Local language (formal language)

A local language (in the context of formal languages) refers to a formal language defined by a set of constraints that apply only to adjacent symbols or substrings within the language's strings. In other words, a language is local if membership in the language can be determined by checking only the local neighborhoods (usually pairs or triples) of symbols within a string.

The key characteristic of a local language is that it doesn't require a global view of the entire string to determine validity. Instead, the "rules" or restrictions governing the language only apply to immediate neighbors.

Local languages are generally a subset of regular languages, and can often be described by finite state automata or regular expressions, although not all regular languages are local. The precise formal definition often involves specifying a set of allowed (or forbidden) substrings of a certain length. If a string is composed only of allowed substrings, then it is in the local language. Conversely, if a string contains any forbidden substring, it is not in the local language.

The concept of locality makes these languages relatively simple to analyze and process. Their properties are particularly useful in areas such as compiler design and formal verification, where efficient validation of syntax is crucial.

While the term "local language" might sometimes be used colloquially to refer to a language spoken in a specific geographic region, in the realm of formal language theory, it carries this precise, technical meaning related to the constraints on symbol adjacency.