Cone (formal languages)

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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[edit]

A cone is a non-empty family \mathcal{S} of languages such that, 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[edit]

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.

See also[edit]

Notes[edit]

References[edit]

  • 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. 
  • 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. 

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