C.D. Page and A.M. Frisch. Generalization and learnability: A study of constrained atoms. In S.H.. Muggleton, editor, Inductive Logic Programming, pages 29--61. 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....terminates for every clause C. Hence, we de ne the hierarchy THEFS(m; k; t; r) of terminating HEFSs. 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 ....

....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 ....

[Article contains additional citation context not shown here]

C. D. Page Jr., A. M. Frisch, Generalization and learnability: A study of constrained atoms, in: S. Muggleton (ed.), Inductive logic programming (Academic Press, 1992) l29-161.

....OGT. Learning of tree patterns dates back to Plotkin [17] where OT with the onto semantics considered and shown to be polynomial time learnable from examples or equivalence query (EQ) alone. Arimura et al. 7] extended Plotkin s algorithm for bounded number of ordered forests. Page and Frisch [16] showed that a class of OT with background theory is polynomial time learnable by a similar algorithm. Arimura et al. 8] and Amoth et al. 4] showed that ordered forests OF with the onto semantics is learnable using EQ and subset queries (SQs) Frazier and Pitt [11] introduced the notion of ....

C. D. Page and A. M. Frisch, Generalization and learnability: a study of constrained atoms, In Inductive Logic Programming, Academic Press (1992) 29 -- 61.

....of answers is based on [15] which does not rely on the semantics of the constraints in the answers. As illustrated in Section 2. 3 and Section 3, the user can provide by hand propagation rules to take into account (partially) this semantics, but, as it has been pointed out to us, approaches like [14] can be used to embed this semantics in a more general way and directly in the computation of the least upper bound. Another complementary aspect that needs to be investigated is the completeness of the solvers generated. It is clear that in general this property cannot be guaranteed, but in some ....

C. Page and A. Frisch. Generalization and learnability: a study of constrained atoms. In Inductive Logic Programming, pages 29-61. London: Academic Press, 1992.

....stated, we are unaffected by the fact that negative examples may not exist. 4 Relation to other work The problem of the limitations in numerical capabilities of existing ILP systems has been addressed variously in the literature by either restricting the language of hypotheses to logical atoms [12], using built in definitions for inequalities [3, 33, 35] or regression and numerical derivatives [13, 14] transformations to propositional level [17] or constraint satisfaction problems [20, 38] or using background knowledge for qualitative regression like predicates [27] This paper ....

....program. The technique of lazy evaluation is not specific to any particular predicate and allows arbitrary functions (statistical, numerical or even other propositional learners) to be used as background knowledge. To this extent, there has been no need to restrict the hypothesis language (as in [12]) use built in predicates, single out regressionlike predicates for special attention (as in [14] or perform transformations. The resulting ILP program should in principle, be capable of learning theories of the form reported in [9, 20] or the more restricted theories produced by ....

C.D. Page Jr. and A.M. Frisch. Generalization and Learnability: A Study of Constrained Atoms. In S. Muggleton, editor, Inductive Logic Programming, pages 29--56. Academic Press, London, 1992.

....of both learning theory and inductive logic programming, Dzeroski et al. 11] have first shown the learnability of (first order) definite programs called ij determinate. Furthermore, the series of 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 ....

C. D. Page Jr., A. M. Frisch, Generalization and learnability: A study of constrained atoms, in: S. Muggleton (ed.), Inductive logic programming (Academic Press, 1992), l29--161.

....David Page has been an active researcher in ILP since its inception in 1990, contributing to both theory and applications. David s Ph.D. work at the University of Illinois included the development of ecient least general generalization algorithms, the rst PAC learnability results for ILP [36, 37], and early results for PAClearning classes of recursive logic programs [12] While a postdoctoral research scientist at Oxford University, David became involved in applications of ILP to knowledge discovery in biology (collaborating with Imperial Cancer Research Fund) chemistry (with P zer ....

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In S. H. Muggleton, editor, Inductive Logic Programming, pages 29-61. Academic Press, London, 1992. First presented at the First International Workshop on Inductive Logic Programming (ILP'91).

....logic programs from examples and queries has attracted a lot of attention in the last fteen years. Many techniques and systems for learning logic programs are developed and used in many applications. See [13] for a survey. In this paper, we consider the framework of learning from entailment [1 5,7,8,15,16] and present a polynomial time algorithm to learn a rich class of logic programs that allow local variables and include many standard programs from Sterling and Shapiro s book [20] This is a revised and extended version of [11] Address for correspondence: M.R.K. Krishna Rao, Institute of ....

C.D. Page and A.M. Frish (1992), Generalization and learnability: a study of constrained atoms, in Muggleton (ed.) Inductive Logic programming, pp. 29-61.

.... queries when the subtrees of the tree pattern are ordered and are mapped to subtrees of the instance in the same order by a one to one onto mapping (Ko et al. 1990; Goldman and Kwek, 1999) Finite unions or forests of ordered tree patterns are learnable from equivalence and membership queries (Page and Frisch, 1992; Page, 1993; Arimura et al. 1995; Amoth et al. 1998) The learning problem for one to one onto mapping is considerably more difficult when the subtrees are not ordered. We first show that unordered tree patterns are not learnable from equivalence and subset queries when the mapping between ....

Page, C. D. and A. M. Frisch: 1992, `Generalization and Learnability: A Study of Constrained Atoms'. In: S. H. Muggleton (ed.): Inductive Logic Programming. Academic Press, pp. 29--61.

.... system is a simple model of functional programming languages, and furthermore, closely related to the recent development of tree rewriting technique used for semi structured data such as XML Translations or XSLT language[6] The learning model considered in this paper is learning from entailment [1, 9, 12]. Learning from entailment is a variant of query learning model [2] tailored for learning first order formulas, in which examples are formulas in the firstorder language which are implied or not implied. In this framework, a number of efficient learnability results are obtained for various ....

....for learning first order formulas, in which examples are formulas in the firstorder language which are implied or not implied. In this framework, a number of efficient learnability results are obtained for various fragments of first order logic have been shown to be learnable using this framework [1, 4, 9 12, 15, 16]. Recently, it is found that a class of learning algorithms for subclasses of first order Horn programs has a common scheme [16, 22] which has its origin in a monotone Boolean DNF learner in [18] Reddy and Tadepalli [16] proposed an algorithm with subsumption membership queries and saturation ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch, "Generalization and learnability: a study of constrained atoms," Inductive Logic Programming, Academic Press, pp.l29--161, 1992.

....to learning PRMs and SLPs are brie y reviewed in Section 5. The paper is summarised and future research directions are discussed in Section 6. 2. BAYESIAN LEARNING FRAMEWORKS A variety of positive and negative PAC learnability results exist for subsets of definite clause logic [Haussler 1990; Page and Frisch 1992; D zeroski et al. 1992; Kietz 1993; Cohen 1993; Raedt and D zeroski 1994; Cohen and Page 1995] However, in contrast to experimentally demonstrated abilities of ILP systems in applications (see [Muggleton 1999a; 1999b] the positive PAC results are rather weak, and even highly restricted forms ....

Page, D. and Frisch, A. 1992. Generalization and learnability: A study of constrained atoms. In Inductive Logic Programming, S. Muggleton, Ed. Academic Press, London.

....top down method. A possible way to overcome this difficulty is to allow a learner to use additional information on the target EFS H 3 . The 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 ....

....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 ....

[Article contains additional citation context not shown here]

C. D. Page Jr., A. M. Frisch, Generalization and learnability: A study of constrained atoms, in: Inductive logic programming (Academic Press, 1992) l29--161.

.... substitutions: an illustration on the mutagenesis problem is presented in [21] 8 Discussion and Perspectives This section first discusses our choice of a maximally discriminant induction, then situates this work with respect to some previous works devoted to generalization of constraints [16, 12] or reformulation of ILP problems [8, 25] 8.1 Generalization Choices This work first extends the frame of induction to constraint logic programming; an application, hopefully demonstrating the potentialities of this frame, is presented in [21] Note that this frame does not allow to learn ....

....saturation of the examples: ICP cannot use the domain knowledge in order to guide the exploration of the search space, as ML Smart [1] or PROGOL do. 8. 2 Generalization from constraints As far as we know, the generalization from constraints has only been addressed so far by Page and Frisch [16] and Mizoguchi and Ohwada [12] In [16] the goal is to generalize constrained atoms. Constrained atoms are handled as definite clauses whose antecedents express the constraints. Constrained generalizations of two atoms are built from the sorted generalizations defined on their arguments. In both ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In S. Muggleton, editor, Proceedings of ILP'91, pages 29--61, 1991.

....Notes in Artificial Intelligence c #Springer Verlag. ## This work is partially supported by Japan Society for the Promotion of Science, Grants in Aid for Encouragement of Young Scientists (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, ....

Page Jr., C. D. and Frisch, A. M: Generalization and learnability: A study of constrained atoms, in [25], l29--161.

....If sentence C is a positive example of sentence C , then we say that C subsumes C, because it has a larger denotation than C. This and similar viewpoints are also supported by previous work in inductive logic programming(Dzeroski et al. 1992; Frazier Page, 1993; Muggleton, 1991; Page Frisch, 1992) and learning from entailment where positive examples of an unknown formula are clauses or other formulas that are entailed by the unknown formula (Angluin, 1988a; Angluin, 1988c; Frazier Pitt, 1993) Besides allowing for flexibility in representation of objects, this approach also has another ....

....results are known. Cohen (1993b) gives a PAC learning algorithm for function free, two clause, closed, linearly recursive, ij determinant logic programs; he also shows (Cohen, 1993a) that when the condition of linear recursiveness is relaxed, the learning problem becomes cryptographically hard. Page and Frisch (1992) show that constrained atoms (a typed logic) are efficiently learnable. Frazier and Page (1993) provide a learning algorithm for a syntactically restricted subclass of first order Horn formulas Dzeroski et al. 1992) provide a learning algorithm for a different restriction of first order Horn ....

Page, C. D. & Frisch, A. M. (1992). Generalization and learnability: A study of constrained atoms. In S. H. Muggleton (Ed.), Inductive Logic Programming chapter 2. London: Academic Press.

....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 give ....

....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 ( 4] 21] H is acyclic ifther exists a terHBBM #] rrHBB over atoms such thath (C) B holdsfor all instances C of clauses in H and all B # ....

[Article contains additional citation context not shown here]

C.D. Page and A.M. Frisch, "Generalization and learnability: A study of constrained atoms," Inductive Logic Programming, pp.129--161, Academic Press, 1992.

....to the learner s vocabulary. This process is known as predicate invention. Several early ILP authors including Plotkin [12] and Shapiro [16] proved learning in the limit results. Recently, ILP learnability results have been proved within Valiant s PAC framework for learning a single de nite clause [11] and in [3] for learning a multiple clause predicate de nition assuming the examples are picked from a simple distribution. 3 Applications ILP is rapidly developing towards being a widely applied technology. In the scienti c area, the ILP system Golem [10] was used to nd rules relating the ....

D. Page and A. Frisch. Generalization and learnability: A study of constrained atoms. In S.H. Muggleton, editor, Inductive Logic Programming. Academic Press, London, 1992.

....stated, we are unaffected by the fact that negative examples may not exist. 4 Relation to other work The problem of the limitations in numerical capabilities of existing ILP systems has been addressed variously in the literature by either restricting the language of hypotheses to logical atoms [12], using built in definitions for inequalities [3, 33, 35] or regression and numerical derivatives [13, 14] transformations to propositional level [18] or constraint satisfaction problems [21, 41] or using background knowledge for qualitative regression like predicates [28] The aims of this ....

....program. The technique of lazy evaluation is not specific to any particular predicate and allows arbitrary functions (statistical, numerical or even other propositional learners) to be used as background knowledge. To this extent, there has been no need to restrict the hypothesis language (as in [12]) use built in predicates, single out regression like predicates for special attention (as in [14] or perform transformations. The resulting ILP program should in principle, be capable of learning directly theories of the form reported in [8, 21] or first order definitions that achieve the ....

C.D. Page Jr. and A.M. Frisch. Generalization and Learnability: A Study of Constrained Atoms. In S. Muggleton, editor, Inductive Logic Programming, pages 29--56. Academic Press, London, 1992.

....see Amoth, Cull and Tadepalli [3] The study of tree patterns is motivated by natural language processing [15] and symbolic integration [34] where instances are represented as parse trees and expressions [34] respectively. Tree patterns are also closely related to logic program representations [10, 23]. Using the exact learning model with membership and equivalence queries, Arimura, Ishizaka and Shinohara [11] showed that ordered forests with bounded number of trees can be learned eciently. Subsequently, Amoth, Cull and Tadepalli [2] showed that ordered forests with an in nite alphabet are ....

C. Page Jr. and A. Frisch. Generalization and learnability: A study of constrained atoms. In Inductive Logic Programming, pages 29-61, 1992.

.... non hybrid reasoners and their completeness proofs [15, 20] Within the substitutional framework we have studied reasoning systems for knowledge retrieval, constraint logic programming, modal logic deduction [21] parsing feature based grammars [16] inductive learning with background information [25], and planning in temporally rich domains. Constraint Solving Alan Frisch In contrast to our results on deduction with constraints, which have been ob tained by abstracting away from algorithmic issues and concentrating on architectural issues, we are taking a growing interest in ....

Page Jr., C. D. and Frisch, A. M. Generalization and Learnability: A Study of Constrained Atoms. In Muggleton, S. H., editor, Inductive Logic Programming, chapter 2, pp. 29--61. Academic Press, London, 1992.

....exclude the possibility of PAC results for first order logic. The situation has been improved by recent positive results in significant sized subsets of definite clause logic. These results have been possible for particular language biases (see Section 7) Namely, single constrained Horn clauses [94] (depth = 0, level = 0 in Section 7.1) and k clause ij determinate non recursive single predicate logic programs [33] under simple distributions. k denotes maximum number of clauses in hypotheses, i denotes level, j denotes the maximum arity of predicates in the background knowledge and simple ....

D. Page and A. Frisch. Generalization and learnability: A study of constrained atoms. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, London, 1992.

....PAC result concerning existentially quantified formulae seemed initially to exclude the possibility of PAC results for first order logic. The situation has been improved by recent positive results in significant sized subsets of definite clause logic. Namely, single constrained Horn clauses [46] and k clause ij determinate logic programs [16] Recent results by Kietz [24] indicate that every proper superset of the kclause ij determinate language is not PAC learnable. This seems to indicate a ceiling to extensions of present approaches. As the ILP applications areas show, Horn clause ....

....of a confirmation function to guide multi layered predicate invention. The resulting algorithm is a non backtracking version of an A search. This approach is effective for guiding deep predicate invention, with multiple layers. 3. 2 Language restrictions (bias) Recent results in PAC learning [46, 16, 24] show that reducing the size of the target language often makes ILP learning more tractable. The main restrictions are on the introduction of existentially quantified variables in the bodies of definite clauses. CLINT [9] places a finite limit on the number of such variables that are allowed to be ....

D. Page and A. Frisch. Generalization and learnability: A study of constrained atoms. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, London, 1992.

....unless the rule representation is restricted the learning problem is intractable. We brie y discuss some of the common restrictions studied and their relation to the use of quanti ed propositions. A Horn clause is constrained if all the variables in the consequent also appear in the antecedent [ Page and Frisch, 1992 ] This is a special case of determinacy de ned as follows. Assume one imposes an order (left to right) on the predicates in a rule s antecedent. A predicate is determinate if in any example and given a binding for the consequent and the rst i 1 predicates, there is at most one binding that ....

