Operator Grammar

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Operator Grammar is a mathematical theory of human language that explains how language carries information. This theory is the culmination of the life work of Zellig Harris, with major publications toward the end of the last century. Operator Grammar proposes that each human language is a self-organizing system in which both the syntactic and semantic properties of a word are established purely in relation to other words. Thus, no external system (metalanguage) is required to define the rules of a language. Instead, these rules are learned through exposure to usage and through participation, as is the case with most social behavior. The theory is consistent with the idea that language evolved gradually, with each successive generation introducing new complexity and variation.
Operator Grammar posits three universal constraints: Dependency (certain words depend on the presence of other words to form an utterance), Likelihood (some combinations of words and their dependents are more likely than others) and Reduction (words in high likelihood combinations can be reduced to shorter forms, and sometimes omitted completely). Together these provide a theory of language information: dependency builds a predication structure; likelihood creates distinct meanings; reduction allows compact forms for communication.
Contents
1 Dependency
2 Likelihood
3 Reduction
4 Information
5 Bibliography
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Dependency
The fundamental mechanism of Operator Grammar is the dependency constraint: certain words (operators) require that one or more words (arguments) be present in an utterance. In the sentence John wears boots, the operator wears requires the presence of two arguments, such as John and boots. (This definition of dependency differs from other dependency grammars in which the arguments are said to depend on the operators.)
In each language the dependency relation among words gives rise to syntactic categories in which the allowable arguments of an operator are defined in terms of their dependency requirements. Class N contains words (e.g. John, boots) that do not require the presence of other words. Class ON contains the words (e.g. sleeps) that require exactly one word of type N. Class ONN contains the words (e.g. wears) that require two words of type N. Class OOO contains the words (e.g. because) that require two words of type O, as in John stumbles because John wears boots. Other classes include OO (is possible), ONNN (put), OON (with, surprise), ONO (know), ONNO (ask) and ONOO (attribute).
The categories in Operator Grammar are universal and are defined purely in terms of how words relate to other words, and do not rely on an external set of categories such as noun, verb, adjective, adverb, preposition, conjunction, etc. The dependency properties of each word are observable through usage and therefore learnable.
Likelihood
The dependency constraint creates a structure (syntax) in which any word of the appropriate class can be an argument for a given operator. The likelihood constraint places additional restrictions on this structure by making some operator/argument combinations more likely than others. Thus, John wears hats is more likely than John wears snow which in turn is more likely than John wears vacation. The likelihood constraint creates meaning (semantics) by defining each word in terms of the words it can take as arguments, or of which it can be an argument.
Each word has a unique set of words with which it has been observed to occur called its selection. The coherent selection of a word is the set of words for which the dependency relation has above average likelihood. Words that are similar in meaning have similar coherent selection. This approach to meaning is self-organizing in that no external system is necessary to define what words mean. Instead, the meaning of the word is determined by its usage within a population of speakers. Patterns of frequent use are observable and therefore learnable. New words can be introduced at any time and defined through usage.
Reduction
The reduction constraint acts on high likelihood combinations of operators and arguments and makes more compact forms. Certain reductions allow words to be omitted completely from an utterance. For example, I expect John to come is reducible to I expect John, because to come is highly likely under expect. The sentence John wears boots and John wears hats can be reduced to John wears boots and hats because repetition of the first argument John under the operator and is highly likely. John reads things can be reduced to John reads, because the argument things has high likelihood of occurring under any operator.
Certain reductions reduce words to shorter forms, creating pronouns, suffixes and prefixes (morphology). John wears boots and John wears hats can be reduced to John wears boots and he wears hats, where the...(and so on)

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