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ω-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 infinite ordinal number, modeling a 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 → 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 \mathbb 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Omega-regular Language
In computer science and formal language theory, the ω-regular languages are a class of ω-languages that generalize the definition of regular languages to infinite words. As regular languages accept finite strings (such as strings beginning in an ''a'', or strings alternating between ''a'' and ''b''), ω-regular languages accept infinite words (such as, infinite sequences beginning in an ''a'', or infinite sequences alternating between ''a'' and ''b''). Formal definition An ω-language ''L'' is ω-regular if it has the form * ''A''ω where ''A'' is a regular language not containing the empty string * ''AB'', the concatenation of a regular language ''A'' and an ω-regular language ''B'' (Note that ''BA'' is ''not'' well-defined) * ''A'' ∪ ''B'' where ''A'' and ''B'' are ω-regular languages (this rule can only be applied finitely many times) The elements of ''A''ω are obtained by concatenating words from ''A'' infinitely many times. Note that if ''A'' is regular, ''A''ω is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |