Plotkin, G. D. 1971. A note on inductive generalisation. Machine Intelligence 6:101--124.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....level at a time, starting from the most general patterns and iterating between the candidate generation and candidate evaluation phases. The high level algorithm of SPADA implements the aforementioned levelwise method (see Figure 5) The pattern space is structured according to the q subsumption [28]. Many ILP systems adopt q subsumption as the generality order for clause spaces. In this context we need to adapt the framework to the case of atomsets. More precisely, the restriction of q subsumption to Datalog queries (i.e. existentially quantified conjunctions of Datalog atoms) is of ....

Plotkin, G., `A note on inductive generalisation', Machine intelligence, No 5, 1970, pp. 153--163.

....but this falls outside the scope of the paper. 3. 2 A Specialised Rule Stretching Method Since Virtual Predict represents the rules of a hypothesis as terms it is possible to compute the minimal generalisation of a rule and the proof of an example by computing the least general generalisation [12]. De nition 1. An atom c is a generalisation of atoms a and b if there exists substitutions 1 and 2 such that c 1 = a and c 2 = b. De nition 2. A generalisation c for two atoms a and b is a least general generalisation (lgg) if for each other generalisation c i of a and b there exists a ....

G. D. Plotkin. A note on inductive generalisation. Machine Intelligence 5, pages 153163, 1970.

....may be missed. If P does not capture all of the necessary axioms, the Validate process will fail. Once a set of answers has been acquired, ExtractAbsFns partitions the set into subsets of terms that each have the same head symbol. The most speci c generalization (MSG) is computed for each subset [11]. The resulting set of MSGs is returned as the set of abstraction functions. In the lightlike example, SNARK is given the query 9(x 1 x 2 ) lightlike x 1 x 2 ) One result of this query is the pair of bindings x 1 = obj time2event o 1 t) x 2 = obj time2event o 2 (recd o 1 o 2 t) The ....

G. Plotkin, \A Note on Inductive Generalisation," Machine Intelligence 5, M. Meltzer and D. Michie (eds.), Elsevier North-Holland, New York, 1970, pp:153-163.

....call (answer) patterns would be typically stored in a dynamic database so that var(a 1 ) var(a 2 ) Hence ha 1 ; f 1 i (or equivalently ha 2 ; f 2 i) have to be appropriately renamed before the join is calculated. This is achieved as follows. Plotkin s anti uni cation algorithm [22] is used to compute the most speci c atom a that generalises a 1 and a 2 . But observe that if a 1 j= a 2 , a 2 is a most speci c generalisation of the atoms. The basic idea is to reformulate a 1 as a pair ha 1 i which satis es two properties: a 1 is a syntactic variant of a; the pair ....

G. Plotkin. A Note on Inductive Generalisation. Machine Intelligence, 5:153-163, 1970.

....up to renaming of variables. Then a k tuple encoding f can be renamed to a k tuple encoding f 0 in O(k log(k) steps. Note that k v P . 7 Join: The most general subsumer of two k tuples that encode EP os formulae, can be computed in O(k log(k) time using Plotkin s anti uni cation algorithm [19]. Observe that k aP . Equivalence check: The test for equivalence of EP os formulae over k variables can be implemented in O(k) simple uni cations (that do not involve compound terms) Let f; f 0 2 EP os be represented as tuples ht 1 ; t k i and ht 1 0 ; t k 0 i. The ....

G. Plotkin. A Note on Inductive Generalisation. Machine Intelligence, 5:153-163, 1970.

....must be done just from positive translation examples because negative examples will not be available most of the time. Just learning from positive examples may cause overgeneralization of examples because there are no restrictions imposed by negative examples. For example, just using Plotkin s [18, 19] the relative least general generalization (RLGG) schema in learning of translation templates from given translation examples between two natural languages will be disastrous. From two unrelated translation examples, an over generalized clause will be created, and this clause will satisfy almost ....

....example, the match sequence of aXbcY d and efbcZ will be (aX; ef)bc(Y d; Z) The reader can observe that a variable cannot appear in the similarity of a minimal match sequence. 3 A Specific Least General Generalization of Strings Plotkin s relative least general generalization (RLGG) technique [18, 19] is used by many ILP systems[13] In that technique, least general generalization (LGG) of terms are used in the generalization process of clauses. For example, the GOLEM system, which uses RLGG schema, generalizes following two clauses p( b,a] p( c,d,a] by creating the following general ....

