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Cryptographic Limitations on Learning Boolean Formulae and Finite Automata
 PROCEEDINGS OF THE TWENTYFIRST ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1989
"... In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntact ..."
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Cited by 311 (16 self)
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In this paper we prove the intractability of learning several classes of Boolean functions in the distributionfree model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent, in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of wellknown publickey cryptosystems to the learning problems. We prove that a polynomialtime learning algorithm for Boolean formulae, deterministic finite automata or constantdepth threshold circuits would have dramatic consequences for cryptography and number theory: in particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a generalization of graph coloring.
Learning DFA from Simple Examples
, 1997
"... Efficient learning of DFA is a challenging research problem in grammatical inference. It is known that both exact and approximate (in the PAC sense) identifiability of DFA is hard. Pitt, in his seminal paper posed the following open research problem: "Are DFAPACidentifiable if examples are drawn ..."
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Cited by 20 (6 self)
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Efficient learning of DFA is a challenging research problem in grammatical inference. It is known that both exact and approximate (in the PAC sense) identifiability of DFA is hard. Pitt, in his seminal paper posed the following open research problem: "Are DFAPACidentifiable if examples are drawn from the uniform distribution, or some other known simple distribution?" [25]. We demonstrate that the class of simple DFA (i.e., DFA whose canonical representations have logarithmic Kolmogorov complexity) is efficiently PAC learnable under the Solomonoff Levin universal distribution. We prove that if the examples are sampled at random according to the universal distribution by a teacher that is knowledgeable about the target concept, the entire class of DFA is efficiently PAC learnable under the universal distribution. Thus, we show that DFA are efficiently learnable under the PACS model [6]. Further, we prove that any concept that is learnable under Gold's model for learning from characteristic samples, Goldman and Mathias' polynomial teachability model, and the model for learning from example based queries is also learnable under the PACS model.
On the Complexity of Small Description and Related Topics
, 1992
"... The class P/poly is known to be the class of sets with small descriptions, more specifically, polynomial size circuits. In this paper, we discuss the problem of obtaining the polynomial size circuits for a given set in P/poly by using the set as an oracle. Recent results on upper and lower bounds of ..."
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Cited by 1 (0 self)
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The class P/poly is known to be the class of sets with small descriptions, more specifically, polynomial size circuits. In this paper, we discuss the problem of obtaining the polynomial size circuits for a given set in P/poly by using the set as an oracle. Recent results on upper and lower bounds of the relative complexity of this problem are presented. We also introduce two related research topics  query learning and identity mapping network  and explain how they are related to this problem.