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 whole...as 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.

POSTSCRIPT - May 21st, 2010

Came up with another AHA this AM that I later found in a French university's academic journal. I had the bright idea of "fuzzy frames" and sure enough later found "A theory of Fuzzy Frames" here... http://www.listic.univ-savoie.org/modules.php?name=Busefal&func=AfficherVolume&no_volume=31 I am not sure it is exactly the same since I was thinking of applying neural net type "learning" (of the sort used to recognize handwriting) on semantic network relations. The inspiration was the problem of how to handle all the variations on a "restaurant scenario" frame (see original Schank book).

Different orders of Polymorphism

When most computer science texts refer to polymorphism (in the context of object oriented programming) they are referring to either refining the implementation of a method via subclasses overriding that method, OR, by defining more than one behavior (method) with the same name. The former case (hopefully) preserves the semantics of the method name, the latter overloads multiple semantics on the same name. In either case, it is only the method that is multifaceted, not the entire object much less the class hierarchy.

With Existential Programming, a different order of polymorphism (i.e. "a single entity can take on multiple forms") is at work. Here a single entity in its entirety can take on multiple forms. Heck, it can take on multiple ontologies (aka class hierarchies). This means either supporting multiple (albeit, fundamentally similar) views/conceptions of an entity, OR, trying to knit together very different models where the object instance is in some sort of oscillating Schrodinger's Cat-like state (except that we can look at it and not affect its state :-).

Existential Programming as Quantum States

In reading about Quantum States in Wikipedia...

"In quantum physics, a quantum state is a mathematical object that fully describes a Quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be statistically mixed, corresponding to an experiment involving a random change of the parameters. States obtained in this way are called mixed states, as opposed to pure states, which cannot be described as a mixture of others. When performing a certain measurement on a quantum state, the result generally described by a probability distribution, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. However, unlike in classical mechanics, the result of a measurement on even a pure quantum state is only determined probabilistically. This reflects a core difference between classical and quantum physics.

Mathematically, a pure quantum state is typically represented by a vector in a Hilbert space. In physics, bra-ket notation is often used to denote such vectors. Linear combinations (superpositions) of vectors can describe interference phenomena. Mixed quantum states are described by density matrices."

...I was struck by the analogy with Existential Programming which proposes that objects hold multiple values for various properties (and in fact multiple sets of properties, hence, multiple ontologies) simultaneously.

Unlike Quantum States however, reading one set of values doesn't make the other sets vanish! ;-)