Linear bounded automaton
Linear bounded automata satisfy 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.
As in the definition of Turing machines, it 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. It 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. 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. Since the size of the accessible tape is bounded by some linear function of the length of the input, the linear bounded automaton is computationally equivalent to a nondeterministic Turing machine restricted to the portion of the tape containing the input, hence the name linear bounded automaton.
This limitation makes the LBA a somewhat more accurate model of computers that actually exist than a Turing machine, whose definition assumes unlimited tape.
LBA and Context-sensitive Languages
Linear bounded automata are acceptors for the class of context-sensitive languages. 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.
In 1960, Myhill introduced an automaton model today known as deterministic linear bounded automaton. Shortly thereafter, Landweber proved that the languages accepted by deterministic LBAs are context-sensitive. In 1964, Kuroda introduced the more general model of (nondeterministic) linear bounded automata, noted that Landweber's proof also works for nondeterministic linear bounded automata, and showed that the languages accepted by them are precisely the context-sensitive languages.
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:
The second LBA problem is whether the class of languages accepted by LBA is closed under complement.
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. As of 2011, the first LBA problem still remains open.
- Myhill (1960)
- Landweber (1963)
- Kuroda (1964)
- John Myhill: Linearly Bounded Automata, WADD Technical Note 60-165, Wright-Patterson Air Force Base, Wright Air Development Division, Ohio, June 1960.
- Peter S. Landweber: Three Theorems on Phrase Structure Grammars of Type 1, Information and Control 6(2): 131-136 (1963)
- Sige-Yuki Kuroda: Classes of languages and linear-bounded automata, Information and Control, 7(2): 207–223 (1964)
- Linear Bounded Automata by Forbes D. Lewis
- Linear Bounded Automata slides, part of Context-sensitive Languages by Arthur C. Fleck
- Linear-Bounded Automata, part of Theory of Computation syllabus, by David Matuszek