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==Properties== |
==Properties== |
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The class of counter automata can recognize a proper superset of the [[regular language|regular]]{{#tag:ref|Every regular language ''L'' is accepted by some [[finite automaton]] ''F'' (see [[Regular language#Equivalent formalisms]]). Enriching ''F'' with a two-symbol stack which is ignored by ''F''’s control makes it a counter automaton accepting ''L''.|group=note}} and a |
The class of counter automata can recognize a proper superset of the [[regular language|regular]]{{#tag:ref|Every regular language ''L'' is accepted by some [[finite automaton]] ''F'' (see [[Regular language#Equivalent formalisms]]). Enriching ''F'' with a two-symbol stack which is ignored by ''F''’s control makes it a counter automaton accepting ''L''.|group=note}} and a subset of the [[deterministic context free language]]s.<ref name="Hopcroft.Motwani.Ullman.2003"/>{{rp|352}} |
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For example, the language <math>\{ a^nb^n : n \in \mathbb{N} \}</math> is a non-regular<ref group=note>by the [[pumping lemma for regular languages#Use of the lemma|pumping lemma for regular languages]]</ref> language accepted by a counter automaton: It can use the symbol <math>A</math> to count the number of <math>a</math>s in a given input string <math>x</math> (writing an <math>A</math> for each <math>a</math> in <math>x</math>), after that, it can delete an <math>A</math> for each <math>b</math> in <math>x</math>. |
For example, the language <math>\{ a^nb^n : n \in \mathbb{N} \}</math> is a non-regular<ref group=note>by the [[pumping lemma for regular languages#Use of the lemma|pumping lemma for regular languages]]</ref> language accepted by a counter automaton: It can use the symbol <math>A</math> to count the number of <math>a</math>s in a given input string <math>x</math> (writing an <math>A</math> for each <math>a</math> in <math>x</math>), after that, it can delete an <math>A</math> for each <math>b</math> in <math>x</math>. |
Revision as of 09:58, 20 January 2017
In computer science, more particular in the theory of formal languages, a counter automaton, or counter machine, is a pushdown automaton with only two symbols, and the initial symbol in , the finite set of stack symbols.[1]: 171
Equivalently, a counter automaton is a nondeterministic finite automaton with an additional memory cell that can hold one nonnegative integer number (of unlimited size), which can be incremented, decremented, and tested for being zero.[2]: 351
Properties
The class of counter automata can recognize a proper superset of the regular[note 1] and a subset of the deterministic context free languages.[3]: 352
For example, the language is a non-regular[note 2] language accepted by a counter automaton: It can use the symbol to count the number of s in a given input string (writing an for each in ), after that, it can delete an for each in .
A two-counter automaton, that is, a two-stack Turing machine with a two-symbol alphabet, can simulate an arbitrary Turing machine.[1]: 172
Notes
- ^ Every regular language L is accepted by some finite automaton F (see Regular language#Equivalent formalisms). Enriching F with a two-symbol stack which is ignored by F’s control makes it a counter automaton accepting L.
- ^ by the pumping lemma for regular languages
References
- ^ a b John E. Hopcroft and Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Reading/MA: Addison-Wesley. ISBN 0-201-02988-X.
- ^ John E. Hopcroft and Rajeev Motwani and Jeffrey D. Ullman (2003). Introduction to Automata Theory, Languages, and Computation. Upper Saddle River/NJ: Addison Wesley. ISBN 0-201-44124-1.
- ^ Cite error: The named reference
Hopcroft.Motwani.Ullman.2003
was invoked but never defined (see the help page).