Muggleton, S.: Inverting Implication, Proceedings of the 2nd International Workshop on Inductive Logic Programming (S. Muggleton, K. Furukawa, Eds.), 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....himself realized that generalization under subsumption was incomplete for a certain type of clause known as a recursive clause. Recursion is an important program structure in logic programming. The ability to learn recursive clauses is therefore crucial when using a clausal representation. In [6] it is shown that the incompleteness of generalization under subsumption only concerns one type of generalization of recursive clauses, which is often called an indirect root. Idestam Almquist [2] described the well known technique to compute least general generalizations under subsumption ....

Muggleton, S., "Inverting Implication", Artificial Intelligence Journal, 1993.

....automating the construction of relational knowledge bases had not yet been developed. Now, however, a growing subfield of machine learning research called Inductive Logic Programming (ILP) addresses the problem of learning first order logic descriptions (Prolog programs) Lavrac Dzeroski, 1994; Muggleton, 1992). Due to the expressiveness of first order logic, ILP methods can learn relational and recursive concepts that cannot be represented in the feature based languages assumed by most machine learning algorithms. ILP methods have successfully induced small programs for sorting and list manipulation ....

....process to derive general clauses from specific consequences. The overall effect is a compression of the concept definition, replacing many specific instances with a few general clauses from which the instances can be derived. A successful representative of this class is Muggleton and Feng s Golem (Muggleton Feng, 1992). Like Foil, Golem may be viewed as a greedy covering algorithm, except that new clauses are hypothesized by considering least general generalizations (LGGs) of more specific clauses (Plotkin, 1970) The LGG of clauses C 1 and C 2 is the least general clause which subsumes both C 1 and C 2 . An ....

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Muggleton, S. (1992). Inverting implication. In Proceedings of the Second International Workshop on Inductive Logic Programming Tokyo, Japan.

....constructive learning in the inductive logic programming (ILP) context requires the ability to learn recursive relations. The shortcomings of existing ILP systems for inducing recursive clauses have been discussed in the recent literature, and are by now well understood (Lapointe Matwin, 1992; Muggleton, 1992; Idestam Almquist, 1993) It is generally believed that the subsumption mechanism is the main source of difficulty in inducing recursive clauses in popular ILP systems (e.g. FOIL (Quinlan, 1991) and Golem (Muggleton Feng, 1990) Several authors have initiated work on ILP systems that are ....

....logic programs. Example 1. BC 1 = last of(A; A] RC 1 = last of(A; B; CjD] last of(A; CjD] LP 1 = BC 1 , RC 1 ) belongs to the family SR; the first arguments of the recursive clause satisfy the second condition of Definition 1 and the second arguments satisfy the first condition. Muggleton s (1992) definition of n th powers and n th roots of a clause will be useful for describing the output of our algorithm. First, we need to define the function L, which contains only linear derivations of Robinson s (1965) function R. Definition2. Resolution closure) Let T be a set of clauses. The ....

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Muggleton, S. (1992). Inverting implication. In Proceedings of the First European Workshop on Inductive Logic Programming. Vienna, Austria: Unpublished.

....from most existing ILP learning methods; they all employ an unusual method of generalizing examples called forced simulation. Forced simulation is a simple and analytically tractable alternative to other methods for generalizing recursive programs against examples, such as n th root finding (Muggleton, 1994), sub unification (Aha, Lapointe, Ling, Matwin, 1994) and recursive anti unification (Idestam Almquist, 1993) but it has been only rarely used in experimental ILP systems (Ling, 1991) The paper is organized as follows. After presenting some preliminary definitions, we begin by presenting ....

....can be obtained using very simple learning algorithms. For example, in model of learnability in the limit (Gold, 1967) any language that is both recursively enumerable and decidable (which includes all of Datalog) can be learned by a simple enumeration procedure; in the model of U learnability (Muggleton Page, 1994) any language that is polynomially enumerable and polynomially decidable can be learned by enumeration. The most similar previous work is that of Frazier and Page (1993a, 1993b) They analyze the learnability from equivalence queries of recursive programs with function symbols but without ....

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Muggleton, S. (1994). Inverting implication. To appear in Artificial Intelligence.

....proof of other clauses. Therefore, they are not tuned to inducing recursive relations even though such relations are prevalent in logic programs written in PROLOG. Consequently, both approaches sometimes require that many examples be given as input. LOPSTER is instead based on logical implication (Muggleton, 1992; Lapointe Matwin, 1992; Idestam Almquist, 1993) It requires fewer examples because inverse implication, as exemplified in Figure 1, infers a clause via a chain of resolutions in which a given clause may be used repeatedly. This is more suitable for inducing recursive clauses since the ....

Muggleton, S. (1992). Inverting implication. In Proceedings of the First European Workshop on Inductive Logic Programming. Vienna, Austria: Unpublished.

.... subsumption is decidable but incomplete i.e. there may exist clauses C and D, such that C 6 D but C j= D. This occurs when C is self resolving (recursive) or if D is tautological (Gottlob, 1987) If tautologies and selfresolution are excluded, then C D , C j= D (Gottlob, 1987; Muggleton, 1993; Kietz Lubbe, 1994) The subsumption problem is NP complete in general (Kapur Narendran, 1986) Known algorithms have a worst case time complexity of O(vars(D) vars(C) or O(jDj jCj ) Definition 1 (Substitution) A substitution is a mapping from variables to terms. We denote ....

Muggleton, S. (1993). Inverting implication. Artificial Intelligence Journal.

....the subsumption theorem can be proved starting from the refutation completeness. This establishes that these two results have equal strength. Furthermore, we show that the subsumption theorem does not hold when only input resolution is used, not even in case Sigma contains only one clause. Since [Mug92, Ide93a] assume the contrary, some results (for instance results on nth roots and nth powers) in these articles should perhaps be reconsidered. 1 Introduction Inductive Logic Programming (ILP) investigates methods to learn theories from examples, within the framework of first order logic. In ILP, the ....

....However, subsumption is not enough : if D subsumes C then D j= C, but not always the other way around. So it is desirable to make the step from subsumption to implication, and the subsumption theorem provides an excellent tool for those who want to make this step. It is used for instance in [Mug92, Ide93a] 2 for inverse resolution. In [LN94b] the theorem is used to extend the result of [LN94a] that there does not exist an ideal refinement operator 3 in the set of clauses ordered by subsumption, to the result that there is no ideal refinement operator in the ordering induced by logical ....

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Muggleton, S., `Inverting Implication', in: Muggleton, S. H., and Furukawa, K. (eds.), Proc. of the Second Int. Workshop on Inductive Logic Programming (ILP-92), ICOT Technical Memorandum TM1182, 1992.

....and all previous specifications. Then the user may terminate the incremental learning process at any stage and have confidence as to the behaviour of the current program. In this context, the specification is an example set which should be covered by a complete and correct program. Muggleton [77] has pointed out the correspondence between this incremental learning approach to program synthesis (in an CHAPTER 5. NON MONOTONIC LEARNING 107 Inductive Logic Programming framework) and the deductive approach of formal methods as described by Hoare [37] Under this reading, in both ILP and ....

S. Muggleton. Inverting Implication. In ILP-92: Proc. of the Second Intl. Workshop on Inductive Logic Programming, Tokyo, 1992. ICOT TM 1182 - Institute for New Generation Computer Technology.

....and L n (T ) fC : C is a resolvent of C 1 2 L n Gamma1 (T ) and C 2 2 Tg. Also define L (T ) L 1 (T ) L 2 (T ) Then the subsumption theorem is stated as follows: T j= C iff there exists a clause D 2 L (T ) such that D subsumes C. S is given in [BM92] S 0 is given in [Mug92]. In [Mug92] Muggleton does not prove S 0 , but refers instead to [BM92] In other articles such as [I A93, LN94b, NLT93] the theorem is also given in the form of S 0 . These articles do not give a proof of S 0 , but refer instead to [BM92] or [Mug92] That is, they refer to a proof of S ....

....) fC : C is a resolvent of C 1 2 L n Gamma1 (T ) and C 2 2 Tg. Also define L (T ) L 1 (T ) L 2 (T ) Then the subsumption theorem is stated as follows: T j= C iff there exists a clause D 2 L (T ) such that D subsumes C. S is given in [BM92] S 0 is given in [Mug92] In [Mug92], Muggleton does not prove S 0 , but refers instead to [BM92] In other articles such as [I A93, LN94b, NLT93] the theorem is also given in the form of S 0 . These articles do not give a proof of S 0 , but refer instead to [BM92] or [Mug92] That is, they refer to a proof of S assuming ....

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S. Muggleton. Inverting Implication. In S. H. Muggleton, and K. Furukawa (eds.), Proc. of the Second Int. Workshop on Inductive Logic Programming (ILP92). ICOT Technical Memorandum TM-1182, 1992.

....D j= C, but not always the other way around. So it is desirable to make the step from subsumption to implication, and the subsumption theorem provides an excellent tool for those who want to make this step, since it states that implication = resolution subsumption. It is used for instance in [4, 13] 2 for inverse resolution. In [10] the theorem is used to extend the result of [9] that there does not exist an ideal refinement operator in subsumption, to the result that there is no ideal refinement operator in logical implication. In [15] the theorem is related to several generality ....

....resolvent of C 1 2 L n Gamma1 ( Sigma) and C 2 2 Sigmag. Also define L ( Sigma) L 1 ( Sigma) L 2 ( Sigma) Then the subsumption theorem is stated as follows: Sigma j= C iff there exists a clause D 2 L ( Sigma) such that D subsumes C. S is given in [1] S 0 is given in [13]. In [13] Muggleton does not prove S 0 , but refers instead to [1] In other articles such as [4, 10, 15] the theorem is also given in the form of S 0 . These articles do not give a proof of S 0 , but refer instead to [1] or [13] That is, they refer to a proof of S assuming that this is ....

