D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University Department of Computer Science, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....without negation. The class R is a simple extension of the class of unate DNF formulas. It is also a simple extension of the class of 2 clause CNF formulas, because each clause of a 2 clause CNF formula is unate. The class of 2 clause CNF formulas is properly learnable in polynomial time [2]. The characterization of Hellerstein et al. 9] says that a class C can be properly learned using a polynomial number of polynomial size membership and equivalence queries iff C has polynomial size certificates. Roughly, C has polynomial size certificates if, for all functions g not ....

....of queries is polynomial in jf j and n. 2.3 Monotone Dimension The monotone dimension was introduced by Bshouty. We give the basic definitions. Additional details can be found in [6] Let a and b be assignments to a set of n variables Vn . Let a b denote the sum of vectors a and b over GF[2]. The relation a b means that a(x) b(x) for all x 2 Vn , and a(y) b(y) for some y 2 Vn . Let c be an assignment to Vn . The relation a c b means that a c b c. For a Boolean function f : f0; 1g f0; 1g, the monotone closure of f with respect to c, written M c (f) 6 is a Boolean ....

[Article contains additional citation context not shown here]

Angluin, D. (1987), Learning k-term dnf formulas using queries and counterexamples, Technical Report YALE/DCS/RR-559, Yale University.

....learning result. Finally, note that RkSj DNF may be thought of as a generalization of k term DNF (those DNFs with at most k terms) Every k term DNF formula is trivially an RkSk DNF formula. Thus, our results may also be viewed as an extension of previous results for learning k term DNF formulas [Ang87, BR92, Bsh93] although a true generalization of the strongest of these results would learn RkSj DNF for k j = O(log n) We leave the latter as an interesting open problem. 5 3 Preliminaries Let X = fx 1 ; x n g be a set of n boolean variables. A literal is either a variable x i or ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, August 1987.

....[7] but these techniques also do not seem powerful enough to handle membership queries. The membership and equivalence query model has received much attention, due in part to the discovery of quite different interesting and efficient learning algorithms for a wide variety of types of functions [59, 6, 4, 10, 11, 54, 21, 23, 22]. This stands in contrast to the relatively few learning algorithms in the PAC model, or in the model of learning with equivalence queries only, and to the strong negative results that have been given for learning even apparently simple types of functions. Yet, prior to our results, virtually no ....

....and the other in the learning domain. A complexity theoretic consequence that immediately follows from previously developed learning algorithms is that the representation problem for many classes, such as read once formulas, and DNF formulas with a fixed constant number of terms, are all in co NP [11, 22, 49, 6, 4]. Of course, for particular classes, there may be more direct ways of showing this than developing a learning algorithm. In Section 7, we show for a variety of classes that the representation problem is co NP complete by applying our main theorem, previous learnability results, and simple ....

[Article contains additional citation context not shown here]

D. Angluin, Learning k-term DNF formulas using queries and counterexamples. Technical Report YALE/DCS/RR-559, Department of Computer Science, Yale University, New Haven, CT, August 1987.

....(GIF) 1 2 BEIMEL, GELLER, AND KUSHILEVITZ 1. INTRODUCTION Angluin s model of learning using membership queries and equivalence queries (i.e. the exact learning model) 3] attracted a lot of attention. In particular, various concept classes were shown to be learnable in this model (e.g. [1, 2, 5, 12, 13, 26, 28, 14, 10, 9] and many others) In some of the above, and in several related papers (e.g. 3, 18, 6, 13, 8, 11, 14, 7] the following common approach is used: the input space, f0; 1g n , is viewed as a lattice with the natural partial order, i.e. for u; v 2 f0; 1g n if u[i] v[i] for all i then u v ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, 1987.

....that learn an approximation of what is desired. The approximation differs from the desired result in finitely many anomalous places. Passive approximate inductive inference has been extensively investigated [8,10,11,21,27] The basic paradigm of asking questions has been applied to DNF formulas [1], CNF formulas [4] formulas [15] context free grammars [2] deterministic one counter automata [7] deterministic bottom up tree automata [23] deterministic skeletal automata [22] deterministic languages [16] and prolog programs [24] Valiant also considered the issue briefly. 28] For a ....

