<p>A two-way deterministic (nondeterministic) multi-head finite automaton can be simulated by a logspace DTM (NTM), and vice versa.</p>

<p>So, for including class $ \mathsf{NL} $, you do not need a counter!</p>

<p>The value of the counter belonging to a two-way nondeterministic multi-head finite automaton with one-counter can be bounded by a polynomial, otherwise, the computation enters an infinite loop. Since both such a counter and the input heads can be simulated in logspace, $ \mathsf{LOGCFL} $ upper bound can be replaced with $ \mathsf{NL} $.</p>

<p>Note that any language recognized by a two-way deterministic (nondeterministic) one-head finite automaton with one-counter is in $ \mathsf{L} $ ($ \mathsf{NL} $).</p>

<p>There are many papers on these topics, e.g. <a href="http://arxiv.org/abs/0906.3051">this paper</a>. You can find more after some googling. (Check also Lemma 1 of <a href="http://arxiv.org/abs/1207.3880">this paper</a>.)</p>

CC BY-SA 3.0
2012-09-11T20:56:31.700Z
Thank you, I knew about L (NL) simulation by multihead (nondeterministic) two-way automaton. First I thought that I obtained NL inclusion for deterministic automaton with counter, but it suddenly appeared that in my reduction raized nondeterminism and  I fixed post but not as properly as I should.

I'll study these papers, thank you very much!
Alexander Rubtsov
CC BY-SA 3.0
2012-09-11T20:39:03.573Z
2012-09-11T20:39:03.573Z
Abuzer Yakaryilmaz
<p>Is that known something about languages recognized by two-way deterministic multihead counter automaton or logspace TM with counter (equivalent model)? This class called Aux2DC in my advisor's <a href="http://arxiv.org/pdf/0811.2586.pdf" rel="noreferrer">paper</a>. Or about such nondeterministic class? I've obtained that class of languages recognized by such nondeterministic maschines includes NL and it seems to be included in LOGCFL. Are there any papers on this subject? Is that result trivial?</p>

<cc.complexity-theory><complexity-classes><fl.formal-languages><logspace><counter-automata>
Two way deterministic multihead counter automata or logspace TM with counter
Alexander Rubtsov
# Two way deterministic multihead counter automata or logspace TM with counter

(Asked by: username_0 on 2012-09-11)

Is that known something about languages recognized by two-way deterministic multihead counter automaton or logspace TM with counter (equivalent model)? This class called Aux2DC in my advisor's [paper](http://arxiv.org/pdf/0811.2586.pdf). Or about such nondeterministic class? I've obtained that class of languages recognized by such nondeterministic maschines includes NL and it seems to be included in LOGCFL. Are there any papers on this subject? Is that result trivial?

**Tags:** cc.complexity-theory, complexity-classes, fl.formal-languages, logspace, counter-automata


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## 1 answers

### Answer by username_1 on 2012-09-11. Score: 6 | Accepted answer

A two-way deterministic (nondeterministic) multi-head finite automaton can be simulated by a logspace DTM (NTM), and vice versa.

So, for including class $ \mathsf{NL} $, you do not need a counter!

The value of the counter belonging to a two-way nondeterministic multi-head finite automaton with one-counter can be bounded by a polynomial, otherwise, the computation enters an infinite loop. Since both such a counter and the input heads can be simulated in logspace, $ \mathsf{LOGCFL} $ upper bound can be replaced with $ \mathsf{NL} $.

Note that any language recognized by a two-way deterministic (nondeterministic) one-head finite automaton with one-counter is in $ \mathsf{L} $ ($ \mathsf{NL} $).

There are many papers on these topics, e.g. [this paper](http://arxiv.org/abs/0906.3051). You can find more after some googling. (Check also Lemma 1 of [this paper](http://arxiv.org/abs/1207.3880).)

> **Comments:**
> **username_0**: Thank you, I knew about L (NL) simulation by multihead (nondeterministic) two-way automaton. First I thought that I obtained NL inclusion for deterministic automaton with counter, but it suddenly appeared that in my reduction raized nondeterminism and  I fixed post but not as properly as I should.  I'll study these papers, thank you very much!  

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