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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.
Efficient Learning of Typical Finite Automata from Random Walks
, 1997
"... This paper describes new and efficient algorithms for learning deterministic finite automata. Our approach is primarily distinguished by two features: (1) the adoption of an averagecase setting to model the ``typical'' labeling of a finite automaton, while retaining a worstcase model for the under ..."
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Cited by 48 (10 self)
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This paper describes new and efficient algorithms for learning deterministic finite automata. Our approach is primarily distinguished by two features: (1) the adoption of an averagecase setting to model the ``typical'' labeling of a finite automaton, while retaining a worstcase model for the underlying graph of the automaton, along with (2) a learning model in which the learner is not provided with the means to experiment with the machine, but rather must learn solely by observing the automaton's output behavior on a random input sequence. The main contribution of this paper is in presenting the first efficient algorithms for learning nontrivial classes of automata in an entirely passive learning model. We adopt an online learning model in which the learner is asked to predict the output of the next state, given the next symbol of the random input sequence; the goal of the learner is to make as few prediction mistakes as possible. Assuming the learner has a means of resetting the target machine to a fixed start state, we first present an efficient algorithm that
Inferring Finite Automata with Stochastic Output Functions and an Application to Map Learning
, 1995
"... It is often useful for a robot to construct a spatial representation of its environment from experiments and observations, in other words, to learn a map of its environment by exploration. In addition, robots, like people, make occasional errors in perceiving the spatial features of their environmen ..."
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Cited by 41 (4 self)
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It is often useful for a robot to construct a spatial representation of its environment from experiments and observations, in other words, to learn a map of its environment by exploration. In addition, robots, like people, make occasional errors in perceiving the spatial features of their environments. We formulate map learning as the problem of inferring from noisy observations the structure of a reduced deterministic finite automaton. We assume that the automaton to be learned has a distinguishing sequence. Observation noise is modeled by treating the observed output at each state as a random variable, where each visit to the state is an independent trial and the correct output is observed with probability exceeding 1=2. We assume no errors in the state transition function. Using this framework, we provide an exploration algorithm to learn the correct structure of such an automaton with probability 1 \Gamma ffi , given as inputs ffi , an upper bound m on the number of states, a disti...
Grammar Inference, Automata Induction, and Language Acquisition
 Handbook of Natural Language Processing
, 2000
"... The natural language learning problem has attracted the attention of researchers for several decades. Computational and formal models of language acquisition have provided some preliminary, yet promising insights of how children learn the language of their community. Further, these formal models als ..."
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Cited by 22 (3 self)
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The natural language learning problem has attracted the attention of researchers for several decades. Computational and formal models of language acquisition have provided some preliminary, yet promising insights of how children learn the language of their community. Further, these formal models also provide an operational framework for the numerous practical applications of language learning. We will survey some of the key results in formal language learning. In particular, we will discuss the prominent computational approaches for learning different classes of formal languages and discuss how these fit in the broad context of natural language learning.
Learning Dynamics: System Identification for Perceptually Challenged Agents
, 1995
"... From the perspective of an agent, the input/output behavior of the environment in which it is embedded can be described as a dynamical system. Inputs correspond to the actions executable by the agent in making transitions between states of the environment. Outputs correspond to the perceptual inform ..."
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Cited by 20 (2 self)
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From the perspective of an agent, the input/output behavior of the environment in which it is embedded can be described as a dynamical system. Inputs correspond to the actions executable by the agent in making transitions between states of the environment. Outputs correspond to the perceptual information available to the agent in particular states of the environment. We view dynamical system identification as inference of deterministic finitestate automata from sequences of input/output pairs. The agent can influence the sequence of input/output pairs it is presented by pursuing a strategy for exploring the environment. We identify two sorts of perceptual errors: errors in perceiving the output of a state and errors in perceiving the inputs actually carried out in making a transition from one state to another. We present efficient, highprobability learning algorithms for a number of system identification problems involving such errors. We also present the results of empirical investi...
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.
