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Widths of regular and context-free languages Mestel, David in 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019) (in press) Detailed reference viewed: 54 (5 UL)Quantifying information flow in interactive systems Mestel, David in 2019 IEEE 32nd Computer Security Foundations Symposium (CSF) (2019) We consider the problem of quantifying information flow in interactive systems, modelled as finite-state transducers in the style of Goguen and Meseguer. Our main result is that if the system is ... [more ▼] We consider the problem of quantifying information flow in interactive systems, modelled as finite-state transducers in the style of Goguen and Meseguer. Our main result is that if the system is deterministic then the information flow is either logarithmic or linear, and there is a polynomial-time algorithm to distinguish the two cases and compute the rate of logarithmic flow. To achieve this we first extend the theory of information leakage through channels to the case of interactive systems, and establish a number of results which greatly simplify computation. We then show that for deterministic systems the information flow corresponds to the growth rate of antichains inside a certain regular language, a property called the width of the language. In a companion work we have shown that there is a dichotomy between polynomial and exponential antichain growth, and a polynomial time algorithm to distinguish the two cases and to compute the order of polynomial growth. We observe that these two cases correspond to logarithmic and linear information flow respectively. Finally, we formulate several attractive open problems, covering the cases of probabilistic systems, systems with more than two users and nondeterministic systems where the nondeterminism is assumed to be innocent rather than demonic. [less ▲] Detailed reference viewed: 92 (6 UL)A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics Atashpendar, Arash ; Mestel, David ; et al E-print/Working paper (2018) From the output produced by a memoryless deletion channel from a uniformly random input of known length n, one obtains a posterior distribution on the channel input. The difference between the Shannon ... [more ▼] From the output produced by a memoryless deletion channel from a uniformly random input of known length n, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length m, and it is natural to ask for which outputs this is extremized. This question was posed in a previous work, where it was conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant strings 𝟶𝟶𝟶… and 𝟷𝟷𝟷… and the alternating strings 𝟶𝟷𝟶𝟷… and 𝟷𝟶𝟷𝟶… respectively. In the present work we confirm the minimization conjecture in the asymptotic limit using results from hidden word statistics. We show how the analytic-combinatorial methods of Flajolet, Szpankowski and Vall\'ee for dealing with the hidden pattern matching problem can be applied to resolve the case of fixed output length and n→∞, by obtaining estimates for the entropy in terms of the moments of the posterior distribution and establishing its minimization via a measure of autocorrelation. [less ▲] Detailed reference viewed: 131 (44 UL)From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions Atashpendar, Arash ; ; et al E-print/Working paper (2018) A binary string transmitted via a memoryless i.i.d. deletion channel is received as a subsequence of the original input. From this, one obtains a posterior distribution on the channel input, corresponding ... [more ▼] A binary string transmitted via a memoryless i.i.d. deletion channel is received as a subsequence of the original input. From this, one obtains a posterior distribution on the channel input, corresponding to a set of candidate supersequences weighted by the number of times the received subsequence can be embedded in them. In a previous work it is conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant and the alternating strings, respectively. In this work, in addition to revisiting the entropy minimization conjecture, we also address several related combinatorial problems. We present an algorithm for counting the number of subsequence embeddings using a run-length encoding of strings. We then describe methods for clustering the space of supersequences such that the cardinality of the resulting sets depends only on the length of the received subsequence and its Hamming weight, but not its exact form. Then, we consider supersequences that contain a single embedding of a fixed subsequence, referred to as singletons, and provide a closed form expression for enumerating them using the same run-length encoding. We prove an analogous result for the minimization and maximization of the number of singletons, by the alternating and the uniform strings, respectively. Next, we prove the original minimal entropy conjecture for the special cases of single and double deletions using similar clustering techniques and the same run-length encoding, which allow us to characterize the distribution of the number of subsequence embeddings in the space of compatible supersequences to demonstrate the effect of an entropy decreasing operation. [less ▲] Detailed reference viewed: 130 (42 UL) |
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