Plotkin, G. D., A Note on Inductive Generalisation, Machine Intelligence 5 , M. Meltzer and D. Michie (eds.), Elsevier North-Holland, New York, 1970, pp:153-163.

....call (answer) patterns would be typically stored in a dynamic database so that var(a 1 ) var(a 2 ) Hence ha 1 ; f 1 i (or equivalently ha 2 ; f 2 i) have to be appropriately renamed before the join is calculated. This is achieved as follows. Plotkin s anti unification algorithm [20] is used to compute the most specific atom a that generalises a 1 and a 2 . The basic idea is to reformulate a 1 as a pair ha 0 1 ; f 0 1 i which satisfies two properties: a 0 1 is a syntactic variant of a; the pair represents the same dependency information as ha 1 ; truei. A pair ha 0 2 ....

G. Plotkin. A Note on Inductive Generalisation. Machine Intelligence, 5:153--163, 1970.

....usually employ q subsumption rather than entailment. Idestam Almquist defines a stronger form of entailment called T implication, which remedies some of the shortcomings of entailment [12, 13] 2 This definition, and the term q subsumption, was introduced in the context of induction by Plotkin [32, 33]. In theorem proving the above version is termed subsumption, whereas q subsumption indicates a special case in which the number of literals of the subsumant does not exceed the number of literals of the subsumee [21] 2.2 Bottom up induction While top down approaches successively specialise a ....

G. Plotkin. A note on inductive generalisation. Machine Intelligence 5, B. Meltzer and D. Michie (editors), North-Holland, pp.153--163, 1970.

....by others: if, after generalizing a rule, hamlet nds that the rule is subsumed by another, it deletes it. If it nds that the rule subsumes another rule, then it deletes the second rule. hamlet needs a special de nition for subsumption instead of the usual one employed by most ILP systems [10], and applies the following de nition: R subsumes R 0 if target concept(R) target concept(R 0 ) AND 9 such that: other goals(R) other goals(R 0 ) prior goals(R) prior goals(R 0 ) rest of preconds(R) rest of preconds(R 0 ) e ects(R) e ects(R 0 ) hamlet al..ways ....

G. D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 5, pages 153-163. Elsevier. North Holland, New York, 1970.

....a new foundation for inductive learning that naturally extends the attributevalue framework to account for applications where the individuals to be classified have complex structure. This foundation is in the tradition of, and generalises, other logic based approaches to inductive learning such as [Plo69, Sam81, Sha83, Mic83, MD94]. We use a typed, higher order logic as the knowledge representation formalism. This logic provides highly expressive hypothesis languages, which support a variety of important data types, including sets, multisets, and graphs. Induced definitions in such hypothesis languages are comprehensible, ....

G.D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence, volume 5, pages 153--163, 1969.

....of bias, we focus on the class of ILP systems described in section 2 and 3. 7 In the recent years, there has been a growing interest in this area. One motivation is that new results from the areas of theorem proving and logic programming can be exploited to further investigate approaches as [Plotkin 1970, Plotkin 1971] Another motivation is to provide a more powerful representation language in order to overcome the limitations of the attribute based representation, e.g. as in ID3 [Quinlan 1986] Choosing Horn logic and its subsets instead of other powerful representation approaches, e.g. KL ONE ....

Plotkin, G. (1970). A note on inductive generalisation. In B. Meltzer & D. Michie (Eds.), Machine Intelligence, volume 5 (pp. 153--163). Edinburgh: Edinburgh University Press.

....[3] 4] but the algorithm there is still inefficient and undirected. In inverse resolution or other related topics we should generalize a clause to clauses which are not too general. For example, given a clause, what are the minimal generalizations Given a few clauses what is their supremum ( 5] [6]) Similiar questions can also be asked for partitions on a given clause C. This article tries to solve these questions. Consider a given clause C and all the partitions on C, we can construct the least higher partitions for a given partition and the supremum of some partitions. The constructions ....

....C 3 C 2 or C 3 and C 2 are incomparable. Consider the set of all partitions on a clause C. We can define the least higher partition of a partition in analogous way. If C 1 =C( 1 ) then finding a minimal generalization C 2 correspond with a partition 2 which is least higher than 1 . Reynolds[6] and Plotkin[5] have proved some properties about the 6 lattice structure of atomic formulas and Reynolds[6] has claimed the existence of a total chain of atomic formulas from C to C when C C. He has stated only two kinds of minimal generalizations without mentioning the minimal generalizations ....

