Omega language

In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an ω-length sequence) of symbols, and an ω-language is a set of infinite words. Here, ω refers to the first ordinal number, the set of natural numbers. Let Σ be a set of symbols (not necessarily finite). Following the standard definition from formal language theory, Σ* is the set of all finite words over Σ. Every finite word has a length, which is a natural number. Given a word w of length n, w can be viewed as a function from the set {0,1,...,n−1} → Σ, with the value at i giving the symbol at position i. The infinite words, or ω-words, can likewise be viewed as functions from to Σ. The set of all infinite words over Σ is denoted Σω. The set of all finite and infinite words over Σ is sometimes written Σ∞ or Σ≤ω. Thus an ω-language L over Σ is a subset of Σω. Some common operations defined on ω-languages are: Intersection and union Given ω-languages L and M, both L ∪ M and L ∩ M are ω-languages. Left concatenation Let L be an ω-language, and K be a language of finite words only. Then K can be concatenated on the left, and only on the left, to L to yield the new ω-language KL. Omega (infinite iteration) As the notation hints, the operation is the infinite version of the Kleene star operator on finite-length languages. Given a formal language L, Lω is the ω-language of all infinite sequences of words from L; in the functional view, of all functions . Prefixes Let w be an ω-word. Then the formal language Pref(w) contains every finite prefix of w. Limit Given a finite-length language L, an ω-word w is in the limit of L if and only if Pref(w) ∩ L is an infinite set. In other words, for an arbitrarily large natural number n, it is always possible to choose some word in L, whose length is greater than n, and which is a prefix of w. The limit operation on L can be written Lδ or . The set Σω can be made into a metric space by definition of the metric as: where |x| is interpreted as "the length of x" (number of symbols in x), and inf is the infimum over sets of real numbers.