A most general refinement operator for reduced sentences

Many learning systems use the space of logic formulas as the search space of hypotheses. To build efficient systems, the set of first order logic formulas can be reduced in many ways. Most systems restrict themselves to (subsets of) Horn clauses. In this paper we investigate the space of reduced first order sentences, which has the same expressive power as an arbritrary first order logic. Shapiro [1981] has used the subset of reduced first order sentences to define a most general refinement operator. His operator is claimed to be complete, i.e., all reduced sentences are derivable from the empty sentence. In this article we will show that his operator is not complete and propose a new, complete refinement operator for reduced first order sentences

Simple improvements of a simple solution for inverting resolution

In this paper we address some simple improvements of the algorithm of Rouveirol and Puget [1989] for inverting resolution. Their approach is based on automatic change of representation called flattening and unflattening of clauses in a logic program. This enables a simple implementation of operators, such as Absorption, presented in Muggleton and Buntine [1988]. Unfortunately both the algorithms of MB and RP are incomplete. We analyze the reasons of the incompleteness of the RP algorithm and present an improved Absorption operator. It appears that flat tree epresentations of clauses and predicate calculus with equality provide an appropriate context for these matters

Constructing refinement operators by decomposing logical implication

Inductive learning models [Plotkin 1971; Shapiro 1981] often use a search space of clauses, ordered by a generalization hierarchy. To find solutions in the model, search algorithms use different generalization and specialization operators. In this article we will decompose the quasi-ordering induced by logical implication into six increasingly weak orderings. The difference between two successive orderings will be small, and can therefore be understood easily. Using this decomposition, we will describe upward and downward refinement operators for all orderings, including $theta$-subsumption and logical implication

A most general refinement operator for reduced sentences

Many learning systems use the space of logic formulas as the search space of hypotheses. To build efficient systems, the set of first order logic formulas can be reduced in many ways. Most systems restrict themselves to (subsets of) Horn clauses. In this paper we investigate the space of reduced first order sentences, which has the same expressive power as an arbritrary first order logic. Shapiro [1981] has used the subset of reduced first order sentences to define a most general refinement operator. His operator is claimed to be complete, i.e., all reduced sentences are derivable from the empty sentence. In this article we will show that his operator is not complete and propose a new, complete refinement operator for reduced first order sentencesMachine-learning, Logisch programmeren

Simple improvements of a simple solution for inverting resolution

In this paper we address some simple improvements of the algorithm of Rouveirol and Puget [1989] for inverting resolution. Their approach is based on automatic change of representation called flattening and unflattening of clauses in a logic program. This enables a simple implementation of operators, such as Absorption, presented in Muggleton and Buntine [1988]. Unfortunately both the algorithms of MB and RP are incomplete. We analyze the reasons of the incompleteness of the RP algorithm and present an improved Absorption operator. It appears that flat tree representations of clauses and predicate calculus with equality provide an appropriate context for these mattersLogisch programmeren, Algoritmen

Subsumption and refinement in model inference

In his famous Model Inference System, Shapiro [1981] uses so-called refinement operators to replace too general hypotheses by logically weaker ones. One of these refinement operators works in the search space of reduced first order sentences. In this article we show that this operator is not complete for reduced sentences, as he claims. We investigate the relations between subsumption and refinement as well as the role of a complexity measure. We present an inverse reduction algorithm which is used in a new refinement operator. This operator is complete for reduced sentences. Finally, we will relate our new refinement operator with its dual, a generalization operator, and its possible application in model inference using inverse resolution.

Subsumption and refinement in model inference

In his famous Model Inference System, Shapiro [1981] uses so-called refinement operators to replace too general hypotheses by logically weaker ones. One of these refinement operators works in the search space of reduced first order sentences. In this article we show that this operator is not complete for reduced sentences, as he claims. We investigate the relations between subsumption and refinement as well as the role of a complexity measure. We present an inverse reduction algorithm which is used in a new refinement operator. This operator is complete for reduced sentences. Finally, we will relate our new refinement operator with its dual, a generalization operator, and its possible application in model inference using inverse resolution

Constructing refinement operators by decomposing logical implication

Inductive learning models [Plotkin 1971; Shapiro 1981] often use a search space of clauses, ordered by a generalization hierarchy. To find solutions in the model, search algorithms use different generalization and specialization operators. In this article we will decompose the quasi-ordering induced by logical implication into six increasingly weak orderings. The difference between two successive orderings will be small, and can therefore be understood easily. Using this decomposition, we will describe upward and downward refinement operators for all orderings, including $theta$-subsumption and logical implication.Machine-learning, Logisch programmeren