Nested stack automaton

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


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.
  • 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.
  • q0∈ Q is the initial state,
  • Z0∈ Γ is the initial stack symbol,
  • F⊆ Q is the set of final states.


configuration, or instantaneous description of such an automaton consists in a triple ⟨ q, [a1a2aian-1], [Z1X2XjXm-1] ⟩, where

  • q∈ Q is the current state,
  • [a1a2aian-1] is the input string; for convenience, a0= [ and an = ] is defined[note 3] The current position in the input, viz. i with 0 ≤ i ≤ n, is marked by underlining the respective symbol.
  • [Z1X2XjXm-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 ≤ j ≤ m, 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. ^ Jump up to:ab Aho, Alfred V. (July 1969). “Nested Stack Automata”. Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529. S2CID 685569.
  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. ^Beeri, C. (June 1975). “Two-way nested stack automata are equivalent to two-way stack automata”. Journal of Computer and System Sciences. 10 (3): 317–339. doi:10.1016/s0022-0000(75)80004-3.
  6. ^Shapiro, Robert Gilman Michael (4 December 1998). On groups whose word problem is solved by a nested stack automaton (Technical report). arXiv:math/9812028. CiteSeerX S2CID 12716492.

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