Two Dimensional Finite Automaton

1. Define two dimensional finite automaton (2DIM-DFA) is defined as follows.  The input is an m X n rectangle, for any m, n  2.  The squares along the boundary of the rectangle contain the symbol # and the internal squares contain symbols over the input alphabet .  The transition function is a mapping  Q x   Q x {L,R,U,D} to indicate the next state and the new head position (Left, Right, Up, down).  The machine accepts when it enters one of the designated accept states.  It rejects if it tries to move off the input rectangle or if it never halts.  Two such machines are equivalent if they accept the same rectangles.  Consider the problem of testing whether two of these machines are equivalent.  Formulate this problem as a language, and show that it is undecidable.

See attached file for full problem description.