Khardon, R.: Learning function-free Horn expressions, Machine Learning, 37(3), December 1999, 241--275.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....We also allow a learner to use two types of additional queries for the target EFS H . The rst type of queries is the entailment membership query in the model of the learning from entailment [15, 31] This model is considered to be reasonable for learning the rstorder logic or logic programs [10, 11, 16, 19, 31]. The goal of a learning algorithm is to nd a hypothesis equivalent to the target hypothesis w.r.t. the entailment semantics using the queries. The entailment semantics is de ned in the next section together with other semantics. The second type of queries is the dependency query to determine ....

....in a dependency relation. We design a learning algorithm for THEFS( k; r) with equivalence, entailment membership, and dependency queries. This algorithm adopts the bottom up search strategy by combining three generalization techniques, namely, saturation, rewind and maximal common subsumer [10, 11, 15, 16, 19, 31]. We show that for every k; r 0, this algorithm exactly learns the class THEFS( k; r) in polynomial time using O(pmn ) equivalence queries, O(p ) entailment membership queries, and O(p dependency queries, where m is the number of clauses and n is the length of the ....

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R. Khardon, Learning function-free Horn expressions, Mach. Learn. 35(1) (1999) 241-275.

....with equivalence and subset queries under the into match semantics. We also showed two hardness results which indicates that above two types of queries are necessary to efficiently learn OGT and OGF. Connection among the learnabilities of OGT, OGF, pattern languages [4, 16] and first order logic [11] is a future problem. ....

R. Khardon, Learning function-free Horn expressions, Mach. Learn., 35(1), 241-- 275, 1999.

....The results of this paper depend on the assumption of infinite alphabet. Hence, it is a future problem to investigate the learnability of the class of bounded forests, sets of at most k OGTs, with a finite alphabet. Connection to the learnability of pattern languages [5, 15] or first order logic [12] is another future problem. ....

R. Khardon, Learning function-free Horn expressions, Mach. Learn., 35(1), 241--275, 1999.

....and membership queries. Our problem is identifying an unknown tree translation system H from examples of ordered pairs E 2 M(H ) that are either derived or not derived by H . As a formal model, we employ a variant of exact learning model by Angluin [2] called learning from entailment[3, 4, 9, 14], which is tailored for translation systems. Let H be a class of translation systems to be learned, called hypothesis space, and LR be the set of all ordered pairs, called the domain of learning . In our learning framework, the meaning or the concept represented by H 2 H is the set M(H ) If ....

R. Khardon, \Learning function-free Horn expressions," Proc. COLT'98, pp.154{ 165, 1998.

....their researches by Cohen [5, 6, 7, 9] Dzeroski [10, 12, 23] Kietz [22, 23, 24] and Page [9, 27] have placed the theoretical researches for the learnability of logic programs in one of the main research topics in inductive logic programming. Recently, it has deeply developed by many researchers [1, 20, 25, 26, 30, 31]. On the other hand, a conjunctive query problem in relational database theory [2, 4, 14, 17, 35] is a problem to determine whether or not a tuple belongs to the answer of a conjunctive query over a database. Here, a tuple, a conjunctive query, and a database in 1 relational database theory are ....

....that the language ACQ becomes a natural example that collapses the equivalence between subsumption e#ciency and e#cient paclearnability from both a simple and an extended instances. Various researches have investigated the e#cient learnability by using equivalence and membership queries such as [1, 25, 26, 31, 30]. Our result in this paper implies that ACQ[j B] j # 3) is not polynomial time learnable using equivalence queries alone. It is a future work to analyze the learnability of ACQ[j B] j # 3) by using membership and equivalence queries, and by extending to one containing function symbols or ....

R. Khardon, Learning function-free Horn expressions, Mach. Learn. 35(1) (1999) 241--275.

....This work is based on previous results on learnability of function free Horn expressions and range restricted Horn expressions. The problem of learning range restricted Horn expressions was solved in [Kha99b] by reducing it to the problem of learning function free Horn expressions, solved in [Kha99a]. The algorithm presented here has been obtained by retracing this reduction and using the resulting algorithm as a starting point. However, it has been significantly modified and improved. The algorithm in [Kha99a,Kha99b] uses two main procedures. The first, given a counterexample clause, ....

