Recognizable languages Büchi automaton

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a non-deterministic büchi automaton recognizes (0∪1)*0


the class of deterministic büchi automata not suffice encompass omega-regular languages. in particular, there no deterministic büchi automaton recognizes language (0∪1)*0, contains words in 1 occurs finitely many times. can demonstrate contradiction no such deterministic büchi automaton exists. let suppose deterministic büchi automaton recognize (0∪1)*0 final state set f. accepts 0. so, visit state in f after reading finite prefix of 0, after i0th letter. accepts ω-word 010. therefore, i1, after prefix 010 automaton visit state in f. continuing construction ω-word 01010... generated causes visit state in f infinitely , word not in (0∪1)*0. contradiction.


the class of languages recognizable deterministic büchi automata characterized following lemma.



lemma: ω-language recognizable deterministic büchi automaton iff limit language of regular language.
proof: deterministic büchi automaton can viewed deterministic finite automaton , vice versa, since both types of automaton defined 5-tuple of same components, interpretation of acceptance condition different. show l(a) limit language of l(a ). ω-word accepted iff force visit final states infinitely often. iff, infinitely many finite prefixes of ω-word accepted . hence, l(a) limit language of l(a ).






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