It is well known that a machine with a single stack as only unlimited storage is not Turing complete, if it can only read from the top of the stack. I want a machine which is (slightly) more powerful than a stack machine, but still not Turing complete. (I wonder whether there exists a non-Turing complete machine, which can deterministically simulate any non-deterministic pushdown automata with an only polynomial slow-down.) The most benign (straightforward) extension that came to my mind was a (single) forward read iterator.

Let me elaborate the implementation details, to make it clear what I mean by a forward read iterator. A singly linked list can be used for implementing a stack. Let the list be implemented by a pointer pTop, which is either zero, or points to an SList node. An SList node consists of a payload field value and a pointer field pNext, where pNext is either zero, or points to an SList node. Let the forward read iterator be implemented by a pointer pRead, which is either zero, or points to an SListnode. The pointers pTop and pRead cannot be accessed directly, but can only be used via the following methods:

  • Push(val) creates a new SList node n with n.value = val and n.pNext = pTop, and sets pTop = &n.
  • Pop() aborts if pTop == 0 or pRead == pTop. Otherwise it reads val = pTop->value and pTopNext = pTop->pNext, frees the SList node pointed to by pTop, sets pTop = pTopNext and returns val.
  • ReadBegin() sets pRead = pTop.
  • ReadNext() aborts if pRead == 0. Otherwise it reads val = pRead->value, sets pRead = pRead->pNext and returns val.
  • ReadFinished() returns true if pRead == 0, and false otherwise.
  • $\begingroup$ I should clarify that initially pTop == 0 and pRead == 0. A method ReadCancel() which sets pRead = 0 might also be a good idea, because otherwise the abort of Pop() for pRead == pTop might be annoying. $\endgroup$ Apr 11, 2015 at 13:19
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    $\begingroup$ Between the pushdown automata and the Turing machines in the Chomsky hierarchy sits the Linear-bounded non-deterministic Turing machine, corresponding to context-sensitive language. $\endgroup$
    – Pål GD
    Apr 11, 2015 at 14:06
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    $\begingroup$ There are automata with a stack of stacks, but I forgot the name. Also, I remember our "homebrew" heap automata. $\endgroup$
    – Raphael
    Apr 11, 2015 at 14:38
  • $\begingroup$ @ThomasKlimpel just a heads up that there was a mistake in my answer, but not a catastrophic one, you're still clear from Turing completeness, but your model is a little more powerful than I first thought (I misinterpreted the non-erasing property, in a really dumb way in hindsight). $\endgroup$ Apr 12, 2015 at 1:07
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    $\begingroup$ @ThomasKlimpel, a second heads up, user23013 is correct (see his answer), your model is Turing complete - the clincher is that you have two pointers into the stack, whereas Stack Automata only have one (so it can move around the stack, but can only pop/push at the top). $\endgroup$ Apr 12, 2015 at 9:37

2 Answers 2


Your model is Turing-complete, unfortunately.

You can simulate a queue in your data structure using the following algorithm. It introduced 3 new stack symbols: $d, x, y$.

Enqueue(val) is just Push(val).

For Dequeue():

  1. ReadBegin().
  2. Count the number of anything else - number of $d$ in the whole stack (which should be always non-negative). Push $y$ or pop $x$ for every $d$, and push $x$ or pop $y$ for anything else. Always prefer pop to push. Finally there won't be any $y$ in the stack and the result will be the number of $x$ on the top of the stack.
  3. ReadBegin().
  4. While pTop is a $x$:
    1. Repeat ReadNext() until it returned something other than $x$ and $d$.
    2. Pop().
  5. Push a $d$.
  6. The last result of ReadNext() is returned as the result of Dequeue.

The proof is straightforward. Check the revision history for a more complicated version firstly reducing it to a two-way version.

  • $\begingroup$ Found the difference, you are correct, the OP's model is Turing complete. A Stack Automaton doesn't have a second read head to scan the stack, it can just move its normal head up and down, but can only push at the top. $\endgroup$ Apr 12, 2015 at 9:25

Your model is Turing complete (unlike what I previously thought), see user23013's answer a sketch of the proof (the essence is you can simulate a queue, and queue automata are Turing complete).

