No abstract
Wythoff's Game is a variation of Nim in which players may take an equal number of stones from each pile or make valid Nim moves. W. A. Wythoff proved that the set of P-Positions (losing position), C, for Wythoff's Game is given by C := ( kφ , kφ 2 ), ( kφ 2 , kφ ) : k ∈ Z ≥0 [Wyt07]. An open Wythoff problem remains where players make the valid Nim moves or remove kb stones from each pile, where b is a fixed integer. We denote this as the (b, b) game. For example, regular Wythoff's Game is just the (1, 1) game. In 2009, Duchêne and Gravier [DG09] proved an algorithm to generate the set of P-Positions for the (2, 2) game by exploiting the periodic nature of the differences of stones between the two piles modulo 4. We observe similar cyclic behaviour (see definition 3.2) for any b, where b is a power of 2, modulo b 2 , and construct an algorithm to generate the set of P-Positions for this game. Let a be a power of 2. We prove our algorithm works by first showing that it holds for the first a 2 terms in the (a, a) game. Next, we construct an ordered multiset for the (2a, 2a) game from the a 2 terms, and an inductive proof follows. Moreover, we conjecture that all cyclic games require a to be a power of 2, suggesting that there is no similar structure in the generalised (b, b) game where b isn't a power of 2. Future directions for generalising this result would likely utilise numeration systems, particularly the PV numbers.
Gene expression data provides molecular insights into the functional impact of genetic variation, for example through expression quantitative trait loci (eQTL). With an improving understanding of the association between genotypes and gene expression comes a greater concern that gene expression profiles could be matched to genotype profiles of the same individuals in another dataset, known as a linking attack. Prior works demonstrating such a risk could analyze only a fraction of eQTLs that are independent due to restrictive model assumptions, leaving the full extent of this risk incompletely understood. To address this challenge, we introduce the discriminative sequence model (DSM), a novel probabilistic framework for predicting a sequence of genotypes based on gene expression data. By modeling the joint distribution over all known eQTLs in a genomic region, DSM improves the power of linking attacks with necessary calibration for linkage disequilibrium and redundant predictive signals. We demonstrate greater linking accuracy of DSM compared to existing approaches across a range of attack scenarios and datasets including up to 22K individuals, suggesting that DSM helps uncover a substantial additional risk overlooked by previous studies. Our work provides a unified framework for assessing the privacy risks of sharing diverse omics datasets beyond transcriptomics.
Gene expression data provide molecular insights into the functional impact of genetic variation, for example, through expression quantitative trait loci (eQTLs). With an improving understanding of the association between genotypes and gene expression comes a greater concern that gene expression profiles could be matched to genotype profiles of the same individuals in another data set, known as a linking attack. Prior works show such a risk could analyze only a fraction of eQTLs that is independent owing to restrictive model assumptions, leaving the full extent of this risk incompletely understood. To address this challenge, we introduce the discriminative sequence model (DSM), a novel probabilistic framework for predicting a sequence of genotypes based on gene expression data. By modeling the joint distribution over all known eQTLs in a genomic region, DSM improves the power of linking attacks with necessary calibration for linkage disequilibrium and redundant predictive signals. We show greater linking accuracy of DSM compared with existing approaches across a range of attack scenarios and data sets including up to 22,288 individuals, suggesting that DSM helps uncover a substantial additional risk overlooked by previous studies. Our work provides a unified framework for assessing the privacy risks of sharing diverse omics data sets beyond transcriptomics.
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