Logical Reduction of Metarules

International audienceMany forms of inductive logic programming (ILP) use metarules, second-order Horn clauses, to define the structure of learnable programs and thus the hypothesis space. Deciding which metarules to use for a given learning task is a major open problem and is a trade-off between efficiency and expressivity: the hypothesis space grows given more metarules, so we wish to use fewer metarules, but if we use too few metarules then we lose expressivity. In this paper, we study whether fragments of metarules can be logically reduced to minimal finite subsets. We consider two traditional forms of logical reduction: subsumption and entailment. We also consider a new reduction technique called derivation reduction, which is based on SLD-resolution. We compute reduced sets of metarules for fragments relevant to ILP and theoretically show whether these reduced sets are reductions for more general infinite fragments. We experimentally compare learning with reduced sets of metarules on three domains: Michalski trains, string transformations, and game rules. In general, derivation reduced sets of metarules outperform subsumption and entailment reduced sets, both in terms of predictive accuracies and learning times

From simple predicators to clausal functors : The english modals through time and the primitives of modality

The ultimate goal of this paper is to find a representation of modality compatible with some basic conditions on the syntax-semantic interface. Such conditions are anchored, for instance, in Chomsky's (1995) principle of full interpretation (FI). Abstract interpretation of modality is, however - be it "only" in semantic terms - already a hard nut to crack, way too vast to be dealt with in any comprehensive way here. What is pursued instead is a case-study-centered analysis. The case in point are the English modals (EM) viewed in their development through time - a locus classicus for a number of linguistic theories and frameworks. The idea will be to start out from two lines of research - continuous grammaticalization vs. cataclysmic change - and to explain some of their incongruities. The first non-trivial point here consists in deriving more fundamental questions from this research. The second, possibly even less trivial one consists in answering them. Specifically, I will argue that regardless of the actual numerical rate of change, there is an underlying and more structured way to account for the notions of change and continuity within the modal system, respectively

Semantic Criteria of Correct Formalization

This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context

Review of Ulrich Baltzer, "Erkenntnis als Relationengeflecht: Kategorien bei Charles S. Peirce"

This book arose from the author’s recent dissertation written under the Gerhard Schonrich at Munich. It focuses on Peirce’s theory of categories and his epistemology. According to Baltzer, what is distinctive in Peirce’s theory of knowledge is that he reconstrues objects as “knots in networks of relations.” The phrase may ring a bell. It suggests a structuralist interpretation of Peirce, influenced by the Munich environs. The study aims to shows how Peirce’s theory of categories supports his theory of knowledge and how “question concerning a priori structures of knowledge” are transformed within this relational framework. A chief critical target is David Savan’s semiotics, specifically the idea that “the multiplicity of development of the categories” is “conditioned by nothing but the indefiniteness of the categories.”1 But in contrast with this, if there is any indefiniteness in the categories, they cannot fully direct their own application, and this is to say regarding them “that our knowledge is never absolute but always swims, as it were, in a continuum...”2 If the doctrine of continuity applies to the categories, they also have a continuum to swim in

How Can Brains in Vats Experience a Spatial World? A Puzzle for Internalists

In this chapter, Pautz raises a puzzle about spatial experience for phenomenal internalists like Ned Block. If an accidental, lifelong brain-in-the-void (BIV) should have all the same experiences as you, it would have an experience as of items having various shapes, and be able to acquire concepts of those shapes, despite being cut off from real things with the shapes. Internalists cannot explain this by saying that BIV is presented with Peacocke-style visual field regions having various shapes, because these would have to be non-physical sense data. They might instead explain this by saying that BIV “phenomenally represents” various shape properties. But since BIV lacks any interesting physical relations to shapes, this would imply that phenomenally representation is an irreducible relation

Type classes for efficient exact real arithmetic in Coq

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275