There are several ways to weaken you model to drop to equivalence with linear bound automata or lower.

Ginsburg, Greibach & Harrison [1] give a machine called a "Stack Automaton" which is a PDA with two additional capabilities: 1. The input head can move left an right (so it can scan previously seen parts of the input). 2. The read/write head on the stack can scan through the stack in read-only mode, but pushing and popping still only occur at the top. Note the key difference here with your model, which confused me earlier: the stack only has one head/pointer that it can move up and down the stack, whereas yours has two, which is enough to make your model Turing complete. They also give another model [2], where the input can only be read left-to-right, but the additional read-only stack scanning is still available.

In Figure 2 of [1] they give the containment (proved in Section 5 of the same, perhaps with some parts in [2]) and two-way nondeterministic stack automata languages are strictly contained in $\mathrm{R}$. However they are equivalent to nondeterministic linear bound automata, so they recognise context sensitive-languages.

Two-way and nondeterministic stack automata and two-way deterministic stack automata seem to be equivalent, however changing the input head to one-way makes a significant difference. The set of one-way nondeterministic stack automata languages is a strict subset of context-sensitive languages (I can't put my finger on exactly where yet), and the set of one-way deterministic stack automata (which are equivalent to your model) languages is a strict subset of the set of one-way nondeterministic stack automata languages.

A weaker type again, which falls below these are non-erasing stack automata, which can only write to the stack.

Hopcroft & Ullman show that the languages recognised by non-erasing deterministic stack automata corresponds to $\mathrm{DSPACE}(n\log n)$ and non-erasing nondeterministic stack automata corresponds to $\mathrm{DSPACE}(n^{2})$.


After some more digging, these lecture slides suggest that one-way non-erasing deterministic stack automata are strictly weaker than the two-way version, so recognise something less than $\mathrm{DSPACE}(n\log n)$.

I also found further Hopcroft & Ullman [4,5] papers, which may provide a few more clues, but it seems to be tangential at this point. [5] at least proves some equivalences of some Stack Automata with LBAs.


  1. Seymour Ginsburg, Sheila A. Greibach and Michael A. Harrison, "Stack Automata and Compiling". Journal of the ACM, 14(1):172–201, 1967.
  2. Seymour Ginsburg, Sheila A. Greibach and Michael A. Harrison, "One-Way Stack Automata". Journal of the ACM, 14(2):389–418, 1967.
  3. John E. Hopcroft, Jeffrey D. Ullman, "Nonerasing Stack Automata". JCSS, 1(2):166–186, 1967.
  4. John E. Hopcroft, Jeffrey D. Ullman, "Deterministic Stack Automata and the Quotient Operator", JCSS, 2:1-12, 1968.
  5. John E. Hopcroft, Jeffrey D. Ullman, "Two results on one-way stack automata", Symposium on Switching and Automata Theory (SWAT - but not that SWAT), 1967.
  • $\begingroup$ But the model in this question isn't non-erasing, which should be equivalent to its two-way version (by recording and counting push/pop/up/down operations in the stack). $\endgroup$
    – user23013
    Apr 11, 2015 at 23:20
  • $\begingroup$ @user23013 correct, I misinterpreted the non-erasing part. $\endgroup$ Apr 12, 2015 at 0:53
  • $\begingroup$ Just tried to see why my conclusion doesn't agree with yours, and I'm convinced that this model is in fact Turing complete now. To confirm: where does it say the two-way stack automaton cannot use the stack as a queue (set the extra pointer at the bottom and never moving down)? If it is forbidden somehow, I think that isn't specified in this question. $\endgroup$
    – user23013
    Apr 12, 2015 at 2:03
  • $\begingroup$ @user23013 the second pointer is read only, there's no operation to add anything where pRead is. (as I understand it) $\endgroup$ Apr 12, 2015 at 2:12
  • $\begingroup$ But pTop and pRead are two separate pointers. You can enqueue at pTop and dequeue (by moving up and ignore anything below) at pRead? (If it is two-way.) $\endgroup$
    – user23013
    Apr 12, 2015 at 2:16

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