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
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. [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.
- 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 defined [note 3] 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 [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.
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
- ^ Aho originally used "$", "¢", and "#" instead of "[", "]", and "", respectively. See Aho (1969), p.385 top.
- ^ 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'.
- ^ Aho originally used the left and right stack marker, viz. $ and ¢, as right and left input marker, respectively.
- ^ The top symbol of a (sub)stack together with its preceding left endmarker "[" is viewed as a single symbol.
References
- ^ a b Aho, Alfred V. (July 1969). "Nested Stack Automata". Journal of the ACM. 16 (3): 383–406. doi: 10.1145/321526.321529. S2CID 685569.
- ^ 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.
- ^ John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X. Here:p.390
- ^ Aho (1969), p.385 top
- ^ 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.
- ^ 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.