## Learning Acyclic First-order Horn Sentences From Entailment (1997)

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Citations: | 27 - 4 self |

### BibTeX

@MISC{Arimura97learningacyclic,

author = {Hiroki Arimura},

title = {Learning Acyclic First-order Horn Sentences From Entailment},

year = {1997}

}

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### Abstract

This paper consider the problem of learning an unknown first-order Horn sentence H 3 from examples of Horn clauses that H 3 implies and does not imply. Particularly, we deal with a subclass of first-order Horn sentences ACH(k), called acyclic constrained Horn programs of constant arity k. ACH(k) allow recursions, disjunctive definitions, and the use of function symbols. We present an algorithm that exactly identifies every target Horn program H 3 in ACH(k) in polynomial time in p; m and n using O(pmn k+1 ) entailment equivalence queries and O(pm 2 n 2k+1 ) request for a hint queries, where p is the number of predicates, m is the number of clauses contained in H 3 and n is the size of the longest counterexample. This algorithm combines saturation and least general generalization operators to invert resolution steps. Then, we show that request for hint queries can be replaced by entailment membership queries for a proper subclass of ACH(k). Using this method, we have a polynomi...

### Citations

1913 |
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- LLoyd
- 1987
(Show Context)
Citation Context ...s x; y; : : :. Each symbol P 2 5 [ 6 is associated with a nonnegative integer called an arity . For the definitions of term, atom, formula, model, substitution, we refer to a textbook by, e.g., Lloyd =-=[16]-=-. A positive Horn clause C of H is an expression of the form (b 1 ; : : : ; b n ! a); where b 1 ; : : : ; b n ; a are atoms. The set of atoms fb 1 ; : : : ; b n g and the atom a are called the body an... |

682 | Learning Quickly When Irrelevant Attributes Abound: A New Linear-Threshold Algorithm
- Littlestone
- 1988
(Show Context)
Citation Context ...ploy standard transformations from an exact learning algorithm into either a PAC-learning algorithm (Angluin [3]), or a polynomial time prediction algorithm with polynomial mistake bound (Littlestone =-=[15]-=-) when we allow membership queries in addition to examples. 3 Learning Acyclic Horn Sentences In this section, we prove that there exists a polynomial time algorithm that exactly identifies all Horn p... |

667 |
Queries and concept learning
- Angluin
- 1988
(Show Context)
Citation Context ...5, all type of queries are efficiently computable by a teacher for ACH(k). Hence, we can employ standard transformations from an exact learning algorithm into either a PAC-learning algorithm (Angluin =-=[3]-=-), or a polynomial time prediction algorithm with polynomial mistake bound (Littlestone [15]) when we allow membership queries in addition to examples. 3 Learning Acyclic Horn Sentences In this sectio... |

481 |
A note on inductive generalization
- Plotkin
- 1969
(Show Context)
Citation Context ... C 1 with C 3 at line 9. By iterating the learning process, LEARN ACH eventurally identifies H 3 . 3.2 Saturation and least general generalization Saturation [11, 23] and least general generalization =-=[22]-=- are operations to invert the resolution steps by which positive examples are derived from clauses in H 3 . The saturation of a clause (A ! a) by H is a clause (Closure H (A) ! a) such that ClosureH (... |

221 |
Finding patterns common to a set of strings
- Angluin
- 1980
(Show Context)
Citation Context ... are necessary to efficiently learn ACH(k). Hereditary Elementary Formal Systems (HEFS), introduced by Miyano et al. [18], are logic programs over strings, which are an extension of pattern languagess=-=[1]-=- by recursive definition. Arimura et al. [7] consider a string counterpart of the lgg operation for pattern languages. It is a future problems to investigate the learnability of HEFS using such operat... |

111 | Learning conjunctions of Horn clauses - Angluin, Frazier, et al. - 1992 |

110 |
When won't membership queries help
- Angluin, Kharitonov
- 1995
(Show Context)
Citation Context ...how in Section 6 that entailment cannot be eliminated to learn ACH(k) for every ks1 even with both equivalence and membership queries are allowed by applying a pwm reduction of Angluin and Kharitonov =-=[5]-=-. Any Horn program of ACH(0) is acyclic propositional Horn sentence of Angluin [2] and any single clause of ACH(k) is a constrained atom of Page and Frisch [20]. Thus, our positive result properly gen... |

42 |
Lower bound methods and separation results for on-line learning models
- Maass, TurĂ¡n
- 1992
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Citation Context ... w C t D. Hence, the theorem follows from the proof of Theorem 15. 2 5 A lower bound result We have the following lower bound on query complexity from a general lower bound result of Maass and Tur'an =-=[17]-=-. This theorem says that the query complexity of our learning algorithm is almost the square of the optimal in n. Thus, it is difficult to significantly improve the efficiency. Theorem 17. For every k... |

40 |
Learning from entailment: An application to propositional Horn sentences
- Frazier, Pitt
- 1993
(Show Context)
Citation Context ...n clauses that H 3 either implies or does not imply. This type of learning framework is called learning from entailment and has been introduced in the studies of learning propositional Horn sentences =-=[2, 11]-=-; Frazier and Pitt [11] showed that propositional Horn sentences are polynomial time learnable using equivalence and membership queries in the sense of entailment. The notion of ? A part of this resea... |

33 |
Parallel complexity of logical query programs
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- 1986
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Citation Context ...me polynomial in size(H) and size(C). To decide if ground oe (H) j= Coe, we can compute ground oe (H)[Aoe j= aoe in polynomial time by a standard method in deductive database, e.g., semi-naive method =-=[24]-=-. 2 2.4 Learning problem We employ a variant [2] of exact learning model from equivalence and membership queries of Angluin [3]. Let L be the underlying first-order language and H 3 be the target Horn... |

32 | learning
- Hancock, Dunham
- 2001
(Show Context)
Citation Context ...ed that various interesting fragments of first-order logic are shown to be efficiently learnable from entailment, such as constrained atoms with a background theory [20] and description logic Classic =-=[12, 10]-=-. In this paper, we consider the learnability of a subclass of first-order Horn sentences ACH(k), called acyclic constrained Horn programs of constant arity k. A Horn program is a conjunction H of imp... |

26 | Finding minimal generalizations for unions of pattern languages and its application to inductive inference from positive data
- Arimura, Shinohara, et al.
- 1994
(Show Context)
Citation Context ...Hereditary Elementary Formal Systems (HEFS), introduced by Miyano et al. [18], are logic programs over strings, which are an extension of pattern languagess[1] by recursive definition. Arimura et al. =-=[7]-=- consider a string counterpart of the lgg operation for pattern languages. It is a future problems to investigate the learnability of HEFS using such operators in the entailment learning model. 8 Note... |

25 | Generalization and Learnability: A Study of Constrained Atoms
- Page, Frisch
- 1992
(Show Context)
Citation Context ...irst-order logic and has demonstrated that various interesting fragments of first-order logic are shown to be efficiently learnable from entailment, such as constrained atoms with a background theory =-=[20]-=- and description logic Classic [12, 10]. In this paper, we consider the learnability of a subclass of first-order Horn sentences ACH(k), called acyclic constrained Horn programs of constant arity k. A... |

24 |
Beyond inversion of resolution
- Rouveirol, Puget
- 1990
(Show Context)
Citation Context ...". Hence, we can generalize H by replacing C 1 with C 3 at line 9. By iterating the learning process, LEARN ACH eventurally identifies H 3 . 3.2 Saturation and least general generalization Satura=-=tion [11, 23]-=- and least general generalization [22] are operations to invert the resolution steps by which positive examples are derived from clauses in H 3 . The saturation of a clause (A ! a) by H is a clause (C... |

18 |
Learning from entailment of logic programs with local variables
- Rao, Sattar
- 1998
(Show Context)
Citation Context ...ceedings of the 7th Int'l Workshop on Algorithmic Learning Theory (LNAI 1316, Springer-Verlag, 432-445, 1997) contained a flaw in the proof of Lemma 5, which was pointed out by Krishna Rao and Sattar =-=[14]-=- and is fixed in this version. In the last version, ground oe (H) was carelessly defined as the set of all the ground clauses obtained from H by substituting for the variables in H arbitrary subterms ... |

18 |
Which classes of elementary formal systems are polynomial-time learnable
- Miyano, Shinohara, et al.
- 1991
(Show Context)
Citation Context ...unding preorder for H . Page and Frisch [20] introduced a restricted subclass of local variable-free clauses, called constrained clauses, which is also called hereditary clauses in Miyano et. al 1991 =-=[18]-=-. We denote by Atoms5 (a) the set f P (t 1 ; : : : ; t r ) j P 2 5 and t 1 ; : : : ; t r are subterms of a g. Definition 3. A clause (A ! a) is constrained if every term that occurs in the body A is a... |

16 | Learning unions of tree patterns using queries
- Arimura, Ishizaka, et al.
- 1997
(Show Context)
Citation Context ... from examples in the sense of entailment. However, recent studies of inductive logic programming showed that most generalizations of single nonrecursive clauses are hard to learn from examples alone =-=[8, 10]-=-. For the class of Horn programs consisting of arbitrary number of clauses, it is an open question whether there exists a polynomial time learning algorithm even if both equivalence and membership que... |

13 | Cryptographic limitations on learning one-clause logic programs - Cohen - 1993 |

9 | Learnability of description logics
- Cohen, Hirsh
- 1992
(Show Context)
Citation Context ...ed that various interesting fragments of first-order logic are shown to be efficiently learnable from entailment, such as constrained atoms with a background theory [20] and description logic Classic =-=[12, 10]-=-. In this paper, we consider the learnability of a subclass of first-order Horn sentences ACH(k), called acyclic constrained Horn programs of constant arity k. A Horn program is a conjunction H of imp... |

7 | Finding tree patterns consistent with positive and negative examples with queries
- Arimura, Shinohara
- 1994
(Show Context)
Citation Context ...ner cannot benefit by asking atoms with letters other than 0; 1 such as P (f(1; 2; 0)). We can see that this transformation is a pwm reduction from DNF to ACH(1) in the sense of [5]. For details, see =-=[8, 13]-=-. 2 7 Conclusion In this paper, we investigated active learning of a fragment of first-order Horn sentences, ACH(k), and presented a polynomial time learning algorithm using equivalence queries and re... |

6 |
Learning with hints
- Angluin
- 1988
(Show Context)
Citation Context ...n clauses that H 3 either implies or does not imply. This type of learning framework is called learning from entailment and has been introduced in the studies of learning propositional Horn sentences =-=[2, 11]-=-; Frazier and Pitt [11] showed that propositional Horn sentences are polynomial time learnable using equivalence and membership queries in the sense of entailment. The notion of ? A part of this resea... |

4 | Prediction preserving reduction - Pitt, Warmuth - 1990 |

3 | The subsumption theorem for several forms of resolution
- Nienhuys-Cheng, Wolf
- 1995
(Show Context)
Citation Context ...inimal derivation G of a clause C = (A ! a) from a Horn program H is said to be trivial if all node d of G are contained in A[fag, otherwise nontrivial . The following is due to Nienhuys-Cheng et al. =-=[19]-=-. Lemma 1 (Subsumption theorem). Let H be a Horn program, and C; D be clauses. Then, H j= C if and only if one of the following cases holds: (i) C is a tautology. (ii) C is subsumed by some clause in ... |

1 | Completeness of depth-bounded resolution for weakly reducing programs
- Arimura
- 1991
(Show Context)
Citation Context ...a bounding preorder for all programs H in the class. We denote by ACH(k;?) the largest subclass of ACH(k) that is uniformly acyclic with respect to ?. Hierarchical programs [16] and reducing programs =-=[6]-=- are examples of ACH(k;?). A nontautological positive example D is said to be composite if there exists some nontrivial, minimal derivations of D from H 3 , and otherwise prime. Theorem 16. Let ks0 an... |

1 | When won't membership queires help - Angluin, Kharitonov - 1995 |