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.

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, which is commonly known as Nivat's Theorem: 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 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 $$\{a,b\}$$ 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.