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.

Formal definition

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.

Operations

Some common operations defined on ω-languages are:

 Intersection and union  Given ω-languages L and M, both  and  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  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 .

Distance between ω-words

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. If  then there is no longest prefix x and so . Symmetry is clear. Transitivity follows from the fact that if w and v have a maximal shared prefix of length m and v and u have a maximal shared prefix of length n then the first  characters of w and u must be the same so . Hence d is a metric.

Important subclasses
The most widely used subclass of the ω-languages is the set of ω-regular languages, which enjoy the useful property of being recognizable by Büchi automata. Thus the decision problem of ω-regular language membership is decidable using a Büchi automaton, and fairly straightforward to compute.

If the language Σ is the power set of a set (called the "atomic propositions") then the ω-language is a linear time property, which are studied in model checking.

Bibliography
 Perrin, D. and Pin, J.-E. "Infinite Words: Automata, Semigroups, Logic and Games". Pure and Applied Mathematics Vol 141, Elsevier, 2004.
 Staiger, L. "ω-Languages". In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, Volume 3, pages 339-387. Springer-Verlag, Berlin, 1997.
 Thomas, W. "Automata on Infinite Objects". In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, Volume B: Formal Models and Semantics, pages 133-192. Elsevier Science Publishers, Amsterdam, 1990.

Theory of computation
Formal languages