Cone (formal languages)

In formal language theory, a cone is a set of formal languages that has some desirable closure properties enjoyed by some well-known sets of languages, in particular by the families of regular languages, context-free languages and the recursively enumerable languages.^[1] The concept of a cone is a more abstract notion that subsumes all of these families. A similar notion is the faithful cone, having somewhat relaxed conditions. For example, the context-sensitive languages do not form a cone, but still have the required properties to form a faithful cone.

The terminology cone has a French origin. In the American oriented literature one usually speaks of a full trio. The trio corresponds to the faithful cone.

Definition

A cone is a family ${\mathcal {S}}$ of languages such that ${\mathcal {S}}$ contains at least one non-empty language, and for any $L\in {\mathcal {S}}$ over some alphabet $\Sigma$ ,

if $h$ is a homomorphism from $\Sigma ^{\ast }$ to some $\Delta ^{\ast }$ , the language $h(L)$ is in ${\mathcal {S}}$ ;
if $h$ is a homomorphism from some $\Delta ^{\ast }$ to $\Sigma ^{\ast }$ , the language $h^{{-1}}(L)$ is in ${\mathcal {S}}$ ;
if $R$ is any regular language over $\Sigma$ , then $L\cap R$ is in ${\mathcal {S}}$ .

The family of all regular languages is contained in any cone.

If one restricts the definition to homomorphisms that do not introduce the empty word $\lambda$ then one speaks of a faithful cone; the inverse homomorphisms are not restricted. Within the Chomsky hierarchy, the regular languages, the context-free languages, and the recursively enumerable languages are all cones, whereas the context-sensitive languages and the recursive languages are only faithful cones.

Relation to Transducers

A finite state transducer is a finite state automaton that has both input and output. It defines a transduction $T$ , mapping a language $L$ over the input alphabet into another language $T(L)$ over the output alphabet. Each of the cone operations (homomorphism, inverse homomorphism, intersection with a regular language) can be implemented using a finite state transducer. And, since finite state transducers are closed under composition, every sequence of cone operations can be performed by a finite state transducer.

Conversely, every finite state transduction $T$ can be decomposed into cone operations. In fact, there exists a normal form for this decomposition,^[2] which is commonly known as Nivat's Theorem:^[3] Namely, each such $T$ can be effectively decomposed as $T(L)=g(h^{{-1}}(L)\cap R)$ , where $g,h$ are homomorphisms, and $R$ is a regular language depending only on $T$ .

Altogether, this means that a family of languages is a cone if it is closed under finite state transductions. This is a very powerful set of operations. For instance one easily writes a (nondeterministic) finite state transducer with alphabet $\{a,b\}$ that removes every second $b$ in words of even length (and does not change words otherwise). Since the context-free languages form a cone, they are closed under this exotic operation.

Notes

References

Ginsburg, Seymour; Greibach, Sheila (1967). "Abstract Families of Languages". Conference Record of 1967 Eighth Annual Symposium on Switching and Automata Theory, 18–20 October 1967, Austin, Texas, USA. IEEE. pp. 128–139.

Nivat, Maurice (1968). "Transductions des langages de Chomsky". Annales de l'Institut Fourier. 18 (1): 339–455. doi:10.5802/aif.287.
Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, ISBN 0-7204-2506-9.
John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X. Chapter 11: Closure properties of families of languages.
Mateescu, Alexandru; Salomaa, Arto (1997). "Chapter 4: Aspects of Classical Language Theory". In Rozenberg, Grzegorz; Salomaa, Arto. Handbook of Formal Languages. Volume I: Word, language, grammar. Springer-Verlag. pp. 175–252. ISBN 3-540-61486-9.

External links

Encyclopedia of mathematics: Trio, Springer.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.