a priori in which layers adaptation should be performed, our method is able to automatically learn the degree of feature alignment required at different levels of the deep network. Thorough experiments on different public benchmarks, in the unsupervised setting, confirm the power of our approach.Comment: arXiv admin note: substantial text overlap with arXiv:1702.06332 added supplementary materia

An adjacent Hindman theorem for uncountable groups

Results of Hindman, Leader and Strauss and of the second author and Rinot showed that some natural analogs of Hindman's theorem fail for all uncountable cardinals. Results in the positive direction were obtained by Komj\'ath, the first author, and the second author and Lee, who showed that there are arbitrarily large abelian groups satisfying some Hindman-type property. Inspired by an analog result studied by the first author in the countable setting, we prove a new variant of Hindman's theorem for uncountable cardinals, called the Adjacent Hindman's Theorem: For every $\kappa$ there is a $\lambda$ such that, whenever a group $G$ of cardinality $\lambda$ is coloured in $\kappa$ colours, there exists a $\lambda$-sized injective sequence of elements of $G$ with all finite products of adjacent terms of the sequence of the same colour. We obtain bounds on $\lambda$ as a function of $\kappa$, and prove that such bounds are optimal. This is the first example of a Hindman-type result for uncountable cardinals that we can prove also in the non-Abelian setting and, furthermore, it is the first such example where monochromatic products (or sums) of unbounded length are guaranteed

Sharp thresholds for hypergraph regressive Ramsey numbers

The f-regressive Ramsey number R(f)(reg)(d, n) is the minimum N such that every coloring of the d-tuples of an N-element set mapping each x(1),...,x(d) to a color below f(x(1)) (when f(x(1)) is positive) contains a min-homogeneous set of size n, where a set is called min-homogeneous if every two d-tuples from this set that have the same smallest element get the same color. If f is the identity, then we are dealing with the standard regressive Ramsey numbers as defined by Kanamori and McAloon. The existence of such numbers for hypergraphs or arbitrary dimension is unprovable from the axioms of Peano Arithmetic. In this paper we classify the growth-rate of the regressive Ramsey numbers for hypergraphs in dependence of the growth-rate of the parameter function f. We give a sharp classification of the thresholds at which the f-regressive Ramsey numbers undergo a drastical change in growth-rate. The growth-rate has to be measured against a scale of fast-growing recursive functions indexed by finite towers of exponentiation in base omega (the first limit ordinal). The case of graphs has been treated by Lee, Kojman, Omri and Weiermann. We extend their results to hypergraphs of arbitrary dimension. From the point of view of Logic, our results classify the provability of the Regressive Ramsey Theorem for hypergraphs of fixed dimension in subsystems of Peano Arithmetic with restricted induction principles. (C) 2010 Elsevier Inc. All rights reserved