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Gordon D. Plotkin. A Note on Inductive Generalisation. Machine Intelligence 5, B. Meltzer & D. Michie (eds.), Edinburgh University Press, 1970.

....require for instance that the generated hypotheses satisfy the language bias, that the operators be complete (generate all clauses in the language) etc. 5.2. subsumption We start discussing the simplest model of deduction for ILP: subsumption as introduced by Plotkin 5 . Definition5.3. [99, 100]) A clause c 1 subsumes a clause c 2 if and only if there exists a substitution such that c 1 c 2 . c 1 is a generalisation of c 2 (and c 2 a specialisation of c 1 ) under subsumption. In this definition, clauses are seen as sets of (positive and negative) literals. The subsumption ....

....rule is thus: subsumption: c 2 c 1 where c 1 c 2 . For example, father(X,Y) parent(X,Y) male(X) subsumes father(jef,paul) parent(jef,paul) parent(jef,ann) male(jef ) female(ann) with = fX = jef, Y = anng. 5.2.1. Properties Some properties of subsumption include (see [100, 99]) Implication. If c 1 subsumes c 2 then c 1 j= c 2 . The opposite does not hold for self recursive clauses: let c 1 = p(f(X) p(X) c 2 = p(f(f(Y ) p(Y ) c 1 j= c 2 but c 1 does not subsume c 2 . Therefore deduction using subsumption is not equivalent to implication among ....

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G.D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence 5, pages 153--163. Edinburgh University Press, Edinburgh, 1969.

....to minimize the impact of small disjuncts by specializing all rules added to the decision list. Every rule added to the decision list was replaced by the most specialized rule that covered all positive cases covered by the initial rule. This new rule was created by forming the least generalization (Plotkin, 1970) of the positive cases covered by the rule. The third technique, intern, sought to minimize the impact of small disjuncts by placing each rule developed as deep within the decision list as possible without decreasing the overall classification accuracy of the list. As small disjuncts are developed ....

Plotkin, G. D. (1970). A note on inductive generalisation. In B. Meltzer & D. Mitchie (Eds.), Machine Intelligence 5 (pp. 153-163). Edinburgh: Edinburgh University Press.

....It appears to the author to be the first implemented machine learning program. Strachey s comments show that he had understood the importance of generalisation within machine learning. In machine learning the the patterns of (1,1,0) and (2,2,0) can be generalised to (n,n,0) using Plotkin s [27] least general generalisation operator. Interestingly, Strachey s Nim rule is almost isomorphic to the colinearity rule in the King RookKing illegality domain described in [22] Machine learning of the colinearity rule has been shown [22] to require relational learning (ILP) Later A.L. Samuel, in ....

G. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Mchine Intelligence 5, pages 153--164. Edinburgh University Press, Edinburgh, 1969.

....to limit this space and guide the search. For learning to take place efficiently, it is often crucial to structure the hypotheses space. This can be done with a model of generalization. Search for hypotheses can then be seen as searching for more general clauses given a known specialized clause. Plotkin (1969; 1971a; 1971b) was the first to study in a rigorous manner the notion of generalization based on Theta subsumption. Clause C Theta subsumes clause D iff there exists a substitution oe such that Coe D (i.e. there exists a substitution that makes clause C a subset of clause D) Clause C 1 is ....

....of clauses relative to some background knowledge (rlgg) An lgg is a generalization that is less general, in terms of Theta subsumption, than any other generalization. The lgg algorithm replaces all the different terms that have the same place within compatible literals 4 by new variables. See (Plotkin 1969) for more details. For example, if we have two compatible literals: L 1 = threat(white,rook,square(1,3) black,bishop,square(4,3) L 2 = threat(white,queen,square(1,4) black,bishop,square(7,4) lgg(L 1 ; L 2 ) threat(white,Piece1,square(1,Y) black,bishop,square(X,Y) 3 Meaning that ....

Plotkin, G. D. 1969. A note on inductive generalisation. In B. Meltzer & D. Michie, editors, Machine Intelligence 5, Edinburgh University Press, Edinburgh, pages 153-- 163.

....this space and guide the search. For learning to take place efficiently, it is often crucial to structure the hypothesis space. This can be done with a model of generalization. Searching for hypothesis can then be seen as searching for more general clauses given a known specialized clause. Plotkin [16, 17, 18] was the first to study in a rigorous manner the notion of generalization based on Theta subsumption. Clause C Theta subsumes clause D iff there exists a substitution oe such that Coe D. Clause C 1 is more general than clause C 2 if C 1 Theta subsumes C 2 . Plotkin investigated the existence ....

