World Library  
Flag as Inappropriate
Email this Article

Nested stack automaton

Article Id: WHEBN0009791447
Reproduction Date:

Title: Nested stack automaton  
Author: World Heritage Encyclopedia
Language: English
Subject: Pushdown automaton, Automata theory, Alfred Aho, Range concatenation grammars, Ogden's lemma
Collection: Automata Theory, Models of Computation
Publisher: World Heritage Encyclopedia
Publication
Date:
 

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.

Contents

  • Formal definition 1
    • Automaton 1.1
    • Configuration 1.2
    • Example 1.3
  • Properties 2
  • Notes 3
  • References 4

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.
  • q0Q is the initial state,
  • Z0 ∈ Γ is the initial stack symbol,
  • FQ 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

  • qQ 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 ≤ in, 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 ≤ jm, 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

  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. ^ a b  
  2. ^  
  3. ^ John E. Hopcroft, Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley.   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. 
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.