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

Constraints in binary semantical networks

Information Systems;Management Information Systems;Networks;management information systems

Unilateral versus bilateral upper limb training after stroke: The upper limb training after stroke clinical trial

This article is available open access through the publisher’s website at the link below. Copyright © 2013 American Heart Association, Inc.Background and Purpose — Unilateral and bilateral training protocols for upper limb rehabilitation after stroke represent conceptually contrasting approaches with the same ultimate goal. In a randomized controlled trial, we compared the merits of modified constraint-induced movement therapy, modified bilateral arm training with rhythmic auditory cueing, and a dose-matched conventional treatment. Modified constraint-induced movement therapy and modified bilateral arm training with rhythmic auditory cueing targeted wrist and finger extensors, given their importance for functional recovery. We hypothesized that modified constraint-induced movement therapy and modified bilateral arm training with rhythmic auditory cueing are superior to dose-matched conventional treatment. Methods — Sixty patients, between 1 to 6 months after stroke, were randomized over 3 intervention groups. The primary outcome measure was the Action Research Arm test, which was conducted before, directly after, and 6 weeks after intervention. Results — Although all groups demonstrated significant improvement on the Action Research Arm test after intervention, which persisted at 6 weeks follow-up, no significant differences in change scores on the Action Research Arm test were found between groups postintervention and at follow-up. Conclusions — Modified constraint-induced movement therapy and modified bilateral arm training with rhythmic auditory cueing are not superior to dose-matched conventional treatment or each other in improving upper limb motor function 1 to 6 months after stroke. Clinical Trial Registration — URL: http://www.trialregister.nl. Unique identifier: NTR1665

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

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

Almost maximally almost-periodic group topologies determined by T-sequences

A sequence $\{a_n\}$ in a group $G$ is a {\em $T$-sequence} if there is a Hausdorff group topology $\tau$ on $G$ such that $a_n\stackrel\tau\longrightarrow 0$. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a $T$-sequence, and investigate special sequences in the Pr\"ufer groups $\mathbb{Z}(p^\infty)$. We show that for $p\neq 2$, there is a Hausdorff group topology $\tau$ on $\mathbb{Z}(p^\infty)$ that is determined by a $T$-sequence, which is close to being maximally almost-periodic--in other words, the von Neumann radical $\mathbf{n}(\mathbb{Z}(p^\infty),\tau)$ is a non-trivial finite subgroup. In particular, $\mathbf{n}(\mathbf{n}(\mathbb{Z}(p^\infty),\tau)) \subsetneq \mathbf{n}(\mathbb{Z}(p^\infty),\tau)$. We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a $T$-sequence with non-trivial finite von Neumann radical.Comment: v2 - accepted (discussion on non-abelian case is removed, replaced by new results on direct sums of finite abelian groups

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

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

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