....lgg of clauses relative to some background knowledge or rlgg. That is, generalizations which are less general, in terms of Theta subsumption, than any other generalization. The lgg algorithm replaces all the different terms that have the same place within compatible literals by new variables (see [16] for more details) For example, if we have two compatible literals: L 1 = threat(white,rook,square(1,3) black,bishop,square(4,3) and L 2 = threat(white,queen,square(1,4) black,bishop,square(7,4) then lgg(L 1 ; L 2 ) threat(white,Piece1,square(1,Y) black,bishop,square(X,Y) This ....

G.D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence 5, pages 153-163, Edinburgh University Press, Edinburgh, 1969.

....ground facts. This special case is the example setting. Diverse implementations of ILP algorithms have been devised based on the idea of inversion of the deduction algorithms used in logic programming. Some of the more common ideas are inverse resolution [12] Relative Least General Generalization [14], Inverse Implication and others. 4 ILP to learn and revise program structure 4.1 Learning new concepts In many programming scenarios, we need to define new concepts to be able to satisfy the functional specifications of the program, which can be expressed in terms of the desired outputs of ....

G. D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence 5, pages 153--163. Elsevier North Holland, New York, 1970.

....been previously obtained on a number of real applications and of test cases taken from standard machine learning data bases. 1. Introduction In the recent literature, inducing descriptions of structured concepts from examples is receiving a growing interest. After the first proposed approaches [11,20,28,29], a fundamental methodological contribution came from Michalski, with the INDUCE system [15] and then the topic has been further developed in other systems, such as ML SMART [1,2] RIGEL [8] GOLEM [18] FOIL [22] CLINT [7] FOCL [19] and the systems developed by Kodratoff [14] In developing ....

....consistent description of the target relation. The system needs to be helped by a teacher, who is required to specify a range of possible values for k. For instance, the declaration greater area(x,k: 5,20,0. 2] specifies that k can assume each one of the values obtained by subdividing the interval [5,20] with a step of 0.2. When ML SMART tries to specialize a formula j using such a predicate, it will select for k the value maximizing the information gain. 4. The Artificial Application Domains In order to perform a set of experiments under controlled conditions, an artificial application domain ....

G.D. Plotkin: "A Note on Inductive generalisation", Machine Intelligence, 6, 1970, 101-124.

....this structure is well known in our domain, rdt learns good rules in both applications. But we need a large number of rule schemata to solve the learning task. For the first task, we have one schema for each expected pattern length (see Figure 5) rdt orders the rule schemata by subsumption [Plotkin, 1970, Plotkin, 1971] This allows pruning, if a hypothesis is already too special to be accepted. But the used schemata are not comparable by subsumption, none of two schemata is more general than the other. Therefore, exactly the same partial patterns were tested for each schema. Clearly, this ....

Plotkin, G. D. (1970). A note on inductive generalisation. Machine Intelligence, 5:153 -- 163.

.... a cover of the set of all possible hypotheses, which only needs to contain the most general integrity constraints satisfied by the data (see Definition 3) An important difference in compar3 This definition, and the term subsumption, was introduced in the context of induction by Plotkin [25, 26]. In theorem proving the above version is termed subsumption, whereas subsumption indicates a special case in which the number of literals of the subsumant does not exceed the number of literals of the subsumee [16] 4 An alternative view of induction of integrity constraints is obtained if we ....

G. Plotkin, `A note on inductive generalisation', Machine Intelligence 5, B. Meltzer & D. Michie (eds.), North-Holland, 1970, pp. 153-163. 1970. 22 Peter A. Flach and Iztok Savnik / Database dependency discovery: a machine learning approach

....11. most specific generalisation of terms Let t be a generalisation of the terms t 1 and t 2 . Then t is the most specific generalisation of t 1 and t 2 , denoted msg(t 1 ; t 2 ) if for any generalisation t 0 of t 1 and t 2 , there exists a substitution 0 such that t 0 0 = t [13] [12]. Definition 12. most specific generalisation of set of equations Let H 1 = fx 1 = t 1 ; xn = t n g and H 2 = fy 1 = s 1 ; ym = s m g. Let z 1 ; z k be the variables that occur on the left hand side of equations in both H 1 and H 2 , say z 1 = t 1 ; z k = t k ....

G. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence, Vol.5. Edinburgh University Press, 1974.