D. Page and A. Frisch. Generalization and learnability: A study of constrianed atoms. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, 1992.

.... non hybrid reasoners and their completeness proofs [40, 43] Within the substitutional framework we have studied reasoning systems for knowledge retrieval, constraint logic programming, modal logic deduction [44] parsing feature based grammars [41] inductive learning with background information [92], and planning in temporally rich domains. Constraint Solving In contrast to the above results, which have been obtained by abstracting away from algorithmic issues and concentrating on architectural issues, we are taking a growing interest in constraint solving algorithms. Our previous work has ....

Page Jr., C. D. and Frisch, A. M. Generalization and Learnability: A Study of Constrained Atoms. In Muggleton, S. H., editor, Inductive Logic Programming, chapter 2, pp. 29--61. Academic Press, London, 1992.

....the sense of entailment. The notion of learning from entailment has been successfully applied to first 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 [11] and description logic Classic [10, 6] 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 implications a 1 ; a n a, where a 1 ; a n ; ....

....function symbols, while related computational problems, such as subsumption and entailment, are still efficiently solvable for every fixed k. These are contrasted with that recent studies of inductive logic programming mainly deal with single nonrecursive function free clauses. Page and Frisch [11] showed that single nonrecursive clauses in ACH(k) are polynomial time learnable 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 [4, 10] For the ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch, Generalization and learnability: a study of constrained atoms, In Inductive Logic Programming, Academic Press (1992) 29 -- 61.

....based on inadequate statistics. ples than FOIL, and is less sensitive to the distribution of examples. More importantly, FORCE2 can be proved to paclearn any program in this class. This result is surprising, as previous positive results have either considered only nonrecursive concepts (e.g. Page and Frisch, 1992 ] or have assumed the ability to make membership or subset queries (e.g. Shapiro, 1982; Banerji, 1988; Dzeroski et al. 1992 ] We make use of two additional sources of information, namely the BASECASE and MAXDEPTH functions; however only one of these (the BASECASE function) is necessary ....

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In Inductive Logic Programming. Academic Press, 1992.

....that it can be hard to find a concept in the language consistent with a given set of examples. Similar results have also been obtained for two restricted languages of Horn clauses (Kietz, 1993) a simple description logic (Cohen Hirsh, 1994) and for the language of sorted first order terms (Page Frisch, 1992). All of these results, however, are specific to the model pac learnability, and none can be easily extended to the polynomial predictability model considered here. The results also do not extend to languages more expressive than these specific constrained languages. Finally, none of these ....

Page, C. D., & Frisch, A. M. (1992). Generalization and learnability: A study of constrained atoms. In Inductive Logic Programming. Academic Press.

....E. It is essential also that H should be logically consistent with the constraints in both B and E. Topics of interest within the theory of ILP include the completeness of inductive inference mechanisms [10, 11] the rate at which correct prediction increases with increasing numbers of examples [22, 4, 3, 20] as well as statistical criteria for acceptance of hypotheses [16, 18] 4 Implementations During the 1990s a large number of ILP systems were developed and compared on academic datasets. Most of these implementations and benchtest datasets have recently been collected together and made publicly ....

D. Page and A. Frisch. Generalization and learnability: A study of constrained atoms. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, London, 1992.

....and from Academy of Finland. learning from entailment has been successfully applied to first 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 Horn program is a conjunction H of implications a 1 ; a n a, where a 1 ; a n ....

....with function symbols, while related computational problems, such as subsumption and entailment, are still efficiently solvable for every fixed k. This is a contrast to that recent studies of inductive logic programming mainly deal with single nonrecursive function free clauses. Page and Frisch [20] showed that single nonrecursive clauses in ACH(k) are polynomial time learnable 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 ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch, Generalization and learnability: a study of constrained atoms, In Inductive Logic Programming, Academic Press (1992) 29 -- 61.

