Abstract
We investigate the polynomial-time learnability by using examples and membership queries. Angluin and Kharitonov [1] proved that various concept classes (e.g., Boolean formulae, non-deterministic finite automata) are not polynomial-time learnable in this learning model based on a public-key encryption scheme with a certain security (i.e., IND-CCA1). As a stronger learning model, we consider an a posteriori query learning model, and show that it is indeed stronger than the above learning model if a one-way function exists. Nevertheless, from a secure encryption scheme, we prove that many natural classes containing Boolean formula concept class is not polynomial-time learnable even in this stronger learning model. The security of an encryption scheme used in this paper is weaker than the one used by Angluin and Kharitonov.
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Notes
- 1.
Let n be the length of input. \(NC^i\) is the class computed by a family of polynomial size and \(O(\log ^i(n))\) depth circuits with bounded fan-in, and \(AC^i\) is the similar class except that gates are allowed to have unbounded fan-in.
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Nanashima, M. (2018). Cryptographic Limitations on Polynomial-Time a Posteriori Query Learning. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_24
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DOI: https://doi.org/10.1007/978-3-319-94667-2_24
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