....Definition 3.2 A clause C is subsumption equivalent to a clause D and we write C j s D iff C D and D C. A clause is reduced if it is not subsumption equivalent to any proper subset of itself. Subsumption is the most common ordering over the set of clauses used in ILP systems. subsumption ([Plo69]) is usually the name used in ILP to refer to the concept of subsumption. The ordering over the set of clauses is sometimes called a generalisation model [Bun88] If not stated otherwise, the generalisation ordering assumed in the definitions of the rest of the report is subsumption. The concept ....

....lattice. The top element of the subsumption lattice is 2, the empty clause. The glb of two clauses C and D is called the most general instance (mgi) and is the union of the two clauses mgi(C,D) C [ D. The lub of two clauses C and D is called the least general generalisation (lgg) [Plo69] of C and D. Under subsumption the glb and lub of clauses is unique up to a renaming of variables. Definition 3.7 Two literals are compatible if they have the same predicate symbol and sign. Definition 3.8 Let l 1 = p(t 11 , t 1n ) and l 2 = p(t 21 , t 2n ) be two compatible ....

G. D. Plotkin. A note on inductive generalisation, pages 153--163. Edinburgh University Press, Edinburgh, 1969. eds. Meltzer, B. and Michie, D.

....of the machine learning system in either layer. To this end we believe that it is useful to provide facilities to enable the user to require the inclusion or exclusion of specific cases from a cluster. A simple means to extend a rule to cover a single additional case is least generalisation [17] . We believe that this provides a suitable mechanism for extending a partition to include a case as it extends that partition the least possible amount. The counterpart to least generalisation, for excluding cases from a partition, is least specialisation. However, while there will always be a ....

G. D. Plotkin, A note on inductive generalisation, in: B. Meltzer and D. Mitchie, eds., Machine Intelligence 5 (Edinburgh University Press, Edinburgh, 1970) 153-163.

....is not. This conclusion can be reached by looking for attributes, apart from C, for which both witnesses have different values, i.e. D. This set of attributes is called the disagreement of the two witnesses, and can be obtained by computing their antiunification (the dual of unification) [Plotkin 1970, 1971; Reynolds 1970] The anti unification of r(a1,b1,c1,d1) and r(a1,b1,c2,d2) is r(a1,b1,C,D) suggesting D as an extension to the lefthand side of the fd. In the next iteration, AD C may itself turn out to be contradicted, if r(a1,b2,c1,d2) happens to be in r. But then we obtain ....

G.D. PLOTKIN, `A note on inductive generalisation', in Machine Intelligence 5, B. Meltzer & D. Michie (eds.), Edinburgh University Press, Edinburgh, 153-163.

....of atoms of predicate symbol p. A generalisation of T is an atom t g such that for all t 2 T , t is an instance of t g . A most specific generalisation (or anti unifier) of T is a generalisation t msg such that for all other generalisations t g of T, t msg is an instance of t g . It can be proven [15] that the most specific generalisation of a non empty set of atoms always exists and is unique up to variable renaming. For notational convenience, we write msg(T ) t msg , although the most specific generalisation is not a function. Definition 3.3 (Benefits from Specialisation) Let P be a ....

G. D. Plotkin. A note on inductive generalisation. Machine Intelligence, 5:153--163, 1970.

....and Applications. The author has argued [25] that the development of a formal semantics of ILP (see Section 2) allows the direct derivation of ILP algorithms. Such derivational techniques have been at the centre of specific to general ILP techniques such as Plotkin s least general generalisation [37, 36], Muggleton and Buntine s inverse resolution [27, 23] and Idestam Almquist s inverse implication [15] Two contending semantics of ILP have been developed [31] These are the so called open world semantics [31] and closedworld semantics [13] A number of efficient ILP systems have been ....

G.D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence 5, pages 153--163. Elsevier North Holland, New York, 1970.

....we define minimal proper generalizations and the supremum of clausal forms w.r.t. to the order relation in the second level. For the results about minimal proper generalizations we can look up in [4] For the supremum of compatible clausal forms we need only to generalize the results of Plotkin in [9]. In chapter 5, 6 we use the results of chapter 3, 4 to solve the problems in V operators and Woperators respectively. In chapter 7 finding generalizations by using the integer coding and term partitions shall be quickly explained. We look at the problems V and W operators once more by using this ....

