Above are papers related to the following paper: [Paper_ID] Paper #42207. 	[Title] Asking Questions to Minimize Errors [Abstract] A number of efficient learning algorithms achieve exact identification of an unknown function from some class using membership and equivalence queries. Using a standard transformation such algorithms can easily be converted to on-line learning algorithms that use membership queries. Under such a transformation the number of equivalence queries made by the query algorithm directly corresponds to the number of mistakes made by the on-line algorithm. In this paper we consider several of the natural classes known to be learnable in this setting, and investigate the minimum number of equivalence queries with accompanying counterexamples (or equivalently the minimum number of mistakes in the on-line model) that can be made by a learning algorithm that makes a polynomial number of membership queries and uses polynomial computation time. We are able both to reduce the number of equivalence queries used by the previous algorithms and often to prove matching lower bounds. As an example, consider the class of DNF formulas over n variables with at most k = O(log n) terms. Previously, the algorithm of Blum and Rudich [BR92] provided the best known upper bound of 2 O(k) log n for the minimum number of equivalence queries needed for exact identification. We greatly improve on this upper bound showing that exactly k counterexamples are needed if the learner knows k a priori and exactly k +1 counterexamples are needed if the learner does not know k a priori. This exactly matches known lower bounds [BC92]. For many of our results we obtain a complete characterization of the tradeoff between the number of membership and equivalence queries needed for exact identification. The classes we consider here are monotone DNF formulas, Horn sentences, O(log n)-term DNF formulas, read-k sat-j DNF formulas, read-once formulas over various bases, and deterministic finite automata. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1118209. 	[Title] Teaching a Smarter Learner [Abstract] We introduce a formal model of teaching in which the teacher is tailored to a particular learner, yet the teaching protocol is designed so that no collusion is possible. Not surprisingly, such a model remedies the non-intuitive aspects of other models in which the teacher must successfully teach any consistent learner. We prove that any class that can be exactly identified by a deterministic polynomial-time algorithm with access to a very rich set of example-based queries is teachable by a computationally unbounded teacher and a polynomial-time learner. In addition, we present other general results relating this model of teaching to various previous results. We also consider the problem of designing teacher/learner pairs in which both the teacher and learner are polynomial-time algorithms and describe teacher/learner pairs for the classes of 1-decision lists and Horn sentences. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1121537. 	[Title] How Many Queries are Needed to Learn? [Abstract] We investigate the query complexity of exact learning in the membership and (proper) equivalence query model. We give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model. We give applications of this characterization, including results on learning a natural subclass of DNF formulas, and on learning with membership queries alone. Query complexity has previously been used to prove lower bounds on the time complexity of exact learning. We show a new relationship between query complexity and time complexity in exact learning: If any "honest" class is exactly and properly learnable with polynomial query complexity, but not learnable in polynomial time, then P 6= NP. In particular, we show that an honest class is exactly polynomial-query learnable if and only if it is learnable using an oracle for p Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1104007. 	[Title] Sequential PAC Learning [Abstract] We consider the use of "on-line" stopping rules to reduce the number of training examples needed to pac-learn. Rather than collect a large training sample that can be proved sufficient to eliminate all bad hypotheses a priori, the idea is instead to observe training examples one-at-a-time and decide "on-line" whether to stop and return a hypothesis, or continue training. The primary benefit of this approach is that we can detect when a hypothesizer has actually "converged," and halt training before the standard fixed-sample-size bounds. This paper presents a series of such sequential learning procedures for: distribution-free pac-learning, "mistake-bounded to pac" conversion, and distribution-specific pac-learning, respectively. We analyze the worst case expected training sample size of these procedures, and show that this is often smaller than existing fixed sample size bounds | while providing the exact same worst case pac-guarantees. We also provide lower bounds that show these reductions can at best involve constant (and possibly log) factors. However, empirical studies show that these sequential learning procedures actually use many times fewer training examples in prac tice. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12182. 	[Title] How to Use Expert Advice (Extended Abstract) [Abstract] We analyze algorithms that predict a binary value by combining the predictions of several prediction strategies, called experts. Our analysis is for worst-case situations, i.e., we make no assumptions about the way the sequence of bits to be predicted is generated. We measure the performance of the algorithm by the difference between the expected number of mistakes it makes on the bit sequence and the expected number of mistakes made by the best expert on this sequence, where the expectation is taken with respect to the randomization in the predictions. We show that the minimum achievable difference is on the order of the square root of the number of mistakes of the best expert, and we give efficient algorithms that achieve this. Our upper and lower bounds have matching leading constants in most cases. We give implications of this result on the performance of batch learning algorithms in a PAC setting which improve on the best results currently known in this context. We also extend our analysis to the case in which log loss is used instead of the expected number of mistakes. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #57764. 	[Title] Inductive Learning by Selection of Minimal Complexity Representations [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12198. 	[Title] Exact Identification of Read-once Formulas Using Fixed Points of Amplification Functions [Abstract] In this paper we describe a new technique for exactly identifying certain classes of read-once Boolean formulas. The method is based on sampling the input-output behavior of the target formula on a probability distribution that is determined by the fixed point of the formula's amplification function (defined as the probability that a 1 is output by the formula when each input bit is 1 independently with probability p). By performing various statistical tests on easily sampled variants of the fixed-point distribution, we are able to efficiently infer all structural information about any logarithmic-depth formula (with high probability). We apply our results to prove the existence of short universal identification sequences for large classes of formulas. We also describe extensions of our algorithms to handle high rates of noise, and to learn formulas of unbounded depth in Valiant's model with respect to specific distributions. Most of this research was carried out while all three authors were at MIT Laboratory for Computer Science with support provided by ARO Grant DAAL03-86-K-0171, DARPA Contract N00014-89-J-1988, NSF Grant CCR-88914428, and a grant from the Siemens Corporation. R. Schapire received additional support from AFOSR Grant 89-0506 while at Harvard University. S. Goldman is currently supported in part by a G.E. Foundation Junior Faculty Grant and NSF Grant CCR-9110108. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12199. 	[Title] Learning k-term DNF Formulas with an Incomplete Membership Oracle [Abstract] We consider the problem of learning k-term DNF formulas using equivalence queries and incomplete membership queries as defined by Angluin and Slonim. We demonstrate that this model can be applied to non-monotone classes. Namely, we describe a polynomial-time algorithm that exactly identifies a k-term DNF formula with a k-term DNF hypothesis using incomplete membership queries and equivalence queries from the class of DNF formulas. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #124734. 	[Title] On Learning Visual Concepts and DNF Formulae [Abstract] We consider the problem of learning DNF formulae in the mistake-bound and the PAC models. We develop a new approach, which is called polynomial explainability, that is shown to be useful for learning some new subclasses of DNF (and CNF) formulae that were not known to be learnable before. Unlike previous learnability results for DNF (and CNF) formulae, these subclasses are not limited in the number of terms or in the number of variables per term; yet, they contain the subclasses of k-DNF and k-term-DNF (and the corresponding classes of CNF) as special cases. We apply our DNF results to the problem of learning visual concepts and obtain learning algorithms for several natural subclasses of visual concepts that appear to have no natural boolean counterpart. On the other hand, we show that learning some other natural subclasses of visual concepts is as hard as learning the class of all DNF formulae. We also consider the robustness of these results under various types of noise. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12350. 	[Title] Cryptographic Limitations on Learning Boolean Formulae and Finite Automata [Abstract] In this paper we prove the intractability of learning several classes of Boolean functions in the distribution-free 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 well-known public-key cryptosys- tems to the learning problems. We prove that a polynomial-time learning algorithm for Boolean formulae, deterministic finite automata or constant-depth 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 gener - alization of graph coloring. fl This research was conducted while the author was at Harvard University and supported by an A.T.& T. Bell Laboratories scholarship. y Supported by grants ONR-N00014-85-K-0445, NSF-DCR-8606366 and NSF-CCR-89-02500, DAAL03-86-K-0171, DARPA AFOSR 89-0506, and by SERC. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1102400. 	[Title] An Incremental Interactive Algorithm for Regular Grammar Inference [Abstract] We present provably correct interactive algorithms for learning regular grammars from positive examples and membership queries. A structurally complete set of strings from a language L(G) corresponding to a target regular grammar G implicitly specifies a lattice of finite state automata (FSA) which contains a FSA M G corresponding to G. The lattice is compactly represented as a version-space and M G is identified by searching the version-space using membership queries. We explore the problem of regular grammar inference in a setting where positive examples are provided intermittently. We provide an incremental version of the algorithm along with a set of sufficient conditions for its convergence. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #157761. 	[Title] On the Complexity of Function Learning [Abstract] The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range f0; 1g. Much less is known about the theory of learning functions with a larger range such as IN or IR. In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any "learning" model). Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from IR d into IR. Previously, the class of linear functions from IR d into IR was the only class of functions with multi-dimensional domain that was known to be learnable within the rigorous framework of a formal model for on-line learning. Finally we give a sufficient condition for an arbitrary class F of functions from IR into IR that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from F . This allows us to exhibit a number of further nontrivial classes of functions from IR into IR for which there exist efficient learning algorithms. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #15429. 	[Title] A General Lower Bound on the Number of Examples Needed for Learning [Abstract] We prove a lower bound of ( 1 * ln 1 ffi + VCdim(C) * ) on the number of random examples required for distribution-free learning of a concept class C, where VCdim(C) is the Vapnik-Chervonenkis dimension and * and ffi are the accuracy and confidence parameters. This improves the previous best lower bound of ( 1 * ln 1 ffi + VCdim(C)), and comes close to the known general upper bound of O( 1 ffi + VCdim(C) * ln 1 * ) for consistent algorithms. We show that for many interesting concept classes, including kCNF and kDNF, our bound is actually tight to within a constant factor. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1108167. 	[Title] On Learning Read-k-Satisfy-j DNF [Abstract] We study the learnability of Read-k-Satisfy-j (RkSj) DNF formulas. These are boolean formulas in disjunctive normal form (DNF), in which the maximum number of occurrences of a variable is bounded by k, and the number of terms satisfied by any assignment is at most j. After motivating the investigation of this class of DNF formulas, we present an algorithm that with high probability finds a DNF formula that is logically equivalent to any unknown RkSj DNF formula to be learned. The algorithm uses the well-studied protocol of equivalence and membership queries, and runs in polynomial time for k j = O( log n log log n ), where n is the number of input variables. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #25805. 	[Title] Weakly Learning DNF and Characterizing Statistical Query Learning Using Fourier Analysis [Abstract] We present new results, both positive and negative, on the well-studied problem of learning disjunctive normal form (DNF) expressions. We first prove that an algorithm due to Kushilevitz and Mansour [16] can be used to weakly learn DNF using membership queries in polynomial time, with respect to the uniform distribution on the inputs. This is the first positive result for learning unrestricted DNF expressions in polynomial time in any nontrivial formal model of learning. It provides a sharp contrast with the results of Kharitonov [15], who proved that AC 0 is not efficiently learnable in the same model (given certain plausible cryptographic assumptions). We also present efficient learning algorithms in various models for the read-k and SAT-k subclasses of DNF. For our negative results, we turn our attention to the recently introduced statistical query model of learning [11]. This model is a restricted version of the popular Probably Approximately Correct (PAC) model [23], and practically every class known to be efficiently learnable in the PAC model is in fact learnable in the statistical query model [11]. Here we give a general characterization of the complexity of statistical query learning in terms of the number of uncorrelated functions in the concept class. This is a distribution-dependent quantity yielding upper and lower bounds on the number of statistical queries required for learning on any input distribution. As a corollary, we obtain that DNF expressions and decision trees are not even weakly learnable with fl This research is sponsored in part by the Wright Laboratory, Aeronautical Systems Center, Air Force Materiel Command, USAF, and the Advanced Research Projects Agency (ARPA) under grant number F33615-93-1-1330. Support also is sponsored by the National Science Foundation under Grant No. CC-9119319. Blum also supported in part by NSF National Young Investigator grant CCR-9357793. Views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing official policies or endorsements, either expressed or implied, of Wright Laboratory or the United States Government, or NSF. respect to the uniform input distribution in polynomial time in the statistical query model. This result is information-theoretic and therefore does not rely on any unproven assumptions. It demonstrates that no simple modification of the existing algorithms in the computational learning theory literature for learning various restricted forms of DNF and decision trees from passive random examples (and also several algorithms proposed in the experimental machine learning communities, such as the ID3 algorithm for decision trees [22] and its variants) will solve the general problem. The unifying tool for all of our results is the Fourier analysis of a finite class of boolean functions on the hypercube. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12247. 	[Title] Warning: missing six few referencesfixed in proceedings. Learning with Queries but Incomplete Information (Extended Abstract) [Abstract] We investigate learning with membership and equivalence queries assuming that the information provided to the learner is incomplete. By incomplete we mean that some of the membership queries may be answered by I don't know. This model is a worst-case version of the incomplete membership query model of Angluin and Slonim. It attempts to model practical learning situations, including an experiment of Lang and Baum that we describe, where the teacher may be unable to answer reliably some queries that are critical for the learning algorithm. We present algorithms to learn monotone k-term DNF with membership queries only, and to learn monotone DNF with membership and equivalence queries. Compared to the complete information case, the query complexity increases by an additive term linear in the number of I don't know answers received. We also observe that the blowup in the number of queries can in general be exponential for both our new model and the incomplete membership model. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #1246. 	[Title] DESIGN AND ANALYSIS OF EFFICIENT REINFORCEMENT LEARNING ALGORITHMS [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #219239. 	[Title] Simulating Access to Hidden Information while Learning [Abstract] We introduce a new technique which enables a learner without access to hidden information to learn nearly as well as a learner with access to hidden information. We apply our technique to solve an open problem of Maass and Turan [18], showing that for any concept class F , the least number of queries sufficient for learning F by an algorithm which has access only to arbitrary equivalence queries is at most a factor of 1= log 2 (4=3) more than the least number of queries sufficient for learning F by an algorithm which has access to both arbitrary equivalence queries and membership queries. Previously known results imply that the 1= log 2 (4=3) in our bound is best possible. We describe analogous results for two generalizations of this model to function learning, and apply those results to bound the difficulty of learning in the harder of these models in terms of the difficulty of learning in the easier model. We bound the difficulty of learning unions of k concepts from a class F in terms of the difficulty of learning F . We bound the difficulty of learning in a noisy environment for deterministic algorithms in terms of the difficulty of learning in a noise-free environment. We apply a variant of our technique to develop an algorithm transformation that allows probabilistic learning algorithms to nearly optimally cope with noise. A second variant enables us to improve a general lower bound of Turan [19] for the PAC-learning model (with queries). Finally, we show that logarithmically many membership queries never help to obtain computationally efficient learning algorithms. fl Supported by Air Force Office of Scientific Research grant F49620-92-J-0515. Most of this work was done while this author was at TU Graz supported by a Lise Meitner Fellowship from the Fonds zur Forderung der wissenschaftlichen Forschung (Austria). Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #6125. 	[Title] Boosting a weak learning algorithm by majority To be published in Information and Computation [Abstract] We present an algorithm for improving the accuracy of algorithms for learning binary concepts. The improvement is achieved by combining a large number of hypotheses, each of which is generated by training the given learning algorithm on a different set of examples. Our algorithm is based on ideas presented by Schapire in his paper "The strength of weak learnability", and represents an improvement over his results. The analysis of our algorithm provides general upper bounds on the resources required for learning in Valiant's polynomial PAC learning framework, which are the best general upper bounds known today. We show that the number of hypotheses that are combined by our algorithm is the smallest number possible. Other outcomes of our analysis are results regarding the representational power of threshold circuits, the relation between learnability and compression, and a method for parallelizing PAC learning algorithms. We provide extensions of our algorithms to cases in which the concepts are not binary and to the case where the accuracy of the learning algorithm depends on the distribution of the instances. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #20601. 	[Title] Learning Conjunctions of Horn Clauses [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #20602. 	[Title] Learning Read-Once Formulas with Queries [Abstract] A read-once formula is a boolean formula in which each variable occurs at most once. Such formulas are also called -formulas or boolean trees. This paper treats the problem of exactly identifying an unknown read-once formula using specific kinds of queries. The main results are a polynomial time algorithm for exact identification of monotone read-once formulas using only membership queries, and a polynomial time algorithm for exact identification of general read-once formulas using equivalence and membership queries (a protocol based on the notion of a minimally adequate teacher [1]). Our results improve on Valiant's previous results for read-once formulas [26]. We also show that no polynomial time algorithm using only membership queries or only equivalence queries can exactly identify all read-once formulas. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12155. 	[Title] Exact Learning of -DNF Formulas with Malicious Membership Queries [Abstract]  Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #95225. 	[Title] Learning Boolean Read-Once Formulas over Generalized Bases [Abstract] A read-once formula is one in which each variable appears on at most a single input. Angluin, Hellerstein, and Karpinski give a polynomial time algorithm that uses membership and equivalence queries to identify exactly read-once boolean formulas over the basis fAND; OR; NOTg [AHK93]. The goal of this work is to consider natural generalizations of these gates, in order to develop exact identification algorithms for more powerful classes of formulas. We show that read-once formulas over a basis of arbitrary boolean functions of constant fan-in k or less (i.e. any f : f0; 1g 1ck ! f0; 1g) are exactly identifiable in polynomial time using membership and equivalence queries. We show that read-once formulas over the basis of arbitrary symmetric boolean functions are also exactly identifiable in polynomial time in this model. Given standard cryptographic assumptions, there is no polynomial time identification algorithm for read-twice formulas over either of these bases using membership and equivalence queries. We further show that for any basis class B meeting certain technical conditions, any polynomial time identification algorithm for read-once formulas over B can be extended to a polynomial time identification algorithm for read-once formulas over the union of B and the arbitrary functions of fan-in k or less. As a result, read-once formulas over the union of arbitrary symmetric and arbitrary constant fan-in gates are also exactly identifiable in polynomial time using membership and equivalence queries. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #12158. 	[Title] Randomly Fallible Teachers: Learning Monotone DNF with an Incomplete Membership Oracle [Abstract] We introduce a new fault-tolerant model of algorithmic learning using an equivalence oracle and an incomplete membership oracle, in which the answers to a random subset of the learner's membership queries may be missing. We demonstrate that, with high probability, it is still possible to learn monotone DNF formulas in polynomial time, provided that the fraction of missing answers is bounded by some constant less than one. Even when half the membership queries are expected to yield no information, our algorithm will exactly identify m-term, n-variable monotone DNF formulas with an expected O(mn 2 ) queries. The same task has been shown to require exponential time using equivalence queries alone. We extend the algorithm to handle some one-sided errors, and discuss several other possible error models. It is hoped that this work may lead to a better understanding of the power of membership queries and the effects of faulty teachers on query models of concept learning. Please summarise all the given information.

Paper #42207 has the title `Asking Questions to Minimize Errors  
` and the abstract `A number of efficient learning algorithms achieve exact identification of an unknown function from some class using membership and equivalence queries. Using a standard transformation such algorithms can easily be converted to on-line learning algorithms that use membership queries. Under such a transformation the number of equivalence queries made by the query algorithm directly corresponds to the number of mistakes made by the on-line algorithm. In this paper we consider several of the natural classes known to be learnable in this setting, and investigate the minimum number of equivalence queries with accompanying counterexamples (or equivalently the minimum number of mistakes in the on-line model) that can be made by a learning algorithm that makes a polynomial number of membership queries and uses polynomial computation time. We are able both to reduce the number of equivalence queries used by the previous algorithms and often to prove matching lower bounds. As an example, consider the class of DNF formulas over n variables with at most k = O(log n) terms. Previously, the algorithm of Blum and Rudich [BR92] provided the best known upper bound of 2 O(k) log n for the minimum number of equivalence queries needed for exact identification. We greatly improve on this upper bound showing that exactly k counterexamples are needed if the learner knows k a priori and exactly k +1 counterexamples are needed if the learner does not know k a priori. This exactly matches known lower bounds [BC92]. For many of our results we obtain a complete characterization of the tradeoff between the number of membership and equivalence queries needed for exact identification. The classes we consider here are monotone DNF formulas, Horn sentences, O(log n)-term DNF formulas, read-k sat-j DNF formulas, read-once formulas over various bases, and deterministic finite automata. 
`.
Question: Which category should Paper #42207 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #42207 has the title `Asking Questions to Minimize Errors  
` and the abstract `A number of efficient learning algorithms achieve exact identification of an unknown function from some class using membership and equivalence queries. Using a standard transformation such algorithms can easily be converted to on-line learning algorithms that use membership queries. Under such a transformation the number of equivalence queries made by the query algorithm directly corresponds to the number of mistakes made by the on-line algorithm. In this paper we consider several of the natural classes known to be learnable in this setting, and investigate the minimum number of equivalence queries with accompanying counterexamples (or equivalently the minimum number of mistakes in the on-line model) that can be made by a learning algorithm that makes a polynomial number of membership queries and uses polynomial computation time. We are able both to reduce the number of equivalence queries used by the previous algorithms and often to prove matching lower bounds. As an example, consider the class of DNF formulas over n variables with at most k = O(log n) terms. Previously, the algorithm of Blum and Rudich [BR92] provided the best known upper bound of 2 O(k) log n for the minimum number of equivalence queries needed for exact identification. We greatly improve on this upper bound showing that exactly k counterexamples are needed if the learner knows k a priori and exactly k +1 counterexamples are needed if the learner does not know k a priori. This exactly matches known lower bounds [BC92]. For many of our results we obtain a complete characterization of the tradeoff between the number of membership and equivalence queries needed for exact identification. The classes we consider here are monotone DNF formulas, Horn sentences, O(log n)-term DNF formulas, read-k sat-j DNF formulas, read-once formulas over various bases, and deterministic finite automata. 
`.
Question: Which category should Paper #42207 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Theory