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. 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.
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
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, which is commonly known as Nivat's Theorem: 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 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.
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