Linear bounded automaton

In computer science, a linear bounded automaton (plural linear bounded automata, abbreviated LBA) is a restricted form of Turing machine.

Operation

A linear bounded automaton is a Turing machine that satisfies the following three conditions:

  • Its input alphabet includes two special symbols, serving as left and right endmarkers.
  • Its transitions may not print other symbols over the endmarkers.
  • Its transitions may neither move to the left of the left endmarker nor to the right of the right endmarker.[1]: 225 

In other words: instead of having potentially infinite tape on which to compute, computation is restricted to the portion of the tape containing the input plus the two tape squares holding the endmarkers.

An alternative, less restrictive definition is as follows:

  • Like a Turing machine, an LBA possesses a tape made up of cells that can contain symbols from a finite alphabet, a head that can read from or write to one cell on the tape at a time and can be moved, and a finite number of states.
  • An LBA differs from a Turing machine in that while the tape is initially considered to have unbounded length, only a finite contiguous portion of the tape, whose length is a linear function of the length of the initial input, can be accessed by the read/write head; hence the name linear bounded automaton.[1]: 225 

This limitation makes an LBA a somewhat more accurate model of a real-world computer than a Turing machine, whose definition assumes unlimited tape.

The strong and the weaker definition lead to the same computational abilities of the respective automaton classes,[1]: 225  by the same argument used to prove the linear speedup theorem.

LBA and context-sensitive languages

Linear bounded automata are acceptors for the class of context-sensitive languages.[1]: 225–226  The only restriction placed on grammars for such languages is that no production maps a string to a shorter string. Thus no derivation of a string in a context-sensitive language can contain a sentential form longer than the string itself. Since there is a one-to-one correspondence between linear-bounded automata and such grammars, no more tape than that occupied by the original string is necessary for the string to be recognized by the automaton.

History

In 1960, John Myhill introduced an automaton model today known as deterministic linear bounded automaton.[2] In 1963, Peter Landweber proved that the languages accepted by deterministic LBAs are context-sensitive.[3] In 1964, S.-Y. Kuroda introduced the more general model of (nondeterministic) linear bounded automata, and adapted Landweber's proof to show that the languages accepted by nondeterministic linear bounded automata are precisely the context-sensitive languages.[4][5]

LBA problems

In his seminal paper, Kuroda also stated two research challenges, which subsequently became famously known as the "LBA problems": The first LBA problem is whether the class of languages accepted by LBA is equal to the class of languages accepted by deterministic LBA. This problem can be phrased succinctly in the language of computational complexity theory as:

First LBA problem: Is NSPACE(O(n)) = DSPACE(O(n))?

The second LBA problem is whether the class of languages accepted by LBA is closed under complement.

Second LBA problem: Is NSPACE(O(n)) = co-NSPACE(O(n))?

As observed already by Kuroda, a negative answer to the second LBA problem would imply a negative answer to the first problem. But the second LBA problem has an affirmative answer, which is implied by the Immerman–Szelepcsényi theorem proved 20 years after the problem was raised.[6][7] As of today, the first LBA problem still remains open. Savitch's theorem provides an initial insight, that NSPACE(O(n)) ⊆ DSPACE(O(n2)).[8]

References
  1. John E. Hopcroft; Jeffrey D. Ullman (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 978-0-201-02988-8.
  2. John Myhill (June 1960). Linear Bounded Automata (WADD Technical Note). Wright Patterson AFB, Wright Air Development Division, Ohio.
  3. P.S. Landweber (1963). "Three Theorems on Phrase Structure Grammars of Type 1". Information and Control. 6 (2): 131–136. doi:10.1016/s0019-9958(63)90169-4.
  4. Sige-Yuki Kuroda (Jun 1964). "Classes of languages and linear-bounded automata". Information and Control. 7 (2): 207–223. doi:10.1016/s0019-9958(64)90120-2.
  5. Willem J. M. Levelt (2008). An Introduction to the Theory of Formal Languages and Automata. John Benjamins Publishing. pp. 126–127. ISBN 978-90-272-3250-2.
  6. Immerman, Neil (1988), "Nondeterministic space is closed under complementation" (PDF), SIAM Journal on Computing, 17 (5): 935–938, doi:10.1137/0217058, MR 0961049
  7. Szelepcsényi, Róbert (1988), "The method of forcing for nondeterministic automata", Acta Informatica, 26 (3): 279–284, doi:10.1007/BF00299636, S2CID 10838178
  8. Arora, Sanjeev; Barak, Boaz (2009). Complexity Theory: A Modern Approach. Cambridge University Press. ISBN 978-0-521-42426-4.
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