Counter automaton

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Diagram of a counter automaton. Each cell of its stack either contains an "A" or a space symbol.

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


The class of counter automata can recognize a proper superset of the regular[note 1] and a subset of the deterministic context free languages.[2]: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


  1. ^ 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.
  2. ^ by the pumping lemma for regular languages


  1. ^ 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. 
  2. ^ a b 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.