Abstract
The comprehension of similarity metrics lags behind that of distance metrics. This study aims to address this disparity by synthesizing the properties of similarity metrics and examining them through the lens of weighted epistemic logic. By incorporating these metrics, we analyze knowledge systems in terms of their metric properties. Modal logic techniques, including bisimulation and bounded morphism, are employed to investigate the definable and undefinable properties of similarity. Definable alternatives for undefinable properties are proposed.
Supported by Project of Humanities and Social Sciences, MOE (China).
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Notes
- 1.
As observed in [4, p. 4], D1 follows from D3 and D4: by setting \(y=x\) in D4, we obtain \(0 \le d(x,x)\), and by letting \(x=z\) in D4, we arrive at D1. It remains unclear why the 2016 edition weakens this to suggest that D1 follows from D2 to D4.
- 2.
This condition is not required in an earlier version [14].
- 3.
- 4.
In [2], it is suggested to treat d(x, y) as \(2p(x, y) - p(x, x) - p(y, y)\), which enforces that \(d(x, x) = 0\).
- 5.
The original definition in [3] uses “if and only if” instead of “implies”, however, the other direction is trivial as long as s is a function.
- 6.
It is called “covering inequality” in [6].
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Liang, X., Wáng, Y.N. (2025). Characterization of Similarity Metrics in Epistemic Logic. In: Hadfi, R., Anthony, P., Sharma, A., Ito, T., Bai, Q. (eds) PRICAI 2024: Trends in Artificial Intelligence. PRICAI 2024. Lecture Notes in Computer Science(), vol 15281. Springer, Singapore. https://doi.org/10.1007/978-981-96-0116-5_9
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