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In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an
Formal definition
editLet
Thus an
Operations
editSome common operations defined on
- Intersection and union
- Given
ω -languages L and M, both L ∪ M and L ∩ M areω -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 newω -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, L
ω is theω -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 .
Distance between ω -words
edit
The set
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
editThe most widely used subclass of the
If the language
Bibliography
edit- 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.