Sunday, September 9, 2007

Quantum Math for Fuzzy Ontologies

In my earlier post "Existential Programming as Quantum States", I mused that objects that were simultaneously carrying properties from multiple ontologies (i.e. multiple class hierarchies or data models), were like Quantum States in Quantum Physics.  This led me later to wonder what math had been developed to work with quantum states...i.e. is there some sort of quantum algebra that might be applicable to Existential Programming? It is needed because, in Existential Programming, a property of an object might carry multiple conflicting values simultaneously, each with varying degrees of certainty or confidence or error margins.

I found Wikipedia page on Quantum-indeterminacy which looks applicable.
Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes of measurements of an observable. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution.
Indeterminacy in measurement was not an innovation of quantum mechanics, since it had been established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the later half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.
AHA! It dawns on me that going beyond the mere fuzzy logic idea of values having a probability or certainty factor, Existential Programming could have a fuzziness value for the property as a in "it is not certain that this property even applies to this object"...and even further it could mean "it is not certain that this property even applies to the entire Class".  A FUZZY ONTOLOGY: method of associating attributes/relationships with entities where each entity is not conclusively known. The value of a property may be certain (i.e. not vague or probabilistic), but whether that property belongs to this object is fuzzy.

Why would you want that ability?  How about data mining web pages where several people's names and a single birth-date (or phone number, address, etc) are found.  Even though it isn't known which person's name is associated with the birthday, one could associate the birth-date with each person with some fractional probability.  With enough out of focus wisps of data like this, from many web pages, the confidence factor of the right birthdate with the right person would rise to the top of the list of all possible dates (analogous to the way that very long range telescopes must accumulate lots of individual, seemingly random, photons to build up a picture of the stars/galaxies being imaged).  The fractional probability assigned could be calculated with heuristics like "lexical-distance-between-age-and-name is proportional to the probability assigned". This could make the "value" of a scalar property (like birth-date), in reality, the summarization of a complete histogram of values-by-source-web-pages.

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