H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity Theoretic Hardness Results for Query Learning. Computational Complexity 7(1):19-53, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with zero delays, known as the linear separability problem, was proved to be co NP complete [13] We generalize the coNP hardness result for spiking neurons with arbitrary delays. On the other hand, the RPN is clearly in 2 whereas its hardness for 2 (or for NP) which would imply [1] that the spiking neurons with arbitrary delays are not learnable with membership and equivalence queries (unless NP = co NP ) remains an open problem. Moreover, it was shown [20] that the class of n variable Boolean functions computable by spiking neurons is strictly contained in the class DLLT ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity Theoretic Hardness Results for Query Learning. Computational Complexity 7(1):19-53, 1998.

....polynomial query complexity with respect to subset and superset queries with DNF formulas as hypotheses [4] it further follows that DNF formulas are properly learnable in polynomial time with subset and superset queries and the help of an oracle in 3 . We further consider a concept class of [1] and show that this concept class is not learnable in polynomial time with an oracle in NP using equivalence queries with boolean circuits as hypotheses, unless the polynomial time hierarchy collapses. This implies that the required oracle in our main theorem cannot be replaced by an oracle in a ....

.... ( 4 )t( 5 Non learnability with an oracle in NP In this section we show that the 2 oracle in our main theorem cannot be replaced by an oracle in NP, unless the polynomial hierarchy collapses to 2 . This computational hardness result is based on the representation problem REP(C) [14, 1] for a representation C, REP(C) fh0 ; ci j (9u 2 ) Cn (u) Circn (c) g; where Circ is the circuit representation for boolean functions. Aizenstein et al. 1] showed that there is a representation K 2 P such that its representation problem REP(K) is complete for 2 . The ....

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H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. Computational Complexity, 7:19-53, 1998.

....properties of 1 decision lists and, vices versa, of the intersections of classes of Boolean functions to which they coincide. Furthermore, we study computational problems on 1 decision lists. We consider recognition from a formula (also called membership problem [20] and representation problem [4, 1]) and problems in the context of partially defined Boolean functions. A partially defined Boolean function (pdBf) can be viewed as a pair (T ; F ) of sets T and F of true and false vectors v 2 f0; 1g n , respectively, where T F = It naturally generalizes a Boolean function, by allowing ....

....problem and the learning problem: in the latter problem, an extension is a priori known to exist, while in the former, this is unknown. A learning algorithm might take advantage of this knowledge and find an extension faster. The extension problem itself is known as the consistency problem [4, 1]; it corresponds to learning from a sample which is possibly spoiled with inconsistent examples. In this context, it is also interesting to know whether the pdBf given by a sample uniquely defines a Boolean function in C; if the learner recognizes this fact, she he has identified the function g ....

[Article contains additional citation context not shown here]

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity Theoretic Hardness Results for Query Learning. Journal of Complexity, 7(1):19--53, 1998.

....threshold functions, 2 monotonic positive functions, dual comparable functions and decomposable functions, are discussed in this paper. In computational learning theory, problem EXTENSION(C) is called the consistency problem for C, and has been studied to prove the hardness of learnability [1, 23]. It is known [23] resp. 1] that, given a polynomially size bounded and polynomially reasonable class of functions C, if C is polynomially PAC learnable [26] resp. polynomially exact learnable with equivalence queries alone [3] then EXTENSION(C) is in RP (resp. P) These are used to ....

....positive functions, dual comparable functions and decomposable functions, are discussed in this paper. In computational learning theory, problem EXTENSION(C) is called the consistency problem for C, and has been studied to prove the hardness of learnability [1, 23] It is known [23] resp. [1]) that, given a polynomially size bounded and polynomially reasonable class of functions C, if C is polynomially PAC learnable [26] resp. polynomially exact learnable with equivalence queries alone [3] then EXTENSION(C) is in RP (resp. P) These are used to show that if EXTENSION(C) is ....

