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Chomsky hierarchy

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Hierarchy of classes of formal grammars

TheChomsky hierarchy in the fields offormal language theory,computer science, andlinguistics, is acontainment hierarchy of classes offormal grammars. A formal grammar describes how to form strings from a formal language'salphabet that are valid according to the language's syntax. The linguistNoam Chomsky theorized that four different classes of formal grammars existed that could generate increasingly complex languages. Each class can also completely generate the language of all inferior classes (set inclusive).

History

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The general idea of a hierarchy of grammars was first described by Noam Chomsky in "Three models for the description of language" during the formalization oftransformational-generative grammar (TGG).^[1]Marcel-Paul Schützenberger also played a role in the development of the theory offormal languages; the paper "The algebraic theory of context free languages"^[2] describes the modern hierarchy, including context-free grammars.^[3]

Independently, alongside linguists, mathematicians were developing models of computation (viaautomata). Parsing a sentence in a language is similar to computation, and the grammars described by Chomsky proved to both resemble and be equivalent in computational power to various machine models.^[4]

The hierarchy

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The following table summarizes each of Chomsky's four types of grammars, the class of language it generates, the type of automaton that recognizes it, and the form its rules must have. The classes are defined by the constraints on theproductions rules.

Grammar	Languages	Recognizing automaton	Production rules (constraints)^[a]	Examples^[5]^[6]
Type-3	Regular	Finite-state automaton	$A\rightarrow {\text{a}}$ $A\rightarrow {\text{a}}B$ (right regular) or $A\rightarrow {\text{a}}$ $A\rightarrow B{\text{a}}$ (left regular)	$L=\{a^{n}\mid n>0\}$
Type-2	Context-free	Non-deterministicpushdown automaton	$A\rightarrow \alpha$	$L=\{a^{n}b^{n}\mid n>0\}$
Type-1	Context-sensitive	Linear-bounded non-deterministic Turing machine	$\alpha A\beta \rightarrow \alpha \gamma \beta$	$L=\{a^{n}b^{n}c^{n}\mid n>0\}$
Type-0	Recursively enumerable	Turing machine	$\gamma \rightarrow \alpha$ ( $\gamma$ non-empty)	$L=\{w\mid w$ describes a terminating Turing machine $\}$

^
Meaning of symbols:

Note that the set of grammars corresponding torecursive languages is not a member of this hierarchy; these would be properly between Type-0 and Type-1.

Every regular language is context-free, every context-free language is context-sensitive, every context-sensitive language is recursive and every recursive language is recursively enumerable. These are all proper inclusions, meaning that there exist recursively enumerable languages that are not context-sensitive, context-sensitive languages that are not context-free and context-free languages that are not regular.^[7]

Regular (Type-3) grammars

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Main article:Regular grammar

Type-3 grammars generate theregular languages. Such a grammar restricts its rules to a single nonterminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed by a single nonterminal, in which case the grammar isright regular. Alternatively, all the rules can have their right-hand sides consist of a single terminal, possiblypreceded by a single nonterminal (left regular). These generate the same languages. However, if left-regular rules and right-regular rules are combined, the language need no longer be regular. The rule $S\rightarrow \varepsilon$ is also allowed here if $S {\displaystyle S}$ does not appear on the right side of any rule. These languages are exactly all languages that can be decided by afinite-state automaton. Additionally, this family of formal languages can be obtained byregular expressions. Regular languages are commonly used to define search patterns and the lexical structure of programming languages.

For example, the regular language $L=\{a^{n}\mid n>0\}$ is generated by the Type-3 grammar $G=(\{S\},\{a\},P,S)$ with the productions $P {\displaystyle P}$ being the following.

S →aS

S →a

Context-free (Type-2) grammars

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Main article:Context-free grammar

Type-2 grammars generate thecontext-free languages. These are defined by rules of the form $A\rightarrow \alpha$ with $A {\displaystyle A}$ being a nonterminal and $\alpha$ being a string of terminals and/or nonterminals. These languages are exactly all languages that can be recognized by a non-deterministicpushdown automaton. Context-free languages—or rather its subset ofdeterministic context-free languages—are the theoretical basis for the phrase structure of mostprogramming languages, though their syntax also includes context-sensitivename resolution due to declarations andscope. Often a subset of grammars is used to make parsing easier, such as by anLL parser.

For example, the context-free language $L=\{a^{n}b^{n}\mid n>0\}$ is generated by the Type-2 grammar $G=(\{S\},\{a,b\},P,S)$ with the productions $P {\displaystyle P}$ being the following.

S →aSb

S →ab

The language is context-free but not regular (by thepumping lemma for regular languages).

Context-sensitive (Type-1) grammars

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Main article:Context-sensitive grammar

Type-1 grammars generatecontext-sensitive languages. These grammars have rules of the form $\alpha A\beta \rightarrow \alpha \gamma \beta$ with $A {\displaystyle A}$ a nonterminal and $\alpha$ , $\beta$ and $\gamma$ strings of terminals and/or nonterminals. The strings $\alpha$ and $\beta$ may be empty, but $\gamma$ must be nonempty. The rule $S\rightarrow \epsilon$ is allowed if $S {\displaystyle S}$ does not appear on the right side of any rule. The languages described by these grammars are exactly all languages that can be recognized by alinear bounded automaton (a nondeterministic Turing machine whose tape is bounded by a constant times the length of the input.)

