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11
On the Power of Equivalence Queries
, 1994
"... In 1990, Angluin showed that no class exhibiting a combinatorial property called "approximate fingerprints" can be identified exactly using polynomially many Equivalence queries (of polynomial size). Here we show that this is a necessary condition: every class without approximate fingerpri ..."
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In 1990, Angluin showed that no class exhibiting a combinatorial property called "approximate fingerprints" can be identified exactly using polynomially many Equivalence queries (of polynomial size). Here we show that this is a necessary condition: every class without approximate fingerprints has an identification strategy that makes a polynomial number of Equivalence queries. Furthermore, if the class is "honest" in a technical sense, the computational power required by the strategy is within the polynomialtime hierarchy, so proving nonlearnability is at least as hard as showing P<F NaN> 6= NP. 1 Introduction Learning via queries is a wellstudied model in computational learning. The types of queries that have been used most often in the design of learning algorithms are, by far, Membership and Equivalence queries. In this paper we focus on the second type. This research was partially supported by the ESPRIT Basic Research Actions Program of the EC under contract No. 7141 (proj...
The Complexity of Learning with Queries
, 1994
"... We survey recent research concerning the qualitative complexity of Angluin 's model of learning with queries. In this model, there is a learner that tries to identify a target concept by means of queries to a teacher. Thus, the process can be naturally formulated as an oracle computation. Among ..."
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Cited by 12 (1 self)
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We survey recent research concerning the qualitative complexity of Angluin 's model of learning with queries. In this model, there is a learner that tries to identify a target concept by means of queries to a teacher. Thus, the process can be naturally formulated as an oracle computation. Among the results we review there are: characterizations of the power of different learning protocols by complexity classes of oracle machines; relations between the complexity of learning and the complexity of computing advice functions for nonuniform classes; and combinatorial characterizations of the concept classes that are learnable in specific protocols. 1 Introduction This paper is a survey on recent results and ideas on the qualitative complexity of Angluin's model of learning with queries, also known as query learning or exact learning. This model is quickly becoming one of the most popular ones the computational learning community, receiving almost as much attention as more traditional mode...
SemiMembership Algorithms: Some Recent Advances
 SIGACT News
, 1994
"... A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990 ..."
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A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semimembership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semimembership algorithm M for a set A takes as its input any strings x and y and decides which is "no less likely" to belong to A in the sense that if exactly one of the strings is in A, then M outputs that one string. Semimembership algorithms have been studied in a number of settings. Recursive semimembership algorithms (and the associated semirecursive setsthose sets having recursive semimembership algorithms) were introduced in the 1...
On Pselectivity and Closeness
 Inf. Processing Letters 54
, 1994
"... P/poly, the class of sets with polynomial size circuits, has been the subject of considerable study in complexity theory. Two important subclasses of P/poly are the class of sparse sets [6] and the class of Pselective sets [31]. A large number of results have been proved about both these classes bu ..."
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P/poly, the class of sets with polynomial size circuits, has been the subject of considerable study in complexity theory. Two important subclasses of P/poly are the class of sparse sets [6] and the class of Pselective sets [31]. A large number of results have been proved about both these classes but it has been observed (for example, [17]) that despite their similarity, proofs about one class generally do not translate easily to proofs regarding the other class. In this note, we propose to resolve this asymmetry by investigating the class PSELclose of sets that are polynomially close to Pselective sets; by definition, PSELclose includes both sparse sets and Pselective sets, thereby providing a unifying platform for proving results applicable to both. Intuitively, PSELclose is the class of sets that can in a certain sense be approximated by Pselective sets. We prove several results separating PSELclose from known classes within and including P/poly, and establish its location op...
Selectivity: Reductions, Nondeterminism, and Function Classes
, 1993
"... A set is Pselective [Se179] if there is a polynomialtime semidecision algorithm or the setan algorithm that given any two strings decides which is "more likely" to be in the set. This paper studies two natural generalizations o Pselectivity: the NPselective sets and the sets reducib ..."
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A set is Pselective [Se179] if there is a polynomialtime semidecision algorithm or the setan algorithm that given any two strings decides which is "more likely" to be in the set. This paper studies two natural generalizations o Pselectivity: the NPselective sets and the sets reducible or equivalent to Pselective sets via polynomialtime reductions. We establish a strict hierarchy among the various reductions and equivalences to Pselective sets. We show that the NPselective sets are in (NP coNP)/poly, are extended low, and (those in NP) are Low2; we also show that NPselective sets cannot be NPcomplete unless NP = coNP. By studying more general notions o nondeterministic selectivity, we conclude that all multivalued NP functions have singlevalued NP refinements only if the polynomial hierarchy collapses to its second level.
Boolean operations, joins, and the extended low hierarchy
 Theoretical Computer Science
, 1998
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Polynomialtime multiselectivity
, 1997
"... We introduce a generalization of Selman's Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove ..."
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We introduce a generalization of Selman's Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove the first known (and optimal) lower bounds for generalized selectivitylike classes in terms of EL2, the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara's Pmc (polynomialtime membershipcomparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the Pselective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits; the NP sets in SH are in Low2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that PSel, the class of Pselective sets, is not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the Pselective sets, and yet, in contrast with the Pmc classes, is refined enough to distinguish them.
Coding Complexity: The Computational Complexity of Succinct Descriptions
, 1996
"... For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these prob ..."
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For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these problems. In this paper, we survey how such investigation has proceeded, and explain the current status of our knowledge.
SIGACT News Complexity Theory Column 6
"... A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990 ..."
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A semimembership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent halfdecade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semimembership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semimembership algorithm M for a set A takes as its input any 1 Supported in part by NSF grant CCR8957604, NSF/JSPS grant INT9116781/ENGR207, HC&M grant ERB4050PL930516, and an NSF REU supplement. 2 Department of Computer Science, University of Rochester, Rochester, NY, 14627. 3 Work done in part while visiting the Tokyo Institute of Technology and the University of Amsterdam. 4 Departments of Mathematics and Computer Science, Unive...