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Aizenstein, H., Hegedus, T., Hellerstein, L. and Pitt, L. (1997), Complexity theoretic hardness results for query learning, to appear in Journal of Complexity.

....from the boolean domain, if it exists, is NP complete. It has been shown that in the boolean domain, exact learning from equivalence and membership queries remains NPhard if the algorithm is required to find (as ours does) an intersection with the same number of halfspaces k, for any fixed k 3 [AHHP98, PR94] 1.2.2 Learning Convex Polytopes using Membership Queries One way of making the learning of convex polytopes tractable using membership queries only is to restrict the number of halfspaces, and sometimes, also the dimension. Bultman and Maass [BM91] presented an algorithm that learns a ....

....the learning algorithm may not be feasible. On the other hand, if it is the same as the bit complexity of the initial labeled sample, then the learning task is intractable. This is because learning intersections of k halfspaces for any k 3 using eqivalence and membership queries is NPhard [AHHP98, PR94] Thus, one would like to know what can be learned if the precision of the MQ oracle is allowed to be a few bits more than the bit complexity of the target concept or the initial labeled sample so that the results obtained are more feasible, and the MQ oracle is powerful enough to learn a ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. Computational Complexity, 7:19--53, 1998.

....of halfspaces. When membership queries are allowed, it has been shown that in the boolean domain, exact learning from equivalence and membership queries remains NP hard if the algorithm is required to find (as ours does) an intersection with the same number of halfspaces k, for any fixed k 3 [AHHP96, PR94] Mostly negative results have been obtained for learning in the case that the number of halfspaces is not held constant, and the learning algorithm is allowed to output a hypothesis containing more halfspaces than the target concept. By using simple prediction preserving reductions it can ....

....actually produce no more hyperplanes than the number in the target. This is the task found NP hard when the domain is restricted to the boolean hypercube in the PAC model without membership queries [BR89] and in the more demanding exact learning model with both membership and equivalence queries [AHHP96, PR94] Let S and S Gamma denote the positive and negative examples of S, respectively. To understand the method that Polly uses to find a collection of at most s hyperplanes whose intersection contains all of S but none of S Gamma , for simplicity we consider the two dimensional ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. computational complexity, 1996. To appear.

....properties of 1 decision lists and, vices versa, of the intersections of classes of Boolean functions to which they coincide. Furthermore, we study computational problems on 1 decision lists. We consider recognition from a formula (also called membership problem [19] and representation problem [4, 1]) and problems in the context of partially defined Boolean functions. A partially defined Boolean function (pdBf) can be viewed as a pair (T ; F ) of sets T and F of true and false vectors v 2 f0; 1g n , respectively, where T F = It naturally generalizes a Boolean function, by allowing that ....

....problem and the learning problem: in the latter problem, an extension is a priori known to exist, while in the former, this is unknown. A learning algorithm might take advantage of this knowledge and find an extension faster. The extension problem itself is known as the consistency problem [4, 1]; it corresponds to learning from a sample which is possibly spoiled with inconsistent examples. In this context, it is also interesting to know whether the pdBf given by a sample uniquely defines a Boolean function in C; if the learner recognizes this fact, she he has identified the function g to ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity Theoretic Hardness Results for Query Learning. Journal of Complexity, 1998. to appear.

.... (a polynomial number of queries may suffice, but it is difficult to generate and process them in polynomial time) Aizenstein, Hegedus, Hellerstein, and Pitt introduced the first technique for proving computational hardness results on proper learning in the membership and equivalence query model [1]. Related techniques were then presented by Pillaipakkamnatt and Raghavan [11] They showed that certain classes of DNF formulas, including read thrice DNF, are not properly learnable in polynomial time with membership and equivalence queries unless P = NP. Thus there is a computational barrier to ....