For example, the context-sensitive language $L=\{a^{n}b^{n}c^{n}\mid n>0\}$ is generated by the Type-1 grammar $G=(\{S,A,B,C,W,Z\},\{a,b,c\},P,S)$ with the productions $P {\displaystyle P}$ being the following.

S →aBC

S →aSBC

CB →CZ

CZ →WZ

WZ →WC

WC →BC

aB →ab

bB →bb

bC →bc

cC →cc

The language is context-sensitive but not context-free (by thepumping lemma for context-free languages).A proof that this grammar generates $L=\{a^{n}b^{n}c^{n}\mid n>0\}$ is sketched in the article onContext-sensitive grammars.

Recursively enumerable (Type-0) grammars

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Main article:Unrestricted grammar

Type-0 grammars include all formal grammars. There are no constraints on the productions rules. They generate exactly all languages that can be recognized by aTuring machine, thus any language that is possible to be generated can be generated by a Type-0 grammar.^[8] These languages are also known as therecursively enumerable orTuring-recognizable languages.^[8] Note that this is different from therecursive languages, which can bedecided by analways-halting Turing machine.

Citations

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^Chomsky 1956.
^Chomsky & Schützenberger 1963.
^Allott, Nicholas; Lohndal, Terje; Rey, Georges (27 April 2021). "Synoptic Introduction".A Companion to Chomsky. pp. 1–17.doi:10.1002/9781119598732.ch1.ISBN 9781119598701.S2CID 241301126.
^Kozen, Dexter C. (2007).Automata and computability. Undergraduate Texts in Computer Science. Springer. pp. 3–4.ISBN 978-0-387-94907-9.
^Geuvers, H.; Rot, J. (2016)."Applications, Chomsky hierarchy, and Recap"(PDF).Regular Languages.Archived(PDF) from the original on 2018-11-19.
^Sudkamp, Thomas A. (1997) [1988].Languages and machines: An Introduction to the Theory of Computer Science. Reading, Massachusetts, USA: Addison Wesley Longman. p. 310.ISBN 978-0-201-82136-9.
^Chomsky, Noam (1963). "Chapter 12: Formal Properties of Grammars". In Luce, R. Duncan; Bush, Robert R.; Galanter, Eugene (eds.).Handbook of Mathematical Psychology. Vol. II. John Wiley and Sons, Inc. pp. 323–418.
^^a ^bSipser, Michael (1997).Introduction to the Theory of Computation (1st ed.). Cengage Learning. p. 130.ISBN 0-534-94728-X.The Church-Turing Thesis

References

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Chomsky, Noam (1956)."Three models for the description of language"(PDF).IRE Transactions on Information Theory.2 (3):113–124.doi:10.1109/TIT.1956.1056813.S2CID 19519474.Archived(PDF) from the original on 2016-03-07.
Chomsky, Noam (1959)."On certain formal properties of grammars"(PDF).Information and Control.2 (2):137–167.doi:10.1016/S0019-9958(59)90362-6.
Chomsky, Noam; Schützenberger, Marcel P. (1963). "The algebraic theory of context free languages". In Braffort, P.; Hirschberg, D. (eds.).Computer Programming and Formal Systems(PDF). Amsterdam: North Holland. pp. 118–161.Archived(PDF) from the original on 2011-06-13.

Automata theory:formal languages andformal grammars

Chomsky hierarchy	Grammars	Languages	Abstract machines
Type-0 — Type-1 — — — — — Type-2 — — Type-3 — —	Unrestricted (no common name) Context-sensitive Positiverange concatenation Indexed — Linear context-free rewriting systems Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular — Non-recursive	Recursively enumerable Decidable Context-sensitive Positiverange concatenation^* Indexed^* — Linear context-free rewriting language Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular Star-free Finite	Turing machine Decider Linear-bounded PTIME Turing Machine Nested stack Thread automaton restrictedTree stack automaton Embedded pushdown Nondeterministic pushdown Deterministic pushdown Visibly pushdown Finite Counter-free (with aperiodic finite monoid) Acyclic finite

Each category of languages, except those marked by a^*, is aproper subset of the category directly above it.Any language in each category is generated by a grammar and by an automaton in the category in the same line.

Noam Chomsky

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bibliography

Linguistics	Syntactic Structures (1957) Current Issues in Linguistic Theory (1964) Aspects of the Theory of Syntax (1965) Cartesian Linguistics (1966) Language and Mind (1968) The Sound Pattern of English (1968) "Remarks on Nominalization" (1970) "Conditions on Transformations" (1973) Reflections on Language (1975) The Logical Structure of Linguistic Theory (1975) Lectures on Government and Binding (1981) Knowledge of Language (1986) The Minimalist Program (1995) New Horizons in the Study of Language and Mind (2000)
Politics	The Responsibility of Intellectuals (1967) American Power and the New Mandarins (1969) For Reasons of State (1973) Counter-Revolutionary Violence (1973), withEdward S. Herman The Fateful Triangle (1983) Manufacturing Consent (1988), withEdward S. Herman Necessary Illusions (1989) Deterring Democracy (1991) Letters from Lexington (1993) The Prosperous Few and the Restless Many (1993) World Orders Old and New (1994) Objectivity and Liberal Scholarship (1997) Hegemony or Survival (2003) Failed States (2006) How the World Works (2011) Requiem for the American Dream (2017)
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