#### Concept of Linear Bounded Automata, Finite Automata with External Storage

Linear bounded automata (LBA):

The linear bounded automaton is a finite automaton with read or write access to a tape T of fixed length recognized by the input string w. M has a read or write head which can be moved left or right one square at a time, however can’t be moved off the tape. The name ‘linear bounded’ signifies to the fact that T’s storage capacity is a constant multiple of capacity needed to hold w.

M consists of a working alphabet B that contains A as a subset, and usually has additional characters utilized for scratch work and book-keeping. In the illustration of figure at left, A = {0, 1}, B = {0, 1, %, ...}. The tape of length L has a storage capacity of L log |B| bits that is a constant factor of log |B| / log |A| bigger than the storage capacity needed to store input strings over A of similar length L . This is convenient to think of M’s tape as comprising of several parallel tracks, as shown at right. The designated track comprises the input string to be processed, and may be read-only. The other tracks are read or write tracks employed for scratch work. In this model, M’s alphabet is the Cartesian product A x B’ x B” x ..of input alphabet A and other track alphabets B’, B”, ..

A deterministic LBA has the components: M = (Q, A, B, f: Q x B -> Q x B x {L, R, H,..}, q0, F}.

Q: finite state space; A: input alphabet and B tape alphabet; f: transition function; q0: initial state; F: final states.

{L, R, H, ..}: tape actions: L = move left, R = move right, H = halt and Optional: “stay put”.

Problem: The word problem for LBAs is decidable.

Illustrate the decision procedure which, given an LBA M and a word w, decides whether M accepts w.

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