....It is easy to construct artificial classes that also have these properties, but it can be quite difficult to determine whether such properties hold for more natural classes of functions. Aizenstein et al. did show that similar properties hold for the class of unions of k graphic halfspaces [1]. All functions in this class are expressible as monotone 2 DNF formulas and thus can be easily learned in terms of a 2 DNF representation using a polynomial number of membership queries (and no equivalence queries) However, Aizenstein et al. showed that unions of k graphic halfspaces cannot be ....

Aizenstein, H., Hegedus, T., Hellerstein, L., and Pitt, L. (1996), Complexity theoretic hardness results for query learning, To appear in Computational Complexity.

.... exists for the class of monotone formulas with constant maximum latency [31, 32] The restriction considered in this paper is based on limiting the number of reads (occurrences of each variable) in a formula, a restriction that has been well investigated in the learning theory literature [3, 38, 9, 10, 39, 1, 2]. Previous work has shown that it is possible to identify an arbitrary monotone read once ( formula under a stronger notion of polynomial time [3, 16] We show here (for the conversion problem) that given a read k monotone CNF expression one can efficiently find its DNF expression. Henceforth, ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. Computational Complexity, 7:19--53, 1998.

.... (a polynomial number of queries may suffice, but it is difficult to generate and process them in polynomial time) Aizenstein, Hegedus, Hellerstein, and Pitt introduced the first technique for proving computational hardness results on proper learning in the membership and equivalence query model [1]. Related techniques were then presented by Pillaipakkamnatt and Raghavan [11] They showed that certain classes of DNF formulas, including read thrice DNF, are not properly learnable in polynomial time with membership and equivalence queries unless P = NP. Thus there is a computational barrier to ....

....It is easy to construct artificial classes that also have these properties, but it can be quite difficult to determine whether such properties hold for more natural classes of functions. 1 1 Aizenstein et al. did show that similar properties hold for the class of unions of k graphic halfspaces [1]. All functions in this class are expressible as monotone 2 DNF formulas and thus can be easily learned in terms of a 2 DNF representation using a polynomial number of membership queries (and no equivalence queries) However, Aizenstein et al. showed that unions of k graphic halfspaces cannot be ....

Aizenstein, H., Hegedus, T., Hellerstein, L., and Pitt, L. (1996), Complexity theoretic hardness results for query learning, To appear in Computational Complexity.

....of halfspaces. When membership queries are allowed, it has been shown that in the boolean domain, exact learning from equivalence and membership queries remains NP hard if the algorithm is required to find (as ours does) an intersection with the same number of halfspaces k, for any fixed k 3 [19, 20]. Mostly negative results have been obtained for learning in the case that the number of halfspaces is not held constant, and the learning algorithm is allowed to output a hypothesis containing more halfspaces than the target concept. By using simple prediction preserving reductions it can be ....

....actually produce no more hyperplanes than the number in the target. This is the task found NP hard when the domain is restricted to the boolean hypercube in the PAC model without membership queries [15] and in the more demanding exact learning model with both membership and equivalence queries [19, 20]. Let S and S Gamma denote the positive and negative examples of S, respectively. To understand the method that Polly uses to find a collection of at most s hyperplanes whose intersection contains all of S but none of S Gamma , for simplicity we consider the two dimensional case. ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. Computational Complexity. To appear.

....threshold formulas or matroid formulas. The restriction considered in this paper is based on limiting the number of reads (occurrences of each variable) in a formula, a restriction that has been well investigated in the learning theory literature [AHK93, PR95, PR96, BHH95a, BHH95b, ABK 97, AHHP97] Previous work has shown that it is possible to find the minterms of an arbitrary monotone read once ( formula under a stronger notion of polynomial time [AHK93, GK95] We show here (for the conversion problem) that given a read k monotone CNF expression, one can efficiently find its DNF ....

H. Aizenstein, T. Hegedus, L. Hellerstein, and L. Pitt. Complexity theoretic hardness results for query learning. Computational Complexity, To appear, 1997.

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