Mathematical Methods in Linguistics pp 527-532 | Cite as

# Linear Bounded Automata, Context Sensitive Languages and Type 1 Grammars

## Abstract

A linear bounded automaton (lba) is, in effect, a Turing machine whose computations are restricted to the amount of tape on which the input is written. We can imagine it as consisting of a finite set of states, a finite alphabet (including special right- and left-endmarkers [ and ]), a designated initial state, and a finite set of instructions of the same form as the quadruples for Turing machines. We assume, however, that the input to an lba is given between the designated endmarkers, i.e. as [*w*], and that the lba has no instructions which allow it to move past these endmarkers or to erase or replace them. Thus, the tape head can move only in the portion of the tape originally occupied by *w*, although an equivalent formulation of lba’s sets the limit on usable tape not as *equal* to the length of the input but rather as a *linear function* of the length of the input. (A linear function in a variable *x* is of the form *ax* + *b*, where *a* and *b* are constants. Plotting values of *ax* + *b* for each value of *x* on graph paper gives a straight line, whereas plots of functions involving *x*^{2}, *x*^{3}, etc. gives curves.) This is the source of the name for these automata—the allowed computational space is bounded by a linear function of the length of the input string.

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