Evaluating testing methods by delivered reliability

There are two main goals in testing software: (1) to achieve adequate quality (debug testing), where the objective is to probe the software for defects so that these can be removed, and (2) to assess existing quality (operational testing), where the objective is to gain confidence that the software is reliable. Debug methods tend to ignore random selection of test data from an operational profile, while for operational methods this selection is all-important. Debug methods are thought to be good at uncovering defects so that these can be repaired, but having done so they do not provide a technically defensible assessment of the reliability that results. On the other hand, operational methods provide accurate assessment, but may not be as useful for achieving reliability. This paper examines the relationship between the two testing goals, using a probabilistic analysis. We define simple models of programs and their testing, and try to answer the question of how to attain program reliability: is it better to test by probing for defects as in debug testing, or to assess reliability directly as in operational testing? Testing methods are compared in a model where program failures are detected and the software changed to eliminate them. The “better” method delivers higher reliability after all test failures have been eliminated. Special cases are exhibited in which each kind of testing is superior. An analysis of the distribution of the delivered reliability indicates that even simple models have unusual statistical properties, suggesting caution in interpreting theoretical comparisons

Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

In his 1947 paper that inaugurated the probabilistic method, Erd\H{o}s proved the existence of $2\log{n}$-Ramsey graphs on $n$ vertices. Matching Erd\H{o}s' result with a constructive proof is a central problem in combinatorics, that has gained a significant attention in the literature. The state of the art result was obtained in the celebrated paper by Barak, Rao, Shaltiel and Wigderson [Ann. Math'12], who constructed a $2^{2^{(\log\log{n})^{1-\alpha}}}$-Ramsey graph, for some small universal constant $\alpha > 0$. In this work, we significantly improve the result of Barak~\etal and construct $2^{(\log\log{n})^c}$-Ramsey graphs, for some universal constant $c$. In the language of theoretical computer science, our work resolves the problem of explicitly constructing two-source dispersers for polylogarithmic entropy

The Erdös-Ko-Rado theorem for vector spaces

AbstractLet V be an n-dimensional vector space over GF(q) and for integers k⩾t>0 let mq(n, k, t) denote the maximum possible number of subspaces in a t-intersecting family F of k-dimensional subspaces of V, i.e., dim F ∩ F′ ⩾ t holds for all F, F′ ϵ F. It is shown that mq(n,k,t)=maxn−tk−t, 2k−tk for n⩾2k−t while for n⩽2k−t trivially mq(n,k,t)=nk holds

How (Not) to Cut Your Cheese

It is well known that a line can intersect at most 2n−1 unit squares of the n × n chessboard. Here we consider the three-dimensional version: how many unit cubes of the 3-dimensional cube [0,n]3 can a hyperplane intersect

Induced restricted Ramsey theorems for spaces

AbstractThe induced restricted versions of the vector space Ramsey theorem and of the Graham-Rothschild parameter set theorem are proved

A Characterization of Mixed Unit Interval Graphs

We give a complete characterization of mixed unit interval graphs, the intersection graphs of closed, open, and half-open unit intervals of the real line. This is a proper superclass of the well known unit interval graphs. Our result solves a problem posed by Dourado, Le, Protti, Rautenbach and Szwarcfiter (Mixed unit interval graphs, Discrete Math. 312, 3357-3363 (2012)).Comment: 17 pages, referees' comments adde