Nested stack automaton

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A nested stack automaton has the same devices as a pushdown automaton, but has less restrictions for using them.

In automata theory, a nested stack automaton is a finite automaton that can make use of a stack containing data which can be additional stacks.[1] Like a stack automaton, a nested stack automaton may step up or down in the stack, and read the current symbol; in addition, it may at any place create a new stack, operate on that one, eventually destroy it, and continue operating on the old stack. This way, stacks can be nested recursively to an arbitrary depth; however, the automaton always operates on the innermost stack only.

A nested stack automaton is capable of recognizing an indexed language,[2] and in fact the class of indexed languages is exactly the class of languages accepted by one-way nondeterministic nested stack automata.[1][3]

Nested stack automata should not be confused with embedded pushdown automata, which have less computational power.[citation needed]

Formal definition[edit]


A (nondeterministic two-way) nested stack automaton is a tuple ⟨Q,Σ,Γ,δ,q0,Z0,F,[,],]⟩ where

  • Q, Σ, and Γ is a nonempty finite set of states, input symbols, and stack symbols, respectively,
  • [, ], and ] are distinct special symbols not contained in Σ ∪ Γ,
    • [ is used as left endmarker for both the input string and a (sub)stack string,
    • ] is used as right endmarker for these strings,
    • ] is used as the final endmarker of the string denoting the whole stack.[note 1]
  • An extended input alphabet is defined by Σ' = Σ ∪ {[,]}, an extended stack alphabet by Γ' = Γ ∪ {]}, and the set of input move directions by D = {-1,0,+1}.
  • δ, the finite control, is a mapping from Q × Σ' × (Γ' ∪ [Γ' ∪ {], []}) into finite subsets of Q × D × ([Γ*D), such that δ maps[note 2]
      Q × Σ' × [Γ into subsets of Q × D × [Γ* (pushdown mode),
Q × Σ' × Γ' into subsets of Q × D × D (reading mode),
Q × Σ' × [Γ' into subsets of Q × D × {+1} (reading mode),
Q × Σ' × {]} into subsets of Q × D × {-1} (reading mode),
Q × Σ' × (Γ' ∪ [Γ') into subsets of Q × D × [Γ*] (stack creation mode), and
Q × Σ' × {[]} into subsets of Q × D × {ε}, (stack destruction mode),
Informally, the top symbol of a (sub)stack together with its preceding left endmarker "[" is viewed as a single symbol;[4] then δ reads
  • the current state,
  • the current input symbol, and
  • the current stack symbol,
and outputs
  • the next state,
  • the direction in which to move on the input, and
  • the direction in which to move on the stack, or the string of symbols to replace the topmost stack symbol.
  • q0Q is the initial state,
  • Z0 ∈ Γ is the initial stack symbol,
  • FQ is the set of final states.


A configuration, or instantaneous description of such an automaton consists in a triple ⟨ q, [], [Z1X2...Xj...Xm-1] ⟩, where

  • qQ is the current state,
  • [] is the input string; for convenience, a0 = [ and an = ] is defined[note 3] The current position in the input, viz. i with 0 ≤ in, is marked by underlining the respective symbol.
  • [Z1X2...Xj...Xm-1] is the stack, including substacks; for convenience, X1 = [Z1 [note 4] and Xm = ] is defined. The current position in the stack, viz. j with 1 ≤ jm, is marked by underlining the respective symbol.


An example run (input string not shown):

Action Step Stack
1:       [a b [k ] [p ] c ]  
create substack       2: [a b [k ] [p [r s ] ] c ]
pop 3: [a b [k ] [p [s ] ] c ]  
pop 4: [a b [k ] [p [] ] c ]  
destroy substack 5: [a b [k ] [p ] c ]  
move down 6: [a b [k ] [p ] c ]  
move up 7: [a b [k ] [p ] c ]  
move up 8: [a b [k ] [p ] c ]  
push 9: [a b [k ] [n o p ] c ]  


When automata are allowed to re-read their input ("two-way automata"), nested stacks do not result in additional language recognition capabilities, compared to plain stacks.[5]

Gilman and Shapiro used nested stack automata to solve the word problem in certain groups.[6]


  1. ^ Aho originally used "$", "¢", and "#" instead of "[", "]", and "]", respectively. See Aho (1969), p.385 top.
  2. ^ Juxataposition denotes string (set) concatenation, and has a higher binding priority than set union ∪. For example, [Γ' denotes the set of all length-2 strings starting with "[" and ending with a symbol from Γ'.
  3. ^ Aho originally used the left and right stack marker, viz. $ and ¢, as right and left input marker, respectively.
  4. ^ The top symbol of a (sub)stack together with its preceding left endmarker "[" is viewed as a single symbol.


  1. ^ a b Aho, Alfred (1969). "Nested stack automata". Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529. ISSN 0004-5411.
  2. ^ Partee, Barbara; Alice ter Meulen; Robert E. Wall (1990). Mathematical Methods in Linguistics. Kluwer Academic Publishers. pp. 536–542. ISBN 978-90-277-2245-4.
  3. ^ John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Here:p.390
  4. ^ Aho (1969), p.385 top
  5. ^ C. Beeri (1975). "Two-Way Nested Stack Automata Are Equivalent to Two-Way Stack Automata" (PDF). J. Comp. and System Sciences. 10: 317–339. doi:10.1016/s0022-0000(75)80004-3.
  6. ^ Robert Gilman, Michael Shapiro (Dec 1998). On Groups Whose Word Problem is Solved by a Nested Stack Automaton (PDF) (Technical report). arXiv. p. 16.