....a nonnegative integer k, TP k is the class of sets of at most k tree patterns. Although concepts in TP k are simple, they have characteristics common to a variety of representation frameworks for structured objects such as knowledge representation languages [7, 10] logic programming languages [6, 13], and combinatorial objects like string patterns [1, 3, 5, 9] Furthermore, computational problems related to tree patterns are more efficiently solvable than the other representation frameworks; for example, the membership and the containment problems are polynomial time solvable for tree ....

....least 2 n 0 1 queries in the worst case. 2 6 Application to Learning from Entailment In this section, we describe an application of the results in Section 3 to learning from entailment. Learning from entailment is exact learning suitable for learning logic programs. In learning from entailment [7, 13], a hypothesis is any logic program H and an example is any clause F in the same class. An equivalence test L(G) L(H) and a membership test F 2 L(H) are replaced by a logical equivalence G , H and an entailment H j= F , respectively. Exact learning and PAC learning based on ....

C. D. Page Jr. and A. M. Frisch. Generalization and Learnability: A Study of Constrained Atoms. S. Muggleton, editor, Inductive Logic Programming, 29--61, 1992.

....than the explicit building of G. 4 Discussion and Perspectives This section first discusses the choice of maximally discriminant induction, made in this work. Then, it situates the proposed cooperation of ILP and CLP, with respect to some previous works devoted to generalization of constraints [21, 16] or reformulation of ILP problems [13, 7] The limitation to a unique positive example, together with the problem of noisy data, is then discussed. 4.1 Redundant Learning The presented work differs from all ILP approaches we are aware of, in that it aims at characterizing all maximally ....

....the building of the whole G set of solutions, in the line of Version Spaces, rather than seeking some best solution according to some criterion. 4. 2 Generalization from constraints As far as we know, the generalization from constraints has only been addressed so far by Page and Frisch [21] and Mizoguchi and Ohwada [16] In [21] the goal is to generalize constrained atoms. Constrained atoms are handled as definite clauses whose antecedents express the constraints. Constrained generalizations of two atoms are built from the sorted generalizations defined on their arguments. In both ....

[Article contains additional citation context not shown here]

C.D. Page and A.M. Frisch. Generalization and learnability: A study of constrained atoms. In S. Muggleton, editor, Proceedings of the first International Workshop on Inductive Logic Programming, pages 29--61, 1991.

.... Section 3) include structure activity prediction for drug design [19, 44, 43] protein secondary structure prediction [29] finite element mesh design [9] and learning semantic grammars [45] A variety of positive and negative PAC learnability results exist for subsets of definite clause logic [11, 34, 10, 18, 7, 40]. However, in contrast to experimentally demonstrated abilities of ILP systems in applications, the positive PAC results are rather weak, and even highly restricted forms of logic programs have been shown to be prediction hard [7] Like many other results in PAC learning, positive results are only ....

D. Page and A. Frisch. Generalization and learnability: A study of constrained atoms. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, London, 1992.

....unless the rule representation is restricted the learning problem is intractable. We briefly discuss some of the common restrictions studied and their relation to the use of quantified propositions. A Horn clause is constrained if all the variables in the consequent also appear in the antecedent [ Page and Frisch, 1992 ] This is a special case of determinacy defined as follows. Assume one imposes an order (left to right) on the predicates in a rule s antecedent. A predicate is determinate if in any example and given a binding for the consequent and the first i Gamma 1 predicates, there is at most one ....

D. Page and A. Frisch. Generalization and learnability: A study of constrianed atoms. In S. Muggleton, editor, Inductive Logic Programming. Academic Press, 1992.

....in jjGjj that can be constructed in polynomial time [ Cohen and Hirsh, 1994a, Theorem 1 ] From this it is clear that prediction preserving reducibilities exist from S DFA to CoreClassic concept graphs to CoreClassic. Another well studied problem is the learnability of logic programs [ Page and Frisch, 1992; Dzeroski et al. 1992; Cohen, 1993; Kietz, 1993; Cohen and Jr. 1995 ] A special case that has received much attention is the learnability of determinate non recursive function free bounded arity one clause programs. We will denote this language below a DetLP, where a is the bound on arity. ....