....Given a clause C, there can be infinite many clauses D such that D =C. For example, C=P(x,x) then D 1 = P(x 1 ,x 2 ) D 2 = P(x 1 ,x 2 ) P(x 2 ,x 1 ) D 3 = P(x 1 ,x 2 ) P(x 2 ,x 3 ) P(x 3 ,x 1 ) etc. are all generalizations of C. Two D i s are not equivalent under the definition of Plotkin[9]. Plotkin defines an equivalence relation by using the concept of subsume. A clause D subsumes C if there is a substitution such that C D . We say D C if C subsumes D and D subsumes C, e.g. P(x,y) P(x,y) P(z,u) If j i, then D i subsumes D j , but D j does not subsume D i . The size of the ....

[Article contains additional citation context not shown here]

Gordon D. Plotkin. A Note on Inductive Generalisation. Machine Intelligence 5, B. Meltzer & D. Michie (eds.), Edinburgh University Press, 1970.

....optimum assignment. The system is evaluated on a complex artificial domain, that shows the good potentialities of this approach. I. INTRODUCTION In the recent literature, inducing descriptions of structured concepts from examples is receiving a growing interest. After the first proposed approaches [1,2,3,4], a fundamental methodological contribution came from Michalski, with the system INDUCE [5,6] Then, the topic has been further developed in other systems, such as ML SMART [7] RIGEL [8] GOLEM [9] FOIL [10] CLINT [11] the system developed by Kodratoff [12] and FOCL [13] In developing FOIL, ....

G.D. Plotkin, "A Note on Inductive generalisation", Machine Intelligence, 6, 1970, 101-124.

....rdt learns good rules in both applications. But we need a large number of rule schemata to solve the learning task. For the first task, we need one schema for each expected pattern length (Figure 3) A second problem concerns the time rdt needed. It orders the rule schemata by subsumption (Plotkin, 1970). This allows pruning, if a hypothesis is already too special to be accepted. But the used schemata are not comparable by subsumption, none of two schemata is more general than the other, because of the chain of time points and there binding to the conclusion. Therefore, exactly the same partial ....

Plotkin, G. D. (1970). A note on inductive generalisation. Machine Intelligence, 5:153--163.

....a unified notation. 2.3 Subsumption of concepts Computing whether one concept subsumes is more general than another (recall Expression 4) is a central facility of all structured concept discovery systems. A common method for subsumption computation is based on the so called subsumption (Plotkin, 1971; Gottlob, 1987) I assume familiarity with the idea of substitutions. subsumption usually applies to logical clauses rather than concepts: the modifications made for the following definition are very slight. Definition 1 A structured concept x: 9 v m )C subsumes a structured concept ....

....is saturated to the concept x:small(x) fluffy(x) dog(x pet(x) A subsuming concept is then produced by removing one or more literals from this saturated concept. As an aside, in the field of inductive logic programming (Muggleton, 1992) see also Section 2.6. 1 below) it is common to employ Plotkin s (1971) least general generalization algorithm as a common subsumer method. While the algorithm runs in polynomial time in the size of two input concepts, when used with a background theory it may produce very long expressions. These will likely contain many redundant variables and literals which ....

Plotkin, G. D. 1971. A note on inductive generalisation. Machine Intelligence 6:101--124.

....usually employ subsumption rather than entailment. Idestam Almquist defines a stronger form of entailment called T implication, which remedies some of the shortcomings of entailment [19, 20] 2 This definition, and the term subsumption, was introduced in the context of induction by Plotkin [34, 35]. In theorem proving the above version is termed subsumption, whereas subsumption indicates a special case in which the number of literals of the subsumant does not exceed the number of literals of the subsumee [25] 2.3 Bottom up induction While top down approaches successively specialise a ....

G. Plotkin. A note on inductive generalisation. Machine Intelligence 5, B. Meltzer & D. Michie (editors), North-Holland, pp.153--163, 1970.

No context found.

Plotkin, G. D. 1971. A note on inductive generalisation. Machine Intelligence 6:101--124.

No context found.

G. Plotkin. A note on inductive generalisation. Machine Intelligence 5, B. Meltzer & D. Michie (editors), North-Holland, pp.153--163, 1970.

No context found.

G. Plotkin, `A note on inductive generalisation', Machine Intelligence 5, B. Meltzer & D. Michie (eds.), North-Holland, 1970.

No context found.

G.D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence 5, pages 153--163. Edinburgh University Press, Edinburgh, 1969.

No context found.

G. D. Plotkin. A note on inductive generalisation. In B. Meltzer and D. Michie, editors, Machine Intelligence 5, pages 153--163. Elsevier North-Holland, 1970.

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