Operator Grammar
Encyclopedia
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 predicate-argument structure
; likelihood creates distinct meanings; reduction allows compact forms for communication.
) 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 grammar
s 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.
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.
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 pronoun he is a reduced form of John. Suffixes and prefixes can be obtained by appending other freely occurring words, or variants of these. John is able to be liked can be reduced to John is likeable. John is thoughtful is reduced from John is full of thought, and John is anti-war from John is against war.
Modifiers are the result of several of these kinds of reductions, which give rise to adjectives, adverbs, prepositional phrases, subordinate clauses, etc.
Each language has a unique set of reductions. For example, some languages have morphology and some don’t; some transpose short modifiers and some do not. Each word in a language participates only in certain kinds of reductions. However, in each case, the reduced material can be reconstructed from knowledge of what is likely in the given operator/argument combination. The reductions in which each word participates are observable and therefore learnable, just as one learns a word’s dependency and likelihood properties.
s, since they do not remove any information, just make sentences more compact. Thus, the base sentences contain all the information of the language and the reduced sentences are variants of these. Base sentences are made up of simple words without modifiers and largely without affixes, e.g. Snow falls, Sheep eat grass, John knows sheep eat grass, That sheep eat snow surprises John.
Each operator in a sentence makes a contribution in information according to its likelihood of occurrence with its arguments. Highly expected combinations have low information; rare combinations have high information. The precise contribution of an operator is determined by its selection, the set of words with which it occurs with high frequency. The arguments boots, hats, sheep, grass and snow differ in meaning according to the operators for which they can appear with high likelihood in first or second argument position. For example, snow is expected as first argument of fall but not of eat, while the reverse is true of sheep. Similarly, the operators eat, devour, chew and swallow differ in meaning to the extent that the arguments they select and the operators that select them differ.
Operator Grammar predicts that the information carried by a sentence is the accumulation of contributions of each argument and operator. The increment of information that a given word adds to a new sentences is determined by how it was used before. In turn, new usages stretch or even alter the information content associated with a word. Because this process is based on high frequency usage, the meanings of words are relatively stable over time, but can change in accordance with the needs of a linguistic community.
Language
Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...
that explains how language carries information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
. This theory is the culmination of the life work of Zellig Harris
Zellig Harris
Zellig Sabbettai Harris was a renowned American linguist, mathematical syntactician, and methodologist of science. Originally a Semiticist, he is best known for his work in structural linguistics and discourse analysis and for the discovery of transformational structure in language...
, 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
Syntax
In linguistics, syntax is the study of the principles and rules for constructing phrases and sentences in natural languages....
and semantic
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
properties of a word are established purely in relation to other words. Thus, no external system (metalanguage
Metalanguage
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language...
) 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
Origin of language
The origin of language is the emergence of language in the human species. This is a highly controversial topic. Empirical evidence is so limited that many regard it as unsuitable for serious scholars. In 1866, the Linguistic Society of Paris went so far as to ban debates on the subject...
gradually, with each successive generation introducing new complexity and variation.
Operator Grammar posits three universal
Linguistic universal
A linguistic universal is a pattern that occurs systematically across natural languages, potentially true for all of them. For example, All languages have nouns and verbs, or If a language is spoken, it has consonants and vowels. Research in this area of linguistics is closely tied to the study of...
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 predicate-argument structure
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
; likelihood creates distinct meanings; reduction allows compact forms for communication.
Dependency
The fundamental mechanism of Operator Grammar is the dependency constraint: certain words (operatorsOperator (linguistics)
In linguistics, an operator is a special variety of determiner including the visible interrogatives, the quantifiers, and the hypothetical invisible pronoun denoted Op...
) 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 grammar
Dependency grammar
Dependency grammar is a class of modern syntactic theories that are all based on the dependency relation and that can be traced back primarily to the work of Lucien Tesnière. Dependency grammars are distinct from phrase structure grammars , since they lack phrasal nodes. Structure is determined by...
s 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
Linguistic universal
A linguistic universal is a pattern that occurs systematically across natural languages, potentially true for all of them. For example, All languages have nouns and verbs, or If a language is spoken, it has consonants and vowels. Research in this area of linguistics is closely tied to the study of...
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
Morphology (linguistics)
In linguistics, morphology is the identification, analysis and description, in a language, of the structure of morphemes and other linguistic units, such as words, affixes, parts of speech, intonation/stress, or implied context...
). John wears boots and John wears hats can be reduced to John wears boots and he wears hats, where the pronoun he is a reduced form of John. Suffixes and prefixes can be obtained by appending other freely occurring words, or variants of these. John is able to be liked can be reduced to John is likeable. John is thoughtful is reduced from John is full of thought, and John is anti-war from John is against war.
Modifiers are the result of several of these kinds of reductions, which give rise to adjectives, adverbs, prepositional phrases, subordinate clauses, etc.
- John wears boots; the boots are of leather (two sentences joined by semicolonSemicolonThe semicolon is a punctuation mark with several uses. The Italian printer Aldus Manutius the Elder established the practice of using the semicolon to separate words of opposed meaning and to indicate interdependent statements. "The first printed semicolon was the work of ... Aldus Manutius"...
operator) → - John wears boots which are of leather (reduction of repeated noun to relative pronounRelative pronounA relative pronoun is a pronoun that marks a relative clause within a larger sentence. It is called a relative pronoun because it relates the relative clause to the noun that it modifies. In English, the relative pronouns are: who, whom, whose, whosever, whosesoever, which, and, in some...
) → - John wears boots of leather (omission of high likelihood phrase which are) →
- John wears leather boots (omission of high likelihood operator of, transposition of short modifier to left of noun)
Each language has a unique set of reductions. For example, some languages have morphology and some don’t; some transpose short modifiers and some do not. Each word in a language participates only in certain kinds of reductions. However, in each case, the reduced material can be reconstructed from knowledge of what is likely in the given operator/argument combination. The reductions in which each word participates are observable and therefore learnable, just as one learns a word’s dependency and likelihood properties.
Information
The importance of reductions in Operator Grammar is that they separate sentences that contain reduced forms from those that don’t (base sentences). All reductions are paraphraseParaphrase
Paraphrase is restatement of a text or passages, using other words. The term "paraphrase" derives via the Latin "paraphrasis" from the Greek , meaning "additional manner of expression". The act of paraphrasing is also called "paraphrasis."...
s, since they do not remove any information, just make sentences more compact. Thus, the base sentences contain all the information of the language and the reduced sentences are variants of these. Base sentences are made up of simple words without modifiers and largely without affixes, e.g. Snow falls, Sheep eat grass, John knows sheep eat grass, That sheep eat snow surprises John.
Each operator in a sentence makes a contribution in information according to its likelihood of occurrence with its arguments. Highly expected combinations have low information; rare combinations have high information. The precise contribution of an operator is determined by its selection, the set of words with which it occurs with high frequency. The arguments boots, hats, sheep, grass and snow differ in meaning according to the operators for which they can appear with high likelihood in first or second argument position. For example, snow is expected as first argument of fall but not of eat, while the reverse is true of sheep. Similarly, the operators eat, devour, chew and swallow differ in meaning to the extent that the arguments they select and the operators that select them differ.
Operator Grammar predicts that the information carried by a sentence is the accumulation of contributions of each argument and operator. The increment of information that a given word adds to a new sentences is determined by how it was used before. In turn, new usages stretch or even alter the information content associated with a word. Because this process is based on high frequency usage, the meanings of words are relatively stable over time, but can change in accordance with the needs of a linguistic community.