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In Inductive Logic Programming. Academic Press, 1992.

....of line segments, each of which represents a segment of a blood vessel. For example, the following predicates describe an internal carotid artery: internalcarotidartery(mb1, 1) segment(1, mb1, n, 40, 130, 2] 4 Claude Sammut segment(2, mb1, w, 40, 144, 3] segment(3, mb1, nw, 35, 135, [4, 5]) segment(4, mb1, n, 40, 50, 6, 7] segment(6, mb1, ne, 20, 170, 8, 9] segment(5, mb1b1, e, 10, 100, segment(7, mb1b2, w, 5, 125, segment(8, mb1b3, e, 18, 90, segment(9, mb1b4, n, 15, 100, The blood vessel mb1, which is an Internal Carotid Artery, starts with segment ....

....We use the diameter and intensity predicates to specify the type and origin of the value found in the segment literal. After execution of this rule, the clause is: internalcarotidartery(mb1, 1) segment(1, mb1, n, 40, 130, 2] segment(2, mb1, w, 40, 144, 3] segment(3, mb1, nw, 35, 135, [4, 5]) segment(4, mb1, n, 40, 50, 6, 7] segment(6, mb1, ne, 20, 170, 8, 9] Intentional background knowledge is required if we wish to include in the concept description which segment has, say, the maximum diameter. This is a little tricky because the refinement rule must scan all of the segments ....

[Article contains additional citation context not shown here]

Page, C. D., Frisch, A. M.: Generalization and Learnability: A study of constrained atoms. In S. Muggleton (Eds.): Inductive Logic Programming. Academic Press (1992) 29--61

....arity j of predicates from B, the above condition of determinacy is equivalent to the notion of ij determinacy used in golem [Muggleton and Feng 1990] A similar idea was later used within foil [Quinlan 1991] Given these definitions, we can state the following prior results. Page and Frisch [Page and Frisch 1992] have shown that a single constrained, nonrecursive clause is learnable. Dzeroski and Lavrac [Dzeroski and Lavrac 1992] have shown that a set of constrained, nonrecursive, function free clauses can be transformed into a polynomially larger propositional representation. The work reported in this ....

C. D. Page and A. M. Frisch. Generalization and learnability: a study of constrained atoms. In S. H. Muggleton, editor, Inductive Logic Programming, Academic Press, London, 1992. In press.

....approximately correct for target concepts in L [B] In general, a non learnability result for a language L that is based on consistency hardness does not preclude the existence of a more expressive language L that is pac learnable. As an example of this in an ILP context, Page and Frisch [ Page and Frisch, 1992 ] show that the language of typed atoms is not pac learnable; however, the more general language of constrained atoms is pac learnable. To summarize, while a consistency hardness result for a language L shows that L is hard to pac learn, it gives no indication of how x L pac learnability may ....

....ground Datalog databases (that is, conjunctions of ground atoms) built from predicates of arity at most a. For clarity we will focus our discussion on such background theories, although some of the results we will discuss extend to more general forms of background theories [ Page and Frisch, 1991; Page and Frisch, 1992 ] Unless otherwise speci ed, we will also assume a Datalog representation for examples and constants. When target concepts are non recursive, we will assume that concepts contain no function symbols, and that examples are ground atoms of size n e containing constants but no other function ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In S. H. Muggleton, editor, Inductive Logic Programming, pages 29-61. Academic Press, London, 1992.

....correct for target concepts in L 1 DEPTH [B] In general, a non learnability result for a language L that is based on consistency hardness does not preclude the existence of a more expressive language L 0 that is pac learnable. As an example of this in an ILP context, Page and Frisch [Page and Frisch, 1992] show that the language of typed atoms is not pac learnable; however, the more general language of constrained atoms is pac learnable. To summarize, while a consistency hardness result for a language L shows that L is hard to pac learn, it gives no indication of how fix L pac learnability may ....

