Английская Википедия: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 <math>\mathcal{S}</math> of languages such that <math>\mathcal{S}</math> contains at least one non-empty language, and for any <math>L \in \mathcal{S}</math> over some alphabet <math>\Sigma</math>,
- if <math>h</math> is a homomorphism from <math>\Sigma^\ast</math> to some <math>\Delta^\ast</math>, the language <math>h(L)</math> is in <math>\mathcal{S}</math>;
- if <math>h</math> is a homomorphism from some <math>\Delta^\ast</math> to <math>\Sigma^\ast</math>, the language <math>h^{-1}(L)</math> is in <math>\mathcal{S}</math>;
- if <math>R</math> is any regular language over <math>\Sigma</math>, then <math>L\cap R</math> is in <math>\mathcal{S}</math>.
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 <math>\lambda</math> 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 <math>T</math>, mapping a language <math>L</math> over the input alphabet into another language <math>T(L)</math> 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 <math>T</math> 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 <math>T</math> can be effectively decomposed as <math>T(L) = g(h^{-1}(L) \cap R)</math>, where <math>g, h</math> are homomorphisms, and <math>R</math> is a regular language depending only on <math>T</math>.
Altogether, this means that a family of languages is a cone if and only 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 <math>\{a,b\}</math> that removes every second <math>b</math> 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.
See also
Notes
References
- Шаблон:Cite journal
- Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, Шаблон:ISBN.
- John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley Publishing, Reading Massachusetts, 1979. Шаблон:ISBN. Chapter 11: Closure properties of families of languages.
- Шаблон:Cite book
External links
- Encyclopedia of mathematics: Trio, Springer.