....by reducing it to the problem of learning function free Horn expressions, solved in [Kha99a] The algorithm presented here has been obtained by retracing this reduction and using the resulting algorithm as a starting point. However, it has been significantly modified and improved. The algorithm in [Kha99a,Kha99b] uses two main procedures. The first, given a counterexample clause, minimises the clause while maintaining it as a counterexample. The minimisation procedure used here is stronger, resulting in a clause which includes a syntactic variant of a target clause as a subset. The second procedure ....

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R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241--275, 1999.

....first type of additional information is provided by entailment membership queries in the framework of learning from entailment . Learning from entailment [12, 26] is a modification of exact learning model [3] which is suitable for studying the learnability of fragments of the first order logic [10, 11, 13, 16, 26]. In this model, an example is a (possibly non ground) clause either entailed or not entailed from the target hypothesis, and the entailment equivalence and the entailment membership queries are defined to be the ordinary equivalence and membership queries in this semantics, respectively. The ....

....preorder for the class. In contrast with the first algorithm, this algorithm LEARN BY GEN uses a bottom up search strategy to search the hypothesis space from specific to general by combining three generalization techniques, namely, saturation, minimization and maximal common subsumer [11, 12, 13, 16, 26]. We show that for every k; r 0, the algorithm LEARN BY GEN exactly learns the class THEFS( 3; k; 3; r) in polynomial time with O(pmn 2r 1 ) equivalence queries and O(p 2 m 2 n 4k 4r 1 k k ) membership queries provided the information on the termination , where m and n is the number ....

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R. Khardon, Learning function-free Horn expressions, in: Proc. the 11th Annual Workshop on Computational Learning Theory (1998) 154--165.

....(A) 11780284. of their researches, Cohen [5 7, 9] Dzeroski [11, 12, 21] Kietz [20 22] and Page [9, 26] have placed the theoretical researches for the learnability of logic programs in one of the main research topics in inductive logic programming. Recently, it has been deeply developed as [1, 18, 23, 24, 29, 30]. On the other hand, a conjunctive query problem in relational database theory [2, 4, 14, 16, 34] is a problem to determine whether or not a tuple belongs to the answer of a conjunctive query over a database. Here, a tuple, a conjunctive query, and a database in relational database theory are ....

....from an extended instance and subsumptione #ciency. It also remains open whether or not pac learnability from a simple instance and subsumption e#ciency are equivalent to any language. Various researches have investigated the learnability by using equivalence and membership queries such as [1, 23, 24, 30, 29]. Note that our result in this paper implies that ACQ[j B] j # 3) is not learnable using equivalence queries alone. It is a future work to analyze the learnability of ACQ[j B] j # 3) by using membership and equivalence queries, and by extending to one containing function symbols or ....

Khardon, R.: Learning function-free Horn expressions, Proc. 11th COLT, 154--165, 1998.

....for ulas. In the beginning of 1990 s, the notion of learning from entailment was intr duced to study thelear55CM 2 y of pr4 ositionalHor sentences [1] 7] and has successfully demonstr8M that varH]4 interC5B8M frC5B ts offir 242M 2 logicar e#cientlylearC2B2 frr entailment [4] 5] 8] [11], 16] 21] 22] Hypothesis constrMC42 techniques developed in ILP play an imp or tantr2H to achieve e#cientlear8CM in theselear488 8 ityr2 #]BM In this paper wetr to combinerneM4B independentlyder2 edfr2 the two r ots with the suppor of moder theorB5 of LogicPrcM4 C 8M In Sect. 2 we ....

....Subclasses of Definite PrniteM Let H be a definite prnite and C,D be definite clauses. IngenerBM it is not e#ciently solvable to decide if D # Cor if H = C. Ther#M H] we intr duce subclasses of definitepriteM for which decisions H = C and D # Car e#ciently solvable. Definition 14 ([11]) H is of k variable if any C # H contains at most k distinct varB55HM The arity of H is the maximumar] y of its pr C424M Definition 15 ( 16] C is constrained if everter occur5M in bd(C) is also asubter of anarHH#] t of h (C) H is constr8 BH if so ar all C # H . Definition 16 ....

