149 research outputs found
Constraints in binary semantical networks
Information Systems;Management Information Systems;Networks;management information systems
Least Generalizations and Greatest Specializations of Sets of Clauses
The main operations in Inductive Logic Programming (ILP) are generalization
and specialization, which only make sense in a generality order. In ILP, the
three most important generality orders are subsumption, implication and
implication relative to background knowledge. The two languages used most often
are languages of clauses and languages of only Horn clauses. This gives a total
of six different ordered languages. In this paper, we give a systematic
treatment of the existence or non-existence of least generalizations and
greatest specializations of finite sets of clauses in each of these six ordered
sets. We survey results already obtained by others and also contribute some
answers of our own. Our main new results are, firstly, the existence of a
computable least generalization under implication of every finite set of
clauses containing at least one non-tautologous function-free clause (among
other, not necessarily function-free clauses). Secondly, we show that such a
least generalization need not exist under relative implication, not even if
both the set that is to be generalized and the background knowledge are
function-free. Thirdly, we give a complete discussion of existence and
non-existence of greatest specializations in each of the six ordered languages.Comment: See http://www.jair.org/ for any accompanying file
Term partitions and minimal generalizations of clauses
Term occurrences of any clause C are determined by their positions. The set of all term
partitions defined on subsets of term occurrences of C form a partially ordered set. This poset is isomorphic to the set of all generalizations of C. The structure of this poset can be inferred from the term occurrences in C alone. We can apply these constructions in this poset in machine learning
Flattening, generalizations of clauses and absorption algorithms
In predicate logic, flattening can be used to replace terms with functions by variables.
It can also be used for expressing absorption in inverse resolution. This has been done by
Rouveirol and Puget. In this article three kinds of absorption algorithms are compared
The V- and W-operators in inverse resolutions
This article gives algorithms for V- and W-operators in inverse resolution. It discusses
also the completeness of these algorithms
Generalizing Refinement Operators to Learn Prenex Conjunctive Normal Forms
Inductive Logic Programming considers almost exclusively universally quantied theories. To add expressiveness, prenex conjunctive normal forms (PCNF) with existential variables should also be considered. ILP mostly uses learning with refinement operators. To extend refinement operators to PCNF, we should first do so with substitutions. However, applying a classic substitution to a PCNF with existential variables, one often obtains a generalization rather than a specialization. In this article we define substitutions that specialize a given PCNF and a weakly complete downward refinement operator. Moreover, we analyze the complexities of this operator in different types of languages and search spaces. In this way we lay a foundation for learning systems on PCNF. Based on this operator, we have implemented a simple learning system PCL on some type of PCNF.learning;PCNF;completeness;refinement;substitutions
Complexity dimensions and learnability
A stochastic model of learning from examples has been introduced by Valiant [1984]. This PAC-learning model (PAC = probably approximately correct) reflects differences in complexity of concept classes, i.e. very complex classes are not efficiently PAC-learnable. Blumer et al. [1989] found, that efficient PAC-learnability depends on the size of the Vapnik Chervonenkis dimension [Vapnik & Chervonenkis, 1971] of a class. We will first discuss this dimension and give an algorithm to compute it, in order to provide the reader with the intuitive idea behind it. Natarajan [1987] defines a new, equivalent
dimension is defined for well-ordered classes. These well-ordered classes happen to satisfy a general condition, that is sufficient for the possible construction of a number of equivalent dimensions. We will give this condition, as well as a generalized notion of an equivalent dimension. Also, a relatively efficient algorithm for the calculation of one such dimension for well-ordered classes is given
Towards a proof of the Kahn principle for linear dynamic networks
We consider dynamic Kahn-like data flow networks, i.e. networks consisting of deterministic processes each of which is able to expand into a subnetwork. The Kahn principle states that such networks are deterministic, i.e. that for each network we have that each execution provided with the same input delivers the same output. Moreover, the principle states that the output streams of such networks can be obtained as the smallest fixed point of a suitable operator derived from the network specification.
This paper is meant as a first step towards a proof of this principle. For a specific
subclass of dynamic networks, linear arrays of processes, we define a transition system
yielding an operational semantics which defines the meaning of a net as the set of all
possible interleaved executions. We then prove that, although on the execution level there
is much nondeterminism, this nondeterminism disappears when viewing the system as a
transformation from an input stream to an output stream. This result is obtained from the
graph of all computations. For any configuration such a graph can be constructed. All
computation sequences that start from this configuration and that are generated by the
operational semantics are embedded in it
The specialization problem and the completeness of unfolding
We discuss the problem of specializing a definite program with respect to sets of positive and negative examples, following Bostrom and Idestam-Almquist. This problem is very relevant in the field of inductive learning. First we show that there exist sets of examples that have no correct program, i.e., no program which implies all positive and no negative examples. Hence it only makes sense to talk about specialization problems for which a solution (a correct program) exists.
To solve such problems, we first introduce UD1-specialization, based upon the transformation rule unfolding. We show UD1-specialization is incomplete - some solvable specialization problems do not have a UD1-specialization as solution - and generalize it to the stronger UD2-specialization. UD2 also turns out to be incomplete. An analysis of program specialization, using the subsumption theorem for SLD-resolution, shows the reason for this incompleteness. Based on that analysis, we then define UDS-specialization (a generalization of UD2-specialization), and prove that any specialization problem has a UDS-specialization as a solution. We also discuss the relationship between this specialization technique, and the generalization technique based on inverse resolution. Finally, we go into several more implementational matters, which outline an interesting topic for future research
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 -subsumption and logical implication
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