Learning conjunctions of Horn clauses

Angluin, Dana; Frazier, Michael; Pitt, Leonard

doi:10.1007/bf00992675

Cited by 101 publications

(113 citation statements)

References 11 publications

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“…Moreover, the execution of the repeat loop minimizes the number of ones of the negative example s so that it fulfills the conditions of Lemma 1, and thus allows identifying the antecedent of some clause (or clauses) in F * . Note that a naive approach, consisting of performing a single execution of the inner for loop could fail to achieve that goal, as the following example proves: Suppose the clauses v 1 ∧ v 2 −→ v 4 and v 3 −→ v 2 are in F * , s = 1110 is the negative example selected at line 7, and the for loop checks variables starting by v 1 and ending by v 4 . Then the example obtained would be s = 1010, but the variables in ones(s) would not coincide with the variables of any antecedent in the clauses above.…”

Section: Lemmamentioning

confidence: 97%

“…For instance, the classes monotone DNF [2], k-term DNF or k-clause CNF [1] and read twice DNF [19] have been proved to be learnable using membership and equivalence queries. In [4], an algorithm that learns a subclass of CNF, the class of Horn formulas, is shown. It is known that allowing more than one unnegated literal per clause makes learning much more difficult (or perhaps impossible).…”

Section: Introductionmentioning

confidence: 99%

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Learning Sets of Antecedent-restricted Functional and Multivalued Dependencies with Queries

Puente

2015

Theory Comput Syst

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Identifying dependencies that hold in relational databases is essential to produce good databases designs. In particular, functional and multivalued dependencies are used to obtain relation schemes that satisfy the 4th normal form, a property that is considered satisfactory for most applications. It is known that the class of sets of functional dependencies is learnable in the exact model of learning with queries. Also a subclass of multivalued dependencies, the class of consequent-restricted multivalued dependencies, has been shown to be learnable in this model. In this paper, we present an algorithm that learns a class that contains sets of both functional and multivalued dependencies under some restrictions imposed on the antecedents of dependencies. We also show, as a by-product, an algorithm that learns a non-trivial subclass of 2-quasi Horn formulas, closely related to the class just mentioned.

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Section: Lemmamentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Learning Sets of Antecedent-restricted Functional and Multivalued Dependencies with Queries

Puente

2015

Theory Comput Syst

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“…The problem of exactly learning propositional Horn sentences from membership and equivalence queries was shown to have polynomial runtime with a fixed number of variables and a fixed number of clauses [2]. This algorithm was later generalized to learning first-order Horn theories from equivalence and membership queries for several learning settings, including learning from interpretations and learning from entailment [6].…”

Section: Previous Workmentioning

confidence: 99%

Learning First-Order Definite Theories via Object-Based Queries

Selman

Fern

2011

Machine Learning and Knowledge Discovery in Databases

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Abstract. We study the problem of exact learning of first-order definite theories via queries, toward the goal of allowing humans to more efficiently teach first-order concepts to computers. Prior work has shown that first order Horn theories can be learned using a polynomial number of membership and equivalence queries [6]. However, these query types are sometimes unnatural for humans to answer and only capture a small fraction of the information that a human teacher might be able to easily communicate. In this work, we enrich the types of information that can be provided by a human teacher and study the associated learning problem from a theoretical perspective. First, we consider allowing queries that ask the teacher for the relevant objects in a training example. Second, we examine a new query type, called a pairing query, where the teacher provides mappings between objects in two different examples. We present algorithms that leverage these new query types as well as restrictions applied to equivalence queries to significantly reduce or eliminate the required number of membership queries, while preserving polynomial learnability. In addition, we give learnability results for certain cases of imperfect teachers. These results show, in theory, the potential for incorporating object-based queries into first-order learning algorithms in order to reduce human teaching effort.

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“…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.…”

Section: Complexity Theory and Learningmentioning

confidence: 99%

Complexity theoretic hardness results for query learning

et al. 1998

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We investigate the complexity of learning for the well-studied 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 = co-NP, no polynomial-time membership and (proper) equivalence query algorithms exist for exactly learning read-thrice 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 technique introduces the representation problem for a class F of representations (e.g., formulas), which is naturally associated with the learning problem for F . This problem is related to the structural question of how to characterize functions representable by formulas in F , and is a generalization of standard complexity problems such as Satisfiability. While in general the representation problem is in Σ P 2 , we present a theorem demonstrating that for "reasonable" classes F , the existence of a polynomial-time membership and equivalence query algorithm for exactly learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard.

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Learning conjunctions of Horn clauses

Cited by 101 publications

References 11 publications

Learning Sets of Antecedent-restricted Functional and Multivalued Dependencies with Queries

Learning Sets of Antecedent-restricted Functional and Multivalued Dependencies with Queries

Learning First-Order Definite Theories via Object-Based Queries

Complexity theoretic hardness results for query learning

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