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

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A cone is a family   of languages such that   contains at least one non-empty language, and for any   over some alphabet  ,

  • if   is a homomorphism from   to some  , the language   is in  ;
  • if   is a homomorphism from some   to  , the language   is in  ;
  • if   is any regular language over  , then   is in  .

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   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

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A finite state transducer is a finite state automaton that has both input and output. It defines a transduction  , mapping a language   over the input alphabet into another language   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   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   can be effectively decomposed as  , where   are homomorphisms, and   is a regular language depending only on  .

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   that removes every second   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

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Notes

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References

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  • 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.
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