Learning Regular Languages using RFSA
, 2001
"... Residual languages are important and natural components of regular languages. Most approaches in grammatical inference rely on this notion. Classical algorithms such as RPNI try to identify prefixes of positive learning examples which give rise to identical residuals. Here, we study inclusion relati ..."
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Cited by 17 (4 self)
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Residual languages are important and natural components of regular languages. Most approaches in grammatical inference rely on this notion. Classical algorithms such as RPNI try to identify prefixes of positive learning examples which give rise to identical residuals. Here, we study inclusion relations between residual languages. We lead experiments which show that when regular languages are randomly drawn using non deterministic representations, the number of inclusion relations is very important. We introduced in previous articles a new class of automata which is defined using the notion of residual languages: residual finite state automata (RFSA). RFSA representations of regular languages have far less states than DFA representations. We prove that RFSA are not polynomially characterizable. However, we design a new learning algorithm, DeLeTe2, based on the search of inclusion relations between residual languages, which produces a RFSA and have both good theoretical properties and good experimental performances.
A Polynomial Time Incremental Algorithm for Learning DFA
"... We present an efficient incremental algorithm for learning deterministic finite state automata (DFA) from labeled examples and membership queries. This algorithm is an extension of Angluin's ID procedure to an incremental framework. The learning algorithm is intermittently provided with labeled ..."
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Cited by 9 (4 self)
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We present an efficient incremental algorithm for learning deterministic finite state automata (DFA) from labeled examples and membership queries. This algorithm is an extension of Angluin's ID procedure to an incremental framework. The learning algorithm is intermittently provided with labeled examples and has access to a knowledgeable teacher capable of answering membership queries. The learner constructs an initial hypothesis from the given set of labeled examples and the teacher's responses to membership queries. If an additional example observed by the learner is inconsistent with the current hypothesis then the hypothesis is modified minimally to make it consistent with the new example. The update procedure ensures that the modified hypothesis is consistent with all examples observed thus far. The algorithm is guaranteed to converge to a minimum state DFA corresponding to the target when the set of examples observed by the learner includes a live complete set. We prove the convergence of this algorithm and analyze its time and space complexities.
Learning GeometricallyConstrained Hidden Markov Models for Robot Navigation: Bridging the TopologicalGeometrical Gap
 Journal of AI Research
, 2002
"... Hidden Markov models (hmms) and partially observable Markov decision processes (pomdps) provide useful tools for modeling dynamical systems. They are particularly useful for representing the topology of environments such as road networks and office buildings, which are typical for robot navigatio ..."
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Cited by 9 (0 self)
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Hidden Markov models (hmms) and partially observable Markov decision processes (pomdps) provide useful tools for modeling dynamical systems. They are particularly useful for representing the topology of environments such as road networks and office buildings, which are typical for robot navigation and planning. The work presented here describes a formal framework for incorporating readily available odometric information and geometrical constraints into both the models and the algorithm that learns them. By taking advantage of such information, learning hmms/pomdps can be made to generate better solutions and require fewer iterations, while being robust in the face of data reduction. Experimental results, obtained from both simulated and real robot data, demonstrate the effectiveness of the approach.
Inferring Flow of Control in Program Synthesis by Example
"... . We present a supervised, interactive learning technique that infers control structures of computer programs from userdemonstrated traces. A twostage process is applied: #rst, a minimal deterministic # nite automaton #DFA# M labeled by the instructions of the program is learned from a set of ..."
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Cited by 7 (0 self)
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. We present a supervised, interactive learning technique that infers control structures of computer programs from userdemonstrated traces. A twostage process is applied: #rst, a minimal deterministic # nite automaton #DFA# M labeled by the instructions of the program is learned from a set of example traces and membership queries to the user. It accepts all pre#xes of traces of the target program. The number of queries is bounded by O#k #jMj#, with k being the total number of instructions in the initial example traces. In the second step we parse this automaton into a highlevel programming language in O#jM j 2 # steps, replacing jumps by conditional control structures. 1