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25
Learning ReadOnce Formulas with Queries
 J. ACM
, 1989
"... A readonce 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 readonce formula using specific kinds of queries. The main results are a polynomial time al ..."
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Cited by 115 (19 self)
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A readonce 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 readonce formula using specific kinds of queries. The main results are a polynomial time algorithm for exact identification of monotone readonce formulas using only membership queries, and a polynomial time algorithm for exact identification of general readonce 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 readonce formulas [26]. We also show that no polynomial time algorithm using only membership queries or only equivalence queries can exactly identify all readonce formulas. 1 Introduction The goal of computational learning theory is to define and study useful models of learning phenomena from an algorithmic point of view. Since there are a variety ...
Complexity Theoretic Hardness Results for Query Learning
 COMPUTATIONAL COMPLEXITY
, 1998
"... We investigate the complexity of learning for the wellstudied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are no ..."
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Cited by 21 (5 self)
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We investigate the complexity of learning for the wellstudied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may make membership queries. We develop a general technique for proving hardness results for learning with membership and equivalence queries (and for more general query models). We apply the technique to show that, assuming NP != coNP, no polynomialtime membership and (proper) equivalence query algorithms exist for exactly learning readthrice DNF formulas, unions of k 3 halfspaces over the Boolean domain, or some other related classes. Our hardness results are representation dependent, and do not preclude the existence of representation independent algorithms. The general
Readonce Polynomial Identity Testing
"... An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readon ..."
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Cited by 21 (6 self)
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An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readonce formulas. the following are some of the results that we obtain. 1. Given k ROFs in n variables, over a field F, we give a deterministic (non blackbox) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k2). 2. We give an n O(d+k2) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k2) for the sum of k depth d ROFs. If F  is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs. 3. We give a hitting set of size exp ( Õ( √ n + k 2)) for the sum of k ROFs (without depth restrictions). In particular this implies a subexponential time deterministic algorithm for
Factoring and Recognition of ReadOnce Functions using Cographs and Normality and the Readability of Functions Associated with Partial ktrees
, 2004
"... An approach for factoring general boolean functions was described in [15] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring readonce functions which is needed as a dedicated factoring subroutine to handle the lower levels o ..."
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Cited by 18 (0 self)
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An approach for factoring general boolean functions was described in [15] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring readonce functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality. For nonreadonce functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial ktree, then F is a read 2 k function and a read 2 k formula for F can be obtained in polynomial time.
Decision Lists and Related Boolean Functions
, 1998
"... We consider Boolean functions represented by decision lists, and study their relationships to other classes of Boolean functions. It turns out that the elementary class of 1decision lists has interesting relationships to independently defined classes such as disguised Horn functions, readonce fu ..."
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Cited by 12 (1 self)
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We consider Boolean functions represented by decision lists, and study their relationships to other classes of Boolean functions. It turns out that the elementary class of 1decision lists has interesting relationships to independently defined classes such as disguised Horn functions, readonce functions, nested differences of concepts, threshold functions, and 2monotonic functions. In particular, 1decision lists coincide with fragments of the mentioned classes. We further investigate the recognition problem for this class, as well as the extension problem in the context of partially defined Boolean functions (pdBfs). We show that finding an extension of a given pdBf in the class of 1decision lists is possible in linear time. This improves on previous results. Moreover, we present an algorithm for enumerating all such extensions with polynomial delay.
Faster Query Answering in Probabilistic Databases using ReadOnce Functions
"... A boolean expression is in readonce form if each of its variables appears exactly once. When the variables denote independent events in a probability space, the probability of the event denoted by the whole expression in readonce form can be computed in polynomial time (whereas the general problem ..."
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Cited by 8 (2 self)
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A boolean expression is in readonce form if each of its variables appears exactly once. When the variables denote independent events in a probability space, the probability of the event denoted by the whole expression in readonce form can be computed in polynomial time (whereas the general problem for arbitrary expressions is #Pcomplete). Known approaches to checking readonce property seem to require putting these expressions in disjunctive normal form. In this paper, we tell a better story for a large subclass of boolean event expressions: those that are generated by conjunctive queries without selfjoins and on tupleindependent probabilistic databases. We first show that given a tupleindependent representation and the provenance graph of an SPJ query plan without selfjoins, we can, without using the DNF of a result event expression, efficiently compute its cooccurrence graph. From this, the readonce form can already, if it exists, be computed efficiently using existing techniques. Our second and key contribution is a complete, efficient, and simple to implement algorithm for computing the readonce forms (whenever they exist) directly, using a new concept, that of cotable graph, which can be significantly smaller than the cooccurrence graph.
Factoring Logic Functions Using Graph Partitioning
 Proc. IEEE/ACM Int. Conf. Computer Aided Design
, 1999
"... Algorithmic logic synthesis is usually carried out in two stages, the independent stage where logic minimization is performed on the Boolean equations with no regard to physical properties and the dependent stage where mapping to a physical cell library is done. The independent stage includes logic ..."
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Cited by 6 (1 self)
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Algorithmic logic synthesis is usually carried out in two stages, the independent stage where logic minimization is performed on the Boolean equations with no regard to physical properties and the dependent stage where mapping to a physical cell library is done. The independent stage includes logic operations like Decomposition, Extraction, Factoring, Substitution and Elimination. These operations are done with some kind of division (boolean, algebraic), with the goal being to obtain a logically equivalent factored form which minimizes the number of literals. In this paper, we present an algorithm for factoring that uses graph partitioning rather than division. Central to our approach is to combine this with the use of special classes of boolean functions, such as readonce functions, to devise new combinatorial algorithms for logic minimization. Our method has been implemented in the SIS environment, and an empirical evaluation indicates that we usually get significantly better results than algebraic factoring and are quite competitive with boolean factoring but with lower computation costs. 1
Theory revision with queries: Horn, readonce, and parity formulas
 Artificial Intelligence
, 2004
"... A theory, in this context, is a Boolean formula; it is used to classify instances, or truth assignments. Theories can model realworld phenomena, and can do so more or less correctly. The theory revision, or concept revision, problem is to correct a given, roughly correct concept. This problem is co ..."
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Cited by 5 (1 self)
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A theory, in this context, is a Boolean formula; it is used to classify instances, or truth assignments. Theories can model realworld phenomena, and can do so more or less correctly. The theory revision, or concept revision, problem is to correct a given, roughly correct concept. This problem is considered here in the model of learning with equivalence and membership queries. A revision algorithm is considered efficient if the number of queries it makes is polynomial in the revision distance between the initial theory and the target theory, and polylogarithmic in the number of variables and the size of the initial theory. The revision distance is the minimal number of syntactic revision operations, such as the deletion or addition of literals, needed to obtain the target theory from the initial theory. Efficient