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Muggleton, S., `Inverting Implication', in: Muggleton, S. H., and Furukawa, K. (eds.), Proc. of the Second Int. Workshop on Inductive Logic Programming (ILP-92), ICOT Technical Memorandum TM-1182, 1992.

....hopefully improve the efficiency of these methods and incorporate advanced features such as predicate invention, negation as failure, uncertain reasoning, and better methods for revising deeply recursive programs. acknowledgements Thanks to Ross Quinlan for making Foil 5. 1 available, to Steve Muggleton for making Golem 1.0 available, and to Josh Konvisser for helping us run these programs. Also, many thanks to the anonymous reviewers for their helpful comments on the initial draft of this paper. The first author was supported by the Air Force Institute of Technology. This research was also supported by the ....

Muggleton, S. (1992b). Inverting implication. Proceedings of the Second International Workshop on Inductive Logic Programming. Tokyo.

....Proposition 15 Let C be a clause and D a non ambivalent clause. Then C ) D if and only if C D. Proposition 15 has been proved by Gottlob (1987, page 110) It follows from this proposition that an LGG and an LGGI of a set of clauses, including at least one nonambivalent clause, are equivalent. Muggleton (1992) has investigated the relationship between resolution and implication between clauses. He describes the subsumption theorem (Lee, 1967) in terms of input resolution, and gives a corollary about the relationship between subsumption and implication between clauses. Unfortunately, this formulation ....

....theorem) Let T be a set of clauses and C a non tautological clause. Then T j= C if and only if there exists a clause D 2 R n (T ) such that D C for some n 0. Two different recent proofs of Theorem 16 have been presented, one by Nienhuys Cheng and de Wolf (1995) and one by Bain and Muggleton (1992). There also exist at least two different earlier proofs of this theorem in the literature, one by Slagle, Chang and Lee (1969) and one by Kowalski (1970) We are interested in the number of resolutions involved in the computation of a clause, and therefore we introduce the notion of nth ....

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Muggleton, S. (1992). Inverting implication. In Proceedings of the Second International Workshop on Inductive Logic Programming. ICOT Technical Memorandum TM-1182.

....than would be required by ILP systems such as Golem [38] and FOIL [49] Although the operations described by Lapointe and Matwin are shown to work on a number of examples it is not clear how general the mechanism is. Various general properties of implication between clauses are investigated in [33]. In particular it is shown that Lee s subsumption lemma [23] has the following corollary. Corollary 13 Implication and recursion. Let C; D be clauses. C D if and only if either D is a tautology or C D or there is a clause E such that E D where E is constructed by repeatedly self resolving ....

....or C D or there is a clause E such that E D where E is constructed by repeatedly self resolving C. 12 Thus the difference between subsumption and implication between C and D is only pertinent when, as in Example 11, C can self resolve. Attempts were made to a) extend inverse resolution [33] and b) use a mixture of inverse resolution and lgg [14] to solve the problem. The extended inverse resolution method in [33] suffers from the same problems of non determinacy as Cigol. Idestam Almquist s [14] use of lgg suffers from the standard problem of intractably large clauses (see Section ....

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S. Muggleton. Inverting implication. In Proceedings of the Second Inductive Logic Programming Workshop, Tokyo, 1992. ICOT (Technical report TM1182) .

....is based on a definition of most general sub unifiers. Although the operations described by Lapointe and Matwin are shown to work on a number of examples it is not clear how general the mechanism is. A complete though non deterministic algorithm is given for inverting implication in [84]. A complete and deterministic method is given by Idemstam Almquist [50] A new and simple inverse implication technique called forced simulation is described in [26] 5.5.2. Implication and resolution In this section the relationship between resolution and implication between clauses is ....

....the problem of finding the inverse resolvent of a pair of clauses as that of finding the set of quotients of two clauses. Following the same analogy the set c 2 = RL 2 (fcg) might be called the squares of the clause c and c 3 = RL 3 (fcg) the cubes of c. The following definition from [84] captures this idea. Definition5.9. nth powers of a clause) Let c and d be clauses. For n 1, d is an nth power of c if and only if d is an alphabetic variant of a clause in RL n (fcg) Taking the analogy a bit further one might also talk about the nth roots of a clause. Definition5.10. nth ....

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S. Muggleton. Inverting implication. In Proceedings of the Second Inductive Logic Programming Workshop, pages 19--39, Tokyo, 1992. ICOT (Technical report TM1182) .

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Muggleton, S.: Inverting Implication, Proceedings of the 2nd International Workshop on Inductive Logic Programming (S. Muggleton, K. Furukawa, Eds.), 1992.

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