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R. Khardon, "Learning function-free Horn expressions," Proc. COLT'98, pp.154--165, 1998.

....4.1 The learning problem Our problem is identifying an unknown tree translation system H 3 from examples of ordered pairs E 2 M(H 3 ) that are either derived or not derived by H 3 . As a formal model, we employ a variant of exact learning model by Angluin [1] called learning from entailment[2, 3, 7, 8], which is tailored for translation systems. Let H be a class of translation systems to be learned, called hypothesis space, and LR be the set of all ordered pairs, called the domain of learning. In our learning framework, the meaning or the concept represented by H 2 H is the set M(H 3 ) If M(P ....

R. Khardon, "Learning function-free Horn expressions," Proc. COLT'98, pp.154--165, 1998.

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R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241--275, 1999.

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R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241--275, 1999.

....Some results in this model have also been obtained for sub classes of Horn expressions in rst order logic but the picture there is less clear. Except for a monotone like case [24] the query complexity is either exponential in one of the crucial parameters (e.g. universally quanti ed variables) [17, 4] or the algorithms use additional syntax based oracles [5, 25] It is thus interesting to investigate whether this gap is necessary. The current paper takes a rst step in this direction by studying the query complexity in the propositional case. Query complexity can be characterized using the ....

R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241{ 275, 1999.

....Frazier Pitt [6] formalised learning from entailment using equivalence queries and membership queries and showed the learnability of propositional Horn expressions. Generalising this result to the rst order setting is of clear interest. Indeed, several works have been done following this line [9, 3, 20, 19, 10, 11, 2] obtaining algorithms that work for certain subsets of Horn expressions. Learning rst order Horn expressions has become a fundamental problem in Inductive Logic Programming [15] Theoretical results have shown that learning from examples only is feasible for very restricted classes [4] and that, ....

....and membership queries (MIS [23] CLINT [18] for example) Some of the techniques developed in this framework have been adapted for systems that learn from examples only [21, 12] We present an algorithm to learn certain subsets of Horn expressions. The algorithm is related to the ones in [10, 11], which learn Range Restricted Horn expressions. The algorithms in [10, 11] and here use two main procedures. The rst, given a counterexample clause, minimises the clause while maintaining it as a counterexample. The minimisation procedure used here is stronger than those in [10, 11] resulting in ....

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R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241-275, 1999.

....(e.g. 20, 17, 9, 5] Since only limited classes of expressions are learnable from examples [6] heuristics are used to obtain good results in practice. One recent strand of theoretical work has shown that larger classes of expressions are learnable if the learner is allowed to ask questions [12, 21, 4, 22, 16, 14, 13, 3, 15]. These works use standard oracles from learning theory [1] as well as new types of questions appropriate for the rst order setting. Two main challenges remain in this area. One is to further clarify which classes are learnable and with what complexity. The other is to establish applications of ....

....which atom is the rst one to be used when deriving a particular conclusion. Krishna Rao and Sattar [16] use subsumption queries to nd out whether hypothesised clauses syntactically match the concept being learned. Despite the variation in example types and question types the algorithms in [12, 21, 4, 22, 16, 14, 13, 3] share a common structure, which is already re ected in learning propositional expressions [1, 2, 10] A similar structure is used for learning description logic expressions in [11] The algorithms maintain multi clause hypotheses, and learn all clauses simultaneously. Given a new uncovered ....

R. Khardon. Learning function-free Horn expressions. Machine Learning, 37(3):241-275, 1999.

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Khardon, R. (1999a). Learning function-free Horn expressions. Machine Learning, 37, 241--275.

....This work is based on previous results on learnability of function free Horn expressions and range restricted Horn expressions. The learnability of the class of range restricted Horn expressions was solved in [Kha99b] by reducing it to the case of function free Horn expressions, already solved in [Kha99a]. The algorithm presented here has been obtained by retracing this reduction and using the resulting algorithm as a starting point. However, it has been significantly modified and improved. The algorithm in [Kha99a, Kha99b] uses two main procedures. The first, given a counterexample clause, ....

