Nested stack automaton

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.

Formal definition

Automaton

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

      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.

Configuration

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

Example

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 ]  

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.[5]

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

Notes

  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.

References

  1. 1.0 1.1 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.
  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.