....ground Datalog databases (that is, conjunctions of ground atoms) built from predicates of arity at most a. For clarity we will focus our discussion on such background theories, although some of the results we will discuss extend to more general forms of background theories [Page and Frisch, 1991, Page and Frisch, 1992] Unless otherwise specified, we will also assume a Datalog representation for examples and constants. When target concepts are non recursive, we will assume that concepts contain no function symbols, and that examples are ground atoms of size n e containing constants but no other function ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In S. H. Muggleton, editor, Inductive Logic Programming, pages 29--61. Academic Press, London, 1992.

.... and their completeness proofs [ Fri91, FP95 ] Within the substitutional framework we have studied reasoning systems for knowledge retrieval, constraint logic programming, modal logic deduction [ FS91 ] parsing feature based grammars [ Fri95 ] inductive learning with background information [ PF92 ] and planning in temporally rich domains. Parsing, Abduction and Deduction In an effort to unite the study of language parsing and logical deduction, researchers have forwarded several proposals, most notably Definite Clause Grammars, for viewing grammars as logical formulas and parsing as ....

C. David Page Jr. and Alan M. Frisch. Generalization and learnability: A study of constrained atoms. In Stephen H. Muggleton, editor, Inductive Logic Programming, chapter 2, pages 29--61. Academic Press, London, 1992.

....and exploited in automated deduction. And a very general definition of instantiation with built in theories has been presented by Frisch and Page Jr. 1995 ] I conjecture that it is possible for ILP systems to also benefit greatly from using instantiation with built in theories. Frisch and Page Jr. 1990, 1992, 1993 ] have already explored this idea in the context of generalisation. This paper takes a first step towards exploring this idea in the context of refinement. 2 Sorted Logic The formulas with which the paper is concerned are built from the usual function and predicate symbols and, in ....

C. David Page Jr. and Alan M. Frisch. Generalization and learnability: A study of constrained atoms. In Stephen H. Muggleton, editor, Inductive Logic Programming, chapter 2, pages 29--61. Academic Press, London, 1992.

.... loves(x; y) mammal(x) mammal(y) Sigma 1 loves(z; mom(z) elephant(z) To see this consider any substitution that maps x to z and y to mom(z) This characterization of Sigma is in fact the instantiation ordering for a restricted class of equality free constrained formulas that Page and Frisch [ 1992 ] used in their study of constrained generalization. Thus this theorem tell us that the ordering used by Page and Frisch is a special case of the ordering defined in this paper. To see why equality is forbidden, consider an example where Sigma = f8x8y(f(x; y) f(y; x) g. Then p(f(b; x) ....

....computation of least upper bounds for ordinary logic. This foundational work has been extended by Frisch and Page to cover sorted logic (based on Theorem 9) Frisch and Page, 1990 ] and then further to constraint logic (using a special case of the characterizations of Theorem 3 and Theorem 6) Page and Frisch, 1992 ] The extension to constraint logic can also be viewed as an extension of Buntine s [ 1988 ] definition of generalized subsumption. One active area of ILP research over the last four years has been the study of PAC learnability of restricted classes of definite clause concepts, relative to ....

[Article contains additional citation context not shown here]

C. D. Page and A. M. Frisch. Generalization and learnability: A study of constrained atoms. In S. H. Muggleton, editor, Inductive Logic Programming, pages 29--61. Academic Press, London, 1992.

No context found.

C.D. Page and A.M. Frisch. Generalization and learnability: A study of constrained atoms. In S.H.. Muggleton, editor, Inductive Logic Programming, pages 29--61. 1992.

No context found.

C. Page and A. Frisch. Generalization and learnability: a study of constrained atoms. In Inductive Logic Programming, pages 29-61. London: Academic Press, 1992. 20

No context found.

) Page, C. D., Frisch, A. M., "Generalization and Learnability: A Study of Constrained Atoms," in S. H. Muggleton (ed.), Inductive Logic Programming , (pp. 29--61), Academic Press, 1992.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC