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Formal definition

Automaton

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.⁵
• 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 maps⁶

• the current state,
• the current input symbol, and
• the current stack symbol,

• 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

A configuration, or instantaneous description of such an automaton consists in a triple ⟨ q, [a1a2...ai...an-1], [Z1X2...Xj...Xm-1] ⟩, where

• q ∈ Q is the current state,
• [a1a2...ai...an-1] is the input string; for convenience, a0 = [ and an = ] is defined⁸ The current position in the input, viz. i with 0 ≤ i ≤ n, is marked by underlining the respective symbol.
• [Z1X2...Xj...Xm-1] is the stack, including substacks; for convenience, X1 = [Z1 ⁹ and Xm = ] is defined. The current position in the stack, viz. j with 1 ≤ j ≤ m, is marked by underlining the respective symbol.

Example

An example run (input string not shown):

Properties

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.¹⁰

Gilman and Shapiro used nested stack automata to solve the word problem in virtually free groups, similarly to the Muller–Schupp theorem.¹¹

Notes

References

1. Aho, Alfred V. (July 1969). "Nested Stack Automata". Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529. S2CID 685569. /wiki/Alfred_Aho ↩

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. 978-90-277-2245-4 ↩

3. Aho, Alfred V. (July 1969). "Nested Stack Automata". Journal of the ACM. 16 (3): 383–406. doi:10.1145/321526.321529. S2CID 685569. /wiki/Alfred_Aho ↩

4. John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Here:p.390 0-201-02988-X ↩

5. Aho originally used "$", "¢", and "#" instead of "[", "]", and "]", respectively. See Aho (1969), p.385 top. ↩

6. 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'. /wiki/String_concatenation#Concatenation_of_sets_of_strings ↩

7. Aho (1969), p.385 top ↩

8. Aho originally used the left and right stack marker, viz. $ and ¢, as right and left input marker, respectively. ↩

9. The top symbol of a (sub)stack together with its preceding left endmarker "[" is viewed as a single symbol. ↩

10. 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. https://doi.org/10.1016%2Fs0022-0000%2875%2980004-3 ↩

11. 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 10.1.1.236.2029. S2CID 12716492. /wiki/ArXiv_(identifier) ↩