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Semantic Similarity: Foundations
Abstract
This paper investigates measures of semantic similarity between conversations from an axiomatic perspective. We abstract away from real conversations, representing them as sequences of formulas, equipped with a notion of semantic interpretation that maps them into a different space. An example we use to illustrate our approach is the language of propositional logic with its classical semantics. We introduce and study a range of different candidate properties for metrics on such conversations, for the structure of the semantic space, and for the behavior of the interpretation function, and their interactions. We define four different metrics and explore their properties in this setting.
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The problem of quantitatively comparing the performance of different broad-coverage grammars of English has to date resisted solution. Prima facie, known English grammars appear to disagree strongly with each other as to the elements of even the simplest sentences. For instance, the grammars of Steve Abney (Bellcore), Ezra Black (IBM), Dan Flickinger (Hewlett Packard), Claudia Gdaniec (Logos), Ralph Grishman and Tomek Strzalkowski (NYU), Phil Harrison (Boeing), Don Hindle (AT&T), Bob Ingria (BBN), and Mitch Marcus (U. of Pennsylvania) recognize in common only the following constituents, when each grammarian provides the single parse which he/she would ideally want his/her grammar to specify for three sample Brown Corpus sentences:The famed Yankee Clipper, now retired, has been assisting (as (a batting coach)).One of those capital-gains ventures, in fact, has saddled him (with Gore Court).He said this constituted a (very serious) misuse (of the (Criminal court) processes).
This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gärdenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourrón and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of its axiomatic bases), that fails to imply the proposition being eliminated.
In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate “partial meet contraction functions”, which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gärdenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are “relational” and “transitively relational”, are studied in detail, and their connections with certain “supplementary postulates” of Gàrdenfors investigated, with a further representation theorem established.
We consider the problem of measuring the similarity or distance between two finite sets of points in a metric space, and computing the measure. This problem has applications in, e.g., computational geometry, philosophy of science, updating or changing theories, and machine learning. We review some of the distance functions proposed in the literature, among them the minimum distance link measure, the surjection measure, and the fair surjection measure, and supply polynomial time algorithms for the computation of these measures. Furthermore, we introduce the minimum link measure, a new distance function which is more appealing than the other distance functions mentioned. We also present a polynomial time algorithm for computing this new measure. We further address the issue of defining a metric on point sets. We present the metric infimum method that constructs a metric from any distance functions on point sets. In particular, the metric infimum of the minimum link measure is a quite int...
A vast and interesting family of natural semantics for belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula ff as the theory defined by the set of all those models of ff that are closest, by d, to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates describe properties of iterated revisions. 1 Introduction 1.1 Overview and related work The aim of this paper is to investigate semantics and logical properties of theory revisions based on an underlying notion of distance between individual models. In many situations it is indeed reasonable to assume that the agent has some natural way to evaluate the distance between any two models of the logical language of interest. The distance between model m and model m 0 is a measure of how far m 0 appears to be from the point of view of m. This distance may me...
Widespread dissatisfaction has been expressed with the measure of parse accuracy used in the Parseval programme, based on the location of constituent boundaries. Scores on the Parseval metric are perceived as poorly correlated with intuitive judgments of goodness of parse; the metric applies only to a restricted range of grammar formalisms; and it is seen as divorced from applications of NLP technology. The present paper defines an alternative metric, which measures the accuracy with which successive words are fitted into parsetrees. (The original statement of this metric is believed to have been the earliest published proposal about quantifying parse accuracy.) The metric defined here gives overall scores that quantify intuitive concepts of good and bad parsing relatively directly, and it gives scores for individual words which enable the location of parsing errors to be pinpointed. It applies to a wider range of grammar formalisms, and is tunable for specific parsing applications.
We review more or less recent results on metrics associated with valuations or (and) minimum weighted paths in posets, metric characterizations of posets and we point out some applications of these results to problems involving uses of distances in social sciences.
Logic and conversation. Studies in the Way of Words
- H P Grice
- HP Grice