Upper bounds on the sizes of variable strength covering arrays using the Lov\'{a}sz local lemma
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Bibliographic record
Abstract
Covering arrays are generalizations of orthogonal arrays that have been widely studied and are used in software testing. The probabilistic method has been employed to derive upper bounds on the sizes of minimum covering arrays and give asymptotic upper bounds that are logarithmic on the number of columns of the array. This corresponds to test suites with a desired level of coverage of the parameter space where we guarantee the number of test cases is logarithmic on the number of parameters of the system. In this paper, we study variable strength covering arrays, a generalization of covering arrays that uses a hypergraph to specify the sets of columns where coverage is required; (standard) covering arrays is the special case where coverage is required for all sets of columns of a fixed size $t$, its strength. We use the probabilistic method to obtain upper bounds on the number of rows of a variable strength covering array, given in terms of parameters of the hypergraph. We then compare this upper bound with another one given by a density-based greedy algorithm on different types of hypergraph such as $t$-designs, cyclic consecutive hypergraphs, planar triangulation hypergraphs, and a more specific hypergraph given by a clique of higher strength on top of a "base strength". The conclusions are dependent on the class of hypergraph, and we discuss specific characteristics of the hypergraphs which are more amenable to using different versions of the Lov\'{a}sz local lemma.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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