ANGLUIN, D. Learning k-term DNF formulas using queries and counter-examples. Department of Computer Science TR-559, Yale University, New Haven, CT, 1987.

.... 2 L They can also be of the kind is this a good model of L and if not, why not Besides asking questions regarding membership and equivalence with another language, Angluin permits questions of subset and superset [5] The basic paradigm of asking questions has been applied to DNF formulas [4], context free grammars [2] deterministic one counter automata [9] deterministic bottom up tree automata [38] deterministic skeletal automata [37] Prolog programs [39] Valiant also considered the issue briefly. 43] For a nice summary of some of these results see [5,6] The focus of this ....

ANGLUIN, D. Learning k-term DNF formulas using queries and counter-examples. Department of Computer Science TR-559, Yale University, New Haven, CT, 1987.

....blurry decision trees, and blurry Horn sentences, is as hard as learning DNF formulas in standard learning models. The learnability of DNF formulas remains a centrally studied unsolved problem; thus, while l term DNF formulas, decision trees, and Horn sentences are learnable in standard models [Ang87a, Bsh95, AFP92], learning these types of concepts from a consistently ignorant teacher would appear to be much more difficult. We also show in Section 5 that the problem of learning blurry DFAs is intractable, given standard cryptographic assumptions. Once again, while DFAs are learnable in standard models ....

....membership query models. Classes known to be learnable under one or both of these models include, for example, deterministic finite automata [Ang87b] read once formulas over various bases [AHK93, BHH95, BHH92] and propositional boolean formulas representable in the following forms: k term DNF [Ang87a, BR92], read twice DNF [AP91, Han91, PR95] and Horn sentences [AFP92] In contrast, Angluin and Kharitonov [AK95] have shown that, under cryptographic assumptions, read thrice formulas, nondeterministic finite automata, and contextfree grammars cannot be learned in the PAC memb model, and that ....

[Article contains additional citation context not shown here]

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, August 1987.

.... of examples of its own choosing a wider collection of DNF subclasses are known learnable, such as monotone DNF [9] and read twice DNF (each variable appears at most twice) 1] 6] However, the only previous advance known for learning k term DNF formulas with queries is an algorithm of Angluin [2] whose running time is worse, O(n k 2 ) but actually finds the original k term DNF. This paper presents an algorithm to learn k term DNF formulas that uses equivalence and membership queries and runs in an expected time, over the coin tosses of the algorithm, of O(n Delta k O(k) Thus ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University Department of Computer Science, 1987.

....algorithms for the class O(log n) term DNF. 1 Introduction Angluin s model of learning using membership queries and equivalence queries (i.e. the exact learning model) Ang88] attracted a lot of attention. In particular, various concept classes were shown to be learnable in this model (e.g. Ang87a, Ang87b, AFP92, BR92, Bsh93, RS93, SS93, Bsh95, BCV96, BBB 96] and many others) In some of the above, and in several related papers (e.g. Ang88, GM92, AHK93, Bsh93, AS94, BCGS95, Bsh95, AKST97] the following common approach is used: the input space, f0; 1g n , is viewed as a lattice ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, 1987.

....parameterize the members of the butterfly family. Using this family of distributions we are able to hit the target of performing proper PAC learning of k term DNF formulas in polynomial time for a constant k 1. Our algorithm uses labelled examples from a unique sample space. Summing up ideas from [1, 8], and arguing along the lines of [4] we can describe our main idea as follows: Let c be an unknown formula that we want to learn within a given class C. We first enumerate all formulas of C. We then draw a set of examples of how c behaves on random inputs and, for each listed formula, we use this ....

....the learning algorithm uses the statistical properties of the labelled samples directly. In this sense the SQ model introduces a metaphor of the oracle STAT(c,P) with which the learning algorithm interacts by asking queries of the form y,a , where y is a function from (X,c c (X) to 0,1 and a [0,1] is an accuracy parameter. The query y,a is intended as a request for the value P y = P(y(X,c c (X) 1) In other words, we assume that some property y of the random variable (X,c c (X) is relevant to our learning task and we check the probability P y . From a labelled sample we might obtain ....

