Pattern theory

Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language.

In addition to the new algebraic vocabulary, its statistical approach is novel in its aim to:

Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and global entropic properties.

The Brown University Pattern Theory Group was formed in 1972 by Ulf Grenander. Many mathematicians are currently working in this group, noteworthy among them being the Fields Medalist David Mumford. Mumford regards Grenander as his "guru" in this subject.

We begin with an example to motivate the algebraic definitions that follow.

If we want to represent language patterns, the most immediate candidate for primitives might be words. However, set phrases, such as “in order to”, immediately indicate the inappropriateness of words as atoms. In searching for other primitives, we might try the rules of grammar. We can represent grammars as finite state automata or context-free grammars. Below is a sample finite state grammar automaton.

The following phrases are generated from a few simple rules of the automaton and programming code in pattern theory:

To create such sentences, rewriting rules in finite state automata act as generators to create the sentences as follows: if a machine starts in state 1, it goes to state 2 and writes the word “the”. From state 2, it writes one of 4 words: prince, boy, princess, girl, chosen at random. The probability of choosing any given word is given by the Markov chain corresponding to the automaton. Such a simplistic automaton occasionally generates more awkward sentences

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