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Monotonic Bit-Invariant Permutation-Invariant Metrics on the Set of All Infinite Binary Sequences

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Uncertainty, Constraints, and Decision Making

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 484))

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Abstract

In a computer, all the information about an object is described by a sequence of 0s and 1s. At any given moment of time, we only have partial information, but as we perform more measurements and observations, we get longer and longer sequence that provides a more and more accurate description of the object. In the limit, we get a perfect description by an infinite binary sequence. If the objects are similar, measurement results are similar, so the resulting binary sequences are similar. Thus, to gauge similarity of two objects, a reasonable idea is to define an appropriate metric on the set of all infinite binary sequences. Several such metrics have been proposed, but their limitation is that while the order of the bits is rather irrelevant—if we have several simultaneous measurements, we can place them in the computer in different order—the distance measured by current formulas change if we select a different order. It is therefore natural to look for permutation-invariant metrics, i.e., distances that do not change if we select different orders. In this paper, we provide a full description of all such metrics. We also explain the limitation of these new metrics: that they are, in general, not computable.

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References

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Acknowledgements

This work was supported in part by the National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI Includes), and by the AT&T Fellowship in Information Technology.

It was also supported by the program of the development of the Scientific-Educational Mathematical Center of Volga Federal District No. 075-02-2020-1478, and by a grant from the Hungarian National Research, Development and Innovation Office (NRDI).

The work of Irina Perfilieva was supported by European Regional Development Fund (ERDF) and European Social Fund (ESF) via the project “Centre for the development of Artificial Intelligence Methods for the Automotive Industry of the region” No. CZ.02.1.01/0.0/0.0/17_049/0008414. The authors are thankful to all the participants of the 19th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’2022, (Milan, Italy, July 11–15, 2022), especially to Alexander Šostak, for valuable discussions.

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Correspondence to Vladik Kreinovich .

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Perfilieva, I., Kreinovich, V. (2023). Monotonic Bit-Invariant Permutation-Invariant Metrics on the Set of All Infinite Binary Sequences. In: Ceberio, M., Kreinovich, V. (eds) Uncertainty, Constraints, and Decision Making. Studies in Systems, Decision and Control, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-031-36394-8_64

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