[Article contains additional citation context not shown here]

ANGLUIN D. Learning k-term-DNF formulas using queries and counterexamples. Tech. Rep. YALEU/RR-559, Yale Univ., Dept. of Comp. Sc, 1987.

.... stems from several facts: on one hand it seems like a simple, natural class that is only slightly above our current state of knowledge, and on the other hand it appears that people like it for representing knowledge [57] Much work was therefore devoted to learning subclasses of DNF formulae [1, 2, 3, 6, 15, 16, 18, 20, 31, 38]. These subclasses are obtained by restricting the DNF formulae in various ways; e.g. by limiting the number of terms in the target formula or by limiting the number of appearances of each variable. The motivation for studying these subclasses is that such results may shed some light on the ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, 1987.

....without negation. The class R 2 is a simple extension of the class of unate DNF formulas. It is also a simple extension of the class of 2 clause CNF formulas, because each clause of a 2 clause CNF formula is unate. The class of 2 clause CNF formulas is properly learnable in polynomial time [2]. The characterization of Hellerstein et al. 9] says that a class C can be properly learned using a polynomial number of polynomial size membership and equivalence queries iff C has polynomial size certificates. Roughly, C has polynomial size certificates if, for all functions g not ....

....number of queries is polynomial in jf j and n. 2.3 Monotone Dimension The monotone dimension was introduced by Bshouty. We give the basic definitions. Additional details can be found in [6] Let a and b be assignments to a set of n variables Vn . Let a b denote the sum of vectors a and b over GF[2]. The relation a b means that a(x) b(x) for all x 2 Vn , and a(y) b(y) for some y 2 Vn . Let c be an assignment to Vn . The relation a c b means that a c b c. For a Boolean function f : f0; 1g V f0; 1g, the monotone closure of f with respect to c, written M c (f) is a Boolean ....

[Article contains additional citation context not shown here]

Angluin, D. (1987), Learning k-term dnf formulas using queries and counterexamples, Technical Report YALE/DCS/RR-559, Yale University.

.... to locate errors in Prolog programs [18] In computational learning theory, several types of active learning queries have also been defined (see for example [3] and compared with Valiant s probably approximately correct (PAC) model of concept identification under random sampling [24] Angluin [2], for example, has shown that there are concept classes that can be efficiently learnt with membership and equivalence queries, but not with random sampling in Valiant s PAC model. Some early connectionist approaches toward active learning include: Ahmad and Omohundro [1] on training networks by ....

D. Angluin. Learning k-term DNF Formulas using Queries and Counterexamples. Technical Report YALU/DCS/RR-559, Yale University, Department of Computer Science, 1987.

....added ability of the learner to make membership queries, a wider collection of DNF subclasses are known to be learnable. These include monotone DNF formulas in [V] and read twice DNF formulas in [AP] and [H] For learning k term DNF formulas with equivalence queries and membership queries, Angluin [Ab] designed an algorithm whose running time is O(d k 2 ) another remarkable result is due to Blum and Rudich [BR] they exhibited an O(d2 O(k) learning algorithm. Enlightened by those positive results on learning k term DNF formulas with queries, we begin to investigate the efficient ....

D. Angluin, "Learning k-term DNF formulas using queries and counterexamples, Technical Report YaleU/DCS/RR-559, Yale University department of Computer Science, 1987.

....with super polynomial number of states. 1 Introduction The exact learning model was introduced by Angluin [5] and since then attracted a lot of attention. In particular, the following classes were shown to be learnable in this model: deterministic automata [4] various types of DNF formulae [1, 2, 3, 6, 16, 17, 18, 20, 29, 33] and multi linear polynomials over GF(2) 44] Learnability in this model also implies learnability in the PAC model with membership queries [46, 5] One of the classes that was shown to be learnable in this model is the class of multiplicity automata [12] 1 and [40] Multiplicity automata are ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, 1987.

....AP92, BR92, B93, SS93, BK 94, B95, BCV96] Of particular relevance to the current note are Angluin s algorithm for learning deterministic automata [A87b] and the algorithms for learning k term DNF formulae. While for the case of constant k, a simple algorithm for learning k term DNF was known [A87], it took some time before powerful techniques were developed that allow constructing algorithms for k = O(log n) terms (where n is the number of variables in the formula) This was first done by Blum and Rudich, using a probabilistic search technique [BR92] then by Bshouty using the beautiful ....