....by reducing it to the case of function free Horn expressions, already solved in [Kha99a] The algorithm presented here has been obtained by retracing this reduction and using the resulting algorithm as a starting point. However, it has been significantly modified and improved. The algorithm in [Kha99a, Kha99b] uses two main procedures. The first, given a counterexample clause, minimises the clause while maintaining it as a counterexample. The minimisation procedure used here is stronger resulting in a clause which includes a syntactic variant of a target clause as a subset. The second procedure ....

[Article contains additional citation context not shown here]

R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241--275, 1999.

....as subterms of more complex terms. And every clause includes in its antecedent all inequalities possible between all terms appearing in it. The paper shows that small modifications to the algorithm and proof of [AK00] yield the learning result. Further background and related work appear in [Ari97, RT98, RS98, MF92, Kha99a, Kha99b]. The rest of the paper is organised as follows. Section 2 gives some preliminary definitions. The learning algorithm is presented in Section 3 and proved correct in Section 4. 2 Preliminaries 2.1 Inequated Range Restricted Horn Expressions We consider a subset of the class of universally ....

R. Khardon. Learning function free Horn expressions. Machine Learning, 37:241--275, 1999.

.... several systems [Sha83,SB86,DRB92,MB92] and results on learnability in the limit were derived [Sha91,DRB92] More recently progress has been made on the problem of learning first order Horn expressions from equivalence and membership queries using additional constraints or other additional queries [Ari97,RT97,Kha98,RT98,RS98]. This work was partly supported by EPSRC Grant GR M21409. In particular [Kha98] shows that universally quantified function free Horn expressions are exactly learnable in several models of learning from equivalence and membership queries. This paper extends these results to a class of ....

....More recently progress has been made on the problem of learning first order Horn expressions from equivalence and membership queries using additional constraints or other additional queries [Ari97,RT97,Kha98,RT98,RS98] This work was partly supported by EPSRC Grant GR M21409. In particular [Kha98] shows that universally quantified function free Horn expressions are exactly learnable in several models of learning from equivalence and membership queries. This paper extends these results to a class of expressions allowing the use of function symbols. In particular, we present algorithms for ....

[Article contains additional citation context not shown here]

R. Khardon. Learning function free Horn expressions. Technical Report ECS-LFCS-98-394, Laboratory for Foundations of Computer Science, Edinburgh University, 1998. A preliminary version of this paper appeared in COLT 1998.

....limit were derived [Sha83, DRB92] More recently Reddy and Tadepalli [RT97, RT98] considered the use of equivalence and membership queries and have shown that Horn definitions (where all clauses have the same unique positive literal) and acyclic Horn expressions are learnable. In previous work [Kha98] we have shown that universally quantified function free Horn expressions are exactly learnable in several models of learning from equivalence and membership queries. This paper extends these results to a class of expressions allowing the use of function symbols. In particular, we present ....

....the following clause and its equivalent form (p 1 (x 1 ; f 1 (x 2 ) p 2 (f 2 ( p 1 (x 1 ; f 2 ( z 1 = f 1 (x 2 ) z 2 = f 2 ( p 1 (x 1 ; z 1 ) p 2 (z 2 ) p 1 (x 1 ; z 2 ) 1 A similar restriction has been used before by several authors. Unfortunately, in a previous version of [Kha98] it was called non generative while in other work it was called generative [MF92] The term range restricted was used in database literature for the function free case [Min88] Here we use a natural generalisation for the case with function symbols. where we have replaced each term t i ....

[Article contains additional citation context not shown here]

R. Khardon. Learning function free Horn expressions. Technical Report ECSLFCS -98-394, Laboratory for Foundations of Computer Science, Edinburgh University, 1998. A preliminary version of this paper appeared in COLT 1998.

No context found.

Khardon, R.: Learning function-free Horn expressions, Machine Learning, 37(3), December 1999, 241--275.

No context found.

R. Khardon, "Learning function-free Horn expressions," Proc. COLT'98 , pp.154--165, 1998.

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