A typical example is shown in Figure 7.1. In many browsers and user interfaces (in the year 2004), if you click on one icon and then click on another, the second will become selected and this will cancel the selection of the first, and if you click on more icons, the one most recently selected becomes the unique active icon, overiding previous selections. However, if you hold down the CTRL key and then click on two different icons, both icons will be selected. And if you hold down the SHIFT key and then click, both icons and those in between will be selected. When using these features, previous selections are not discarded --- all the icons selected while holding down the correct key are combined to give a special set of selected items. The processes of combination correspond to disjunctions (OR operators or joins) in different logics.
The 'CTRL + click' option is essentially a discrete approach
which builds a set whose elements are two isolated points. The
'SHIFT + click' option is a more continuous approach, which
uses the geometry of the surrounding region, building a set
consisting of all points contained in the rectangle spanned by the two
icons the user selected.
|Holding down CTRL and clicking on Folder 1 and Folder 6 highlights these two folders only.||Making the same two clicks while pressing SHIFT highlights Folders 2 to 5 as well|
|Figure 7.1: The first version is a Boolean disjunction, the second is a kind of quantum disjunction|
This chapter describes these two brands of logic, using the algebra of George Boole as the classical example of a discrete (and indeed, binary) approach to logic, and the quantum logic of Garrett Birkhoff and John von Neumann as the standard example of a continuous logic. Geometrically, the differences between these approaches arise from the differences between set theory and vector spaces, which form the underlying spatial models for Boolean logic and quantum logic respectively.
Because vector spaces provide the spatial model for both quantum logic and the Infomap WORDSPACE, it turns out that we can use the same logical operators in both. For me, this story began when I implemented a vector negation operator in WORDSPACE using the concept of orthogonality, just as described by Birkhoff and von Neumann in 1936. This precedent was happily brought to my attention by Professor Lawrence Moss of Indiana University, as the research evolved.
Exploring such operations, comparing their effects on document retrieval, and piecing together the scientific context of the WORDSPACE model proved an interesting journey. Many historic branches of logic, geometry, and the physical sciences became woven into the story, and I shall explain how these traditions have been used in search engines, especially in WORDSPACE.
The 'quantum' thread in this chapter stretches from Aristotle's description in the Metaphysics, through geometric implementations by Euclid, Descartes, and Grassmann, the birth of quantum mechanics itself in the early 1900s, and the outward stimulus this has provided to areas of logic, philosophy, computation, and word sense discrimination. The quantum idea, and the spatial extensions with which it is modelled, is a useful tool both for describing the way a particle may be represented by a combination of possible pure states, and for describing the way an ambiguous word may be represented by a combination of available 'pure' meanings.
First, for necessary context and contrast, we explore some of the traditional combination operations in discrete logical models, focussing on the work of George Boole.
About 100 years later, electronic machines were built that could make Boole's algebra a reality. Computers were built that could store these 0s and 1s electronically, and perform logical operations on them exactly as Boole described. These discrete binary operations are called Boolean operators to this day, and you can perform a "Boolean search" in most information retrieval systems, often using the "Advanced Search" feature of an internet search engine. Boolean "logic gates" have become so familiar through electronics and computer science that it's easy to believe that Boolean logic is logic, just as until the 1830's, people believed that Euclidean geometry was geometry. But just as there are differently shaped spaces, there are differently shaped logics.
|Add to query||Subtract from query||Term||Similarity|
|Add to query||Subtract from query||Term||Similarity|
All of this is happening by manipulating vectors in WORDSPACE so that they are perpendicular or orthogonal to one another. Move the vectors so that they are at right angles to one another, and you effectively make the meanings of two word-vectors irrelevant to one another.
|A single particle emitted from E can interfere with itself as though it is a wave that has passed through the slits at both A and B. But if you observe which slit the particle has gone through, this interference doesn't happen!|
In quantum mechanics, a single particle could potentially be in one of several pure states, and until you make an observation of the particle, it doesn't make sense to claim that it is in one state or another (though you can predict that some states are on the whole more likely than others).
A famous experiment that demonstrated this amazing behaviour is the double-slit experiment, which is described briefly in the diagram on the right. It appears that the particle behaves like a wave, one component of which went through A and another through B. Until you observe the particle, all you can predict is that the particle will be represented by some linear combination of pure states, which is expressed as a vector &lambda A + &mu B.
The equations that are used to represent these particles are exactly the same as those used to represent an ambiguous word in WORDSPACE, and the projection equation used to remove an unwanted meaning from a word in Section 2 is exactly the same as the equation used in quantum mechanics to obtain the pure states of a particle from the observed interference patterns.
This enables a user to strip away unwanted meanings in succession and zoom in on a region in WORDSPACE. When tested in document retrieval experiments, this method removed not only the unwanted keywords, but many more of their neighbours in WORDSPACE and their synonyms in WordNet than was achieved by the alternative Boolean operators.
You find this difference between continuous and discrete interpretations of logic in very simple sentences. If somebody says "you should take bus 60 or bus 70", and you later discover that they meant you to take bus 67, you would think you had been very badly advised. If, however, you were told that the journey would take "60 or 70 minutes", and it took 67 minutes, you would agree that you had been advised perfectly wisely. This is just like the difference between 'CTRL + click' and 'SHIFT + click' in the diagram at the beginning of this chapter.
The following paper describes the quantum connectives and their semantic properties in more detail.
The following paper describes the implementation of the quantum negation operators in WORDSPACE and evaluates their statistical performance in document retrieval experiments.
|Up to Geometry and Meaning|||||Back to Chapter 6|||||On to Chapter 8|