.... k ) In particular, the class of O(log n) term DNF formulae is learnable in the exact learning model in time poly(n) Proof: Based on the construction of Lemma 1, a learning algorithm for learning a k term DNF formula, OE, will use any of the known algorithms for learning deterministic automata [A87, RS93, KV94] as a procedure, and try to learn the automaton A OE . Note that the algorithm is given oracles for the formula OE and not for the automaton A OE . Therefore, whenever this procedure asks a membership query on some string x, if x is of length different than n we answer this membership query with ....

[Article contains additional citation context not shown here]

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, August 1987.

....representation class) In fact, Pitt and Valiant [PV88] have shown that for k 2, the class of k term DNF formulas cannot be exactly identified, with a k term DNF result, in polynomial time using only equivalence queries if P 6= NP . In contrast to this representational hardness result, Angluin [Ang87a] has given an O(n k 2 ) algorithm for learning k term DNF formulas that constructs a k term DNF formula logically equivalent to the target formula using equivalence and membership queries. Also, by modifying the Blum and Rudich algorithm [Blu92] one can obtain an algorithm that builds a k term ....

....first non monotone class that is learnable in polynomial time with incomplete membership queries yet cannot be learned in polynomial time without the membership queries unless P = NP . 2 Definitions We begin by formally describing the model of learning from membership and equivalence queries [Ang87a]. The learner must infer an unknown target concept h chosen from some known concept class C. In this paper, C = S n1 C n is parameterized by the number of variables n, and each h 2 C n represents a DNF formula over the instance space f0; 1g n . Also, we assume that the n variables are x 1 ; x ....

[Article contains additional citation context not shown here]

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University, August 1987.

....extremely difficult. Nevertheless, as we have discussed, there are many situations in which it is extremely important to minimize the number of equivalence queries needed to obtain exact identification. 3 Definitions We now formalize the model of learning from membership and equivalence queries [Ang87a]. The learner must infer an unknown target concept f chosen from some known representation class C, which is a set of representations of functions mapping some domain X into a range Y . We typically parameterize C as C = S n1 C n , where C n is those elements of C that represent functions on n ....

....when a polynomial number of membership queries can be made. Here is a summary of the representation classes we study in this paper. k term DNF This is the class of DNF formulas having at most k terms. Angluin gave a polynomial time identification algorithm for the special case when k is constant [Ang87a], and Blum and Rudich have since given a more efficient algorithm that runs in polynomial time for k = O(log n) BR92] Read k Sat j DNF A DNF formula is a read k sat j DNF formula if every variable appears at most k times, and every assignment satisfies at most j terms in the formula. The class ....

Dana Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University, August 1987.

....algorithms for the class O(logn) term DNF. 1 Introduction Angluin s model of learning using membership queries and equivalence queries (i.e. the exact learning model) Ang88] attracted a lot of attention. In particular, various concept classes were shown to be learnable in this model (e.g. Ang87a, E mail: beimel deas.harvard.edu. http: www.deas.harvard.edu beimel. Supported by grants ONRN00014 96 1 0550 and ARO DAAL 03 92 G0115. y E mail: felix cs.technion.ac.il. z E mail: eyalk cs.technion.ac.il. http: www.cs.technion.ac.il eyalk. Supported by Technion V.P.R. Fund 120 872, by ....

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Department of Computer Science, Yale University, 1987.

No context found.

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University Department of Computer Science, 1987.

No context found.

D. Angluin, "Learning k-term DNF formulas using queries and counterexamples, Technical Report YaleU/DCS/RR-559,Yale University department of Computer Science, 1987.

No context found.

D. Angluin, Learning k-term DNF Formulas Using Queries and Counterexamples, Technical Report, Yale University, YALE/DCS/RR-559, 1987.

No context found.

Dana Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University Department of Computer Science, 1987.

No context found.

D. Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University Department of Computer Science, Aug. 1987.

No context found.

Dana Angluin. Learning k-term DNF formulas using queries and counterexamples. Technical Report YALEU/DCS/RR-559, Yale University, August 1987.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC