Buntine, W. (1988) Generalized Subsumption and Its Applications to Induction and Redundancy, Artificial Intelligence, 36, 149-176.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can contain an extremely large number of (redundant) literals. If we have a background model M and n examples then the maximum number of literals in the created clause can be M n 1. A su#ciently large background model and or a lot of examples can make the creation of the clause intractable [Bun88]. Besides from the obvious deletion of examples (literals without variables in general) from the clause, a few other methods also have been proposed for reducing the number of literals in the induced clause. 4.3.2.6 Reducing the size of the clauses The first method for limiting the number of ....

W. Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36(2):149-- 176, 1988.

....refinements. Hence, the new problem is defining operators that refine whole theories rather than single clauses [9, 1] The resulting extended setting could take into account the possibly available background knowledge, and then it is also comparable to generalized and relative subsumption [4, 14] or implication [13] The remainder of the paper is organized as follows. In Section 2, we present the semantics and proof theory in our framework. Section 3 deals with refinement operators and their properties. Then, in Section 4, we define refinement operators that are proven ideal for the ....

W. Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36(2):149--176, 1988.

.... B ) iff for every example e, covers(S, M(e U B) implies covers(G, M(e U B) iff S U B G iff S B G. B is also called relative implication. It is sometimes also referred to as semantic generality. Weaker versions of B are Plotkin s relative subsumption and Buntine s generalized subsumption (see[Buntine, 1988]) When the background is empty, both reduce to 0 subsumption. It has been shown by [Jung, 1993] that generalized subsumption can be computed by applying saturation and 0 subsumption. The idea of saturation of a query is to extend the conjunction with the heads of the background rules whose body ....

Wray Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36:375-399, 1988.

....) is entailed while neither body(H 1 ) nor body(H 2 ) are entailed (no information about the number of loads of b is present in e) 5 Hypothesis ordering Lastly, let us define the partial ordering that structures our search space. Roughly speaking, it is an extension of generalized subsumption [Bun88] when background knowledge is made of a set of non recursive horn clauses plus a terminological component. A hypothesis is a set of Horn clauses described using the language of the Horn clause component of CARIN ALN and concluding on the target concept. Intuitively, a hypothesis H g is more ....

W. Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36:375--399, 1988.

....algorithms are often implemented using the least general generalization (lgg) algorithms introduced by Plotkin [Plotkin, 1969] Use of lgg (or its extensions e.g. 5 To see that this is true, observer that a clause like p(X,X) B can be equivalently written p(X 1 ,X 2 ) equal(X 1 ,X 2 )B . 28 [Buntine, 1988]) usually leads to algorithms that extend to clauses with arbitrary function symbols, but which are harder to analyze from a learning theory prospective. The language L k LOCAL is more powerful than it would at first appear, as is discussed in the section below. 5 Summary of learning theory ....

W. Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36(2):149--176, 1988.

....ar char acterH]M with Buntine sgener C]88 subsumption. Definition 9: Let A be agr#]2 atom and I be anHerCCHM inter4 #M 58# A definite clause A 0 # A 1 , A m covers A in I ifther is a substitution # such that A 0 # = A and A i # istr in Ifor ever i =1, n. Definition 10 ([6]) Let H and E be two definite clauses. H subsumes E w.r.t. B if,for anyHer#CBM model M of B andfor anygrC82 atom A, H cover A in M whenever E cover A. Theorem 4 ( 10] Let E be a definite clause and B a definite prnite such that B # = E. Then a definite clause K isderH able ....

....thatnonr4855 siveconstr2H ] Hor definitions K # withconstr4HM2 ARIMURA and YAMAMOTO: INDUCTIVE LOGIC PROGRAMMING 17 backgrC88 theor Bar polynomial timelearH25H with the modified EntLearn. This technique is also used in [16] to compute the leastgener8 24M B] w. rr the gener4M C5 subsumption [6] in Sect. 4.2. 5.7Lear##2 AcyclicConstr 4]5 Definite PrniteM Finally, we consider thelearCH2 of a subclass of rM curHB e definitepriteM ACH(k) acyclicconstrHH8B definite prniteM [4] Since H # allows r 25HHM C unforHM C]5H ,for a given positive counter H#C2M E,ther may existmor than one ....

W. Buntine, "Generalized subsumption and its applications to induction and redundancy," Artificial Intelligence, vol.36, pp.149--176, 1988.

....in the broadest sense is concerned with those semantic properties that can be decided or deduced by analyzing a syntactical description of the system. The properties studied in this paper are redundancy and subsumption, as they are known from the area of rule based systems (see, for instance, [2], 8] 9] Consider a high level replacement system, that is, a set of productions, an initial object, and a class of terminal objects. A production is subsumed by another if any application of the former is mimicked by the latter. As a first result, # Partially supported by the EC TMR Network ....

W. Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36(2):149--176, 1988.

....1 C 2 , but not vice versa. In the general case, subsumption between two clauses is NP complete (Garey, Johnson, 1979) but Helft [8] states that he has an n 3 approximation, for reduction under j of function free Hornclauses. To incorporate background knowledge, Plotkin [19] and Buntine [2] define relative subsumption B as subsumption relative to background knowledge B. They show that subsumption and relative subsumption are equivalent if B is empty. But relative subsumption is also equivalent to subsumption between saturated clauses 4 . Hence, it is equivalent to use ....

....if at least one of its arguments is. An argument is linked if either the literal is in the head of the clause or another argument in the same literal is linked [8] 3 A clause C1 is subsumed by a clause C2 (written C1 C2) iff C 1 C 2 for some substitution . Plotkin [19] and Buntine [2] defines relative (generalized) subsumption B as subsumption relative to some background knowledge. 4 This follows immediately from Theorem 4.2 of Buntine [2] and the properties of saturation as described by Rouveirol [21] If saturation is not finite, generalized subsumption is also ....

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Wray Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36:149 -- 176, 1988.

....also based on inverse resolution, is not restricted to unit clauses, but uses restrictions on the inverse substitutions that may be assumed. Restricted background knowledge In Horn clause logic, the generalization relationship between two clauses is easy to handle without background knowledge [Bun88] Thus, the form of allowed background knowledge has an important influence on the complexity of the learning task. One often used restriction is the use of background knowledge only in the form of ground unit clauses (facts) In MARVIN [SB86] this is done by forward applying the available ....

Wray Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36:149 -- 176, 1988.

....= threat(white,Piece1,square(1,Y) black,bishop,square(X,Y) This generalization process is repeated between all the pairs of compatible literals within clauses. That is, the lgg of two clauses C 1 and C 2 is defined as: fl : l 1 2 C 1 and l 2 2 C 2 and l = lgg(l 1 ; l 2 )g. More recently, Buntine [5] defined a model theoretic characterization of Theta subsumption, called generalized subsumption for Horn clauses (see [5] for more details) Buntine also suggested a method for constructing rlggs using Plotkin s lgg algorithm between clauses. The general idea of the rlgg algorithm is to augment ....

....compatible literals within clauses. That is, the lgg of two clauses C 1 and C 2 is defined as: fl : l 1 2 C 1 and l 2 2 C 2 and l = lgg(l 1 ; l 2 )g. More recently, Buntine [5] defined a model theoretic characterization of Theta subsumption, called generalized subsumption for Horn clauses (see [5] for more details) Buntine also suggested a method for constructing rlggs using Plotkin s lgg algorithm between clauses. The general idea of the rlgg algorithm is to augment the body of the example clauses with facts derived from the background knowledge definitions (K) and the current body of ....

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Buntine, W. (1988). "Generalized subsumption and its applications to induction and redundancy ", Artificial intelligence, 36(2), 149--176.

....negative example, it is not necessary to develop any of its generalizations, as none of them will ever meet the correctness criterion anymore. ILP related works have studied different quasi orderings for First Order Logic (FOL) search spaces: subsumption [Plotkin, 1970] generalized subsumption [Buntine, 1988], Timplication [Idestam Almquist, 1995] or logical implication [Nienhuys Cheng and de Wolf, 1996] and have formalized learning operators as refinement operators that go through a quasi ordered space of clauses [Shapiro, 1981, Niblett, 1993, van der Laag, 1995] On the other hand, pruning ....

.... Report LRI Private Properties and Natural Relations in ILP Several generality orderings have been studied within ILP, among those, subsumption [Plotkin, 1970] Definition 5 ( subsumption) A clause C subsumes a clause D iff 9 : C D : Let us also quote generalized subsumption [Buntine, 1988], T implication [Idestam Almquist, 1995] or logical implication [Nienhuys Cheng and de Wolf, 1996] Definition 6 (Refinement Operator) In this paper, we consider a refinement operator as a binary relation on the search space. An operator O is then represented by the set of pairs (H; H 0 ) ....

Buntine, W. (1988). Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36:375--399.

....The order of examples in the series is irrelevant, and other examples in the series can be arbitrary. This eliminates the possibility of coding the target concept by examples. 2 SIM is the inverse process of Shapiro s MIS. Many data driven algorithms [ Banerji, 1988, Ling, 1989, Muggleton and Buntine, 1988, Ishizaka, 1988 ] have been studied. However, most of them are heuristic. They learn faster from some good data, but fail to learn (in the limit) from bad data; little is done to characterize the good data for efficient learning. Some theoretical studies on learning from good examples have ....

....instances of the clause in the form of A i : B 1;i ; B n;i (1 i r) such that all A i , B 1;i ; B n;i are in S, and lgg 1ir (A i : B 1;i ; B n;i ) contains the clause A : B 1 ; Bn . The resulting clauses can be very long. Some techniques of logical reduction of clauses [ Buntine, 1988, Muggleton and Feng, 1990 ] and syntactic restriction (such as ij determination [ Muggleton and Feng, 1990 ] may be applied. Clearly the total number of possible clauses from the lgg of at most r most specific clauses is polynomial, so is the maximum number of faulty clauses in the conjecture. ....

W.L. Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36(2):149--176, 1988.

....all possible instantiations of rule models with domain predicates. For efficiently searching in this hypothesis space a generalization relation on the set of rule models is defined by suitably extending the Theta subsumption for clauses (see [ 12 ] or the generalized Theta subsumption (see [ 13 ] respectively. According to Buntine, a clause C Theta subsumes a clause C 0 (C Theta C 0 ) if the more general clause C can be converted to the clause C 0 by repeatedly turning variables to constants or other terms, adding atoms to the body, or partially evaluating the body of C by ....

W. Buntine, Generalized subsumption and its applications to induction and redundancy, Artificial Intelligence 36 (1988) 149--176.

....exists a substitution oe such that C 1 oe C 2 . in this area. Especially in inductive logic programming(ILP) it is extended at least in the following two dimensions. One is to extend from comparing two single clauses to two clauses with background theory[Plotkin71] MF] and to two programs[Buntine]. Another dimension is to extend the ordering to be considered, i.e. in what sense an object is more general than another object. Two of the extremes are subsumption and logical implication. The weakness of the subsumption is that it goes up too quickly along the generalization hierarchy. ....

....generate more specific generalization. So, generally speaking, the weaker the ordering, the more desirable of the corresponding generalization. Theorem 1 TP is strictly stronger than , and is strictly stronger than . TP corresponds to a particular case of the generalized subsumption in [Buntine]. As for the correspondence between the usual notions of orderings and the above notions, we can summarize with the following theorem. Theorem 2 The arrows go from a stronger ordering to a weaker ordering: TP I Figure 1 oe A question that naturally arises is: what is the more ....

Wray Buntine, Generalized subsumption and its applications to induction and redundancy, Artificial Intelligence, 36(2):149-176, 1988.

....we were now overlapping with two other areas of research, namely, logic programming and automatic programming (Cohen and Sammut, 1980) At about the same time, and unknown to us, Shapiro (1981a, b) was also making the same connections. Generalisation Marvin s generalisation method was used by Buntine (1986, 1988) in his theory of generalised subsumption and this formed the foundation for Muggleton and Buntine s (1988) absorption operator. Rouveirol s (1990) saturation operator is also derived from Marvin. Assuming that concepts are stored as Horn clauses, the basis of Marvin s generalisation operation ....

....(Cohen and Sammut, 1980) At about the same time, and unknown to us, Shapiro (1981a, b) was also making the same connections. Generalisation Marvin s generalisation method was used by Buntine (1986, 1988) in his theory of generalised subsumption and this formed the foundation for Muggleton and Buntine s (1988) absorption operator. Rouveirol s (1990) saturation operator is also derived from Marvin. Assuming that concepts are stored as Horn clauses, the basis of Marvin s generalisation operation was to find a subset of literals, within the current hypothesis, that matched the body of a clause already ....

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Buntine, W. (1988). Generalized Subsumption and its Applications to Induction and Redundancy. Artificial Intelligence, 36, 149-176.

....operational predicates (or base predicates) Definition 3. An example is a ground definite clause containing only operational predicates in its body. A concept description C is said to cover an example E under a theory T if and only if C subsumes E under the theory T , denoted as C T E (see [21]) Also, a theory is said to cover an example if and only if there is a concept description C in T such that C T E, denoted as T j= E. If a concept description (or a theory) covers all positive examples, it is called complete [22] If it covers no negative examples, it is called consistent ....

W. Buntine, Generalized subsumption and its applications to induction and redundancy, Artificial intelligence 36 (1988) 149--176.

....that have arity at most j. Example 5: The clause p(X,W) q(X,W) r(W,Z) p(W,Z) is 32 determinate, provided all literals in its body are determinate. This model of relative subsumption is restricted to ground background knowledge, but was generalised later to any kind of Horn clausal knowledge [15]. Such generalised subsumption is however not used by any of the techniques overviewed here. 2.3.3 Inverse Resolution Another model of generality is inverse resolution, based on inverting one or two resolution steps so as to induce some of its their antecedent(s) from the other antecedent(s) and ....

W. Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence 36(2):149--176, Sept. 1988.

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Buntine, W. (1988) Generalized Subsumption and Its Applications to Induction and Redundancy, Artificial Intelligence, 36, 149-176.

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Buntine, W. 1988. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence 36:149--176.

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W. Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36(2):149--176, 1988.

No context found.

W. Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence 36:149--176, 1988.

No context found.

W. Buntine. Generalized subsumption and its application to induction and redundancy. Artificial Intelligence, 36(2):149--176, 1988.

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W. Buntine, Generalized Subsumption and Its Applications to Induction and Redundancy, Artificial Intelligence 36 (1988) 149-176

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) Buntine, W., "Generalized Subsumption and Its Applications to Induction and Redundancy," Artif. Intell. 36, pp.149--176, 1988.

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Wray Buntine. Generalized subsumption and its applications to induction and redundancy. Artificial Intelligence, 36:149 -- 176, 1988.

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