{{Bots|deny=AWB}}<!--REASON: incorrectly identifies the first "{" in "{empty string{{!}}ε}" as an unbalanced bracket & removes it--> [[File:Pushdown-overview.svg|thumb|right|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 that can be additional stacks.<ref name="aho">{{cite journal |last1=Aho |first1=Alfred V. |s2cid=685569 |authorlink1=Alfred Aho |title=Nested Stack Automata |journal=Journal of the ACM |date=July 1969 |volume=16 |issue=3 |pages=383–406 |doi=10.1145/321526.321529 |doi-access=free }}</ref> 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,<ref>{{cite book | last = Partee | author-link = Barbara Partee | first = Barbara |author2=Alice ter Meulen |author2-link=Alice ter Meulen|author3=Robert E. Wall | title = Mathematical Methods in Linguistics | url = https://archive.org/details/mathematicalmeth00part_211| url-access = limited| year = 1990 | publisher = Kluwer Academic Publishers | pages = [https://archive.org/details/mathematicalmeth00part_211/page/n556 536]–542 | isbn = 978-90-277-2245-4 }}</ref> and in fact the class of indexed languages is exactly the class of languages accepted by one-way nondeterministic nested stack automata.<ref name="aho" /><ref>{{cite book| author=John E. Hopcroft, Jeffrey D. Ullman| title=Introduction to Automata Theory, Languages, and Computation| year=1979| publisher=Addison-Wesley| isbn=0-201-02988-X| url-access=registration| url=https://archive.org/details/introductiontoau00hopc}} Here:p.390</ref>
Nested stack automata should not be confused with embedded pushdown automata, which have less computational power.{{citation needed|reason=The claim is currently supported only by the order in which both notions appear in the 'Automata theory: formal languages and formal grammars' overview table below.|date=February 2014}}
==Formal definition== ===Automaton=== A (nondeterministic two-way) nested stack automaton is a tuple {{angbr|''Q'',Σ,Γ,δ,''q''<sub>0</sub>,''Z''<sub>0</sub>,''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.<ref group=note>Aho originally used "$", "¢", and "#" instead of "[", "]", and "''']'''", respectively. See Aho (1969), p.385 top.</ref> * 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<sup>*</sup> ∪ ''D''), such that δ maps<ref group=note>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'.</ref> {| |- | || ''Q'' × Σ' × [Γ || into subsets of ''Q'' × ''D'' × [Γ<sup>*</sup> || (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<sup>*</sup>] || (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;<ref>Aho (1969), p.385 top</ref> 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. * ''q''<sub>0</sub> ∈ ''Q'' is the initial state, * ''Z''<sub>0</sub> ∈ 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 {{angbr| ''q'', [''a''<sub>1</sub>''a''<sub>2</sub>...<u>''a''<sub>''i''</sub></u>...''a''<sub>''n''-1</sub>], [''Z''<sub>1</sub>''X''<sub>2</sub>...<u>''X''<sub>''j''</sub></u>...''X''<sub>''m''-1</sub>''']''' }}, where * ''q'' ∈ ''Q'' is the current state, * [''a''<sub>1</sub>''a''<sub>2</sub>...<u>''a''<sub>''i''</sub></u>...''a''<sub>''n''-1</sub>] is the input string; for convenience, ''a''<sub>0</sub> = [ and ''a''<sub>''n''</sub> = ] is defined<ref group=note>Aho originally used the left and right stack marker, viz. $ and ¢, as right and left input marker, respectively.</ref> The current position in the input, viz. ''i'' with 0 ≤ ''i'' ≤ ''n'', is marked by underlining the respective symbol. * [''Z''<sub>1</sub>''X''<sub>2</sub>...<u>''X''<sub>''j''</sub></u>...''X''<sub>''m''-1</sub>''']''' is the stack, including substacks; for convenience, ''X''<sub>1</sub> = [''Z''<sub>1</sub> <ref group=note>The top symbol of a (sub)stack together with its preceding left endmarker "[" is viewed as a single symbol.</ref> and ''X''<sub>''m''</sub> = ''']''' 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):
{| class=wikitable |- ! Action ! Step ! colspan=11 | Stack |- | | 1: | style="font-family:monospace"|[''a'' || style="font-family:monospace"| ''b'' || style="font-family:monospace"| [''k'' || style="font-family:monospace"| ] || style="font-family:monospace"| <u>[''p''</u> || style="font-family:monospace"| ] || style="font-family:monospace"| ''c'' || style="font-family:monospace"| ''']''' | colspan=3 | |- | create substack | 2: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| [''p'' || style="font-family:monospace"| <u>[''r''</u> || style="font-family:monospace"| ''s'' || style="font-family:monospace"| ] || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|''c''}} || style="font-family:monospace"| {{color|#808080|''']'''}} |- | pop | 3: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|[''p''}} || style="font-family:monospace"| <u>[''s''</u> || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|''c''}} || style="font-family:monospace"| {{color|#808080|''']'''}} | |- | pop | 4: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|[''p''}} || style="font-family:monospace"| <u>[]</u> || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|''c''}} || style="font-family:monospace"| {{color|#808080|''']'''}} | colspan=2 | |- | destroy substack | 5: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|[''p''}} || style="font-family:monospace"| <u>]</u> || style="font-family:monospace"| {{color|#808080|''c''}} || style="font-family:monospace"| {{color|#808080|''']'''}} | colspan=4 | |- | move down | 6: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|[''p''}} || style="font-family:monospace"| ] || style="font-family:monospace"| <u>''c''</u> || style="font-family:monospace"| {{color|#808080|''']'''}} | colspan=4 | |- | move up | 7: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|[''p''}} || style="font-family:monospace"| <u>]</u> || style="font-family:monospace"| ''c'' || style="font-family:monospace"| {{color|#808080|''']'''}} | colspan=4 | |- | move up | 8: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| <u>[''p''</u> || style="font-family:monospace"| ] || style="font-family:monospace"| {{color|#808080|''c''}} || style="font-family:monospace"| {{color|#808080|''']'''}} | colspan=4 | |- | push | 9: | style="font-family:monospace"|{{color|#808080|[''a''}} || style="font-family:monospace"| {{color|#808080|''b''}} || style="font-family:monospace"| {{color|#808080|[''k''}} || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| <u>[''n''</u> || style="font-family:monospace"| ''o'' || style="font-family:monospace"| ''p'' || style="font-family:monospace"| {{color|#808080|]}} || style="font-family:monospace"| {{color|#808080|''c''}} || style="font-family:monospace"| {{color|#808080|''']'''}} | colspan=2 | |}
==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.<ref>{{cite journal |last1=Beeri |first1=C. |title=Two-way nested stack automata are equivalent to two-way stack automata |journal=Journal of Computer and System Sciences |date=June 1975 |volume=10 |issue=3 |pages=317–339 |doi=10.1016/s0022-0000(75)80004-3 |doi-access=free }}</ref>
Gilman and Shapiro used nested stack automata to solve the word problem in virtually free groups, similarly to the Muller–Schupp theorem.<ref>{{cite tech report |last1=Shapiro |first1=Robert|last2 = Gilman|first2 = Michael |title=On groups whose word problem is solved by a nested stack automaton |date=4 December 1998 |arxiv=math/9812028 |s2cid=12716492 |citeseerx=10.1.1.236.2029 }}</ref>
==Notes== {{Reflist|group=note}}
==References== {{Reflist}}
{{Formal languages and grammars}}
Category:Models of computation Category:Automata (computation)