519.681The equivalence of multitape automata with multi-dimensional tapes is considered. Their heads move monotonically in all directions (their backward movement is impossible). The special case when the dimensions of tapes are less than or equal to 2 is proved to be solvable.Automata with multidimensional tapes in which heads move monotonically in all directions (their backward movement is impossible) are introduced in [1]. As was shown, the equivalence problem in the class of program schemes over a nondegenerate basis of rank unity is reduced to the equivalence of multidimensional multitape automata.In the present article, the equivalence problem of two-dimensional multitape automata is considered. It is proved that this problem is reduced to the equivalence of multitape automata [2].Let us give some definitions from [1] that are necessary for the further consideration. Let r > 0 be an integer, and let N = { } 0 1 , , . . . . We call the set N r an r-dimensional tape, the elements of the set N r (i.e., r-tuples of the form ( , . . . , ) a a r 1 ) the cells of the tape, and numbers a a r 1 , . . . , coordinates of the corresponding cell. The cell ( , . . . , ) 0 0 is initial. Let X be a finite alphabet. Then a mapping N X r ® is called a filling of an r-dimensional tape with symbols from X . A set S n m n m k k = { } ( , ), . . . , ( , ) 1 1 , where n i and m i ( ) 1£ £ i k are natural numbers and n n i j i j = Û = for any 1 £ £ i j k and , is called the signature of a multidimensional multitape automaton. The signature determines the number and arity of tapes as follows: if ( , ) n m S Î , then an automaton with the signature S has exactly m tapes of dimension n. Let us consider the case when n i £ 2 . We assume that S m = {( , ), 1 1 ( , ) 2 2 m }, where m m 1 2 0 0 ³ ³ , , and m m m = + > 1 2 0 . Let A Q Q Q X q Q m F = á = È È ñ 1 0 . . . , , , , , j y be a two-dimensional m-tape automaton of the signature S , where Q is its set of states, Q i contains all the states (and only such states) in which a tape i Q Q i j, Ç =AEis read out if i j ¹ , X is its input alphabet, q 0 is its initial state, Q F is its set of final states, j: Q X Q ® is its next-state function, and y : , Q X ®{ } 1 2 is its function of movement direction.We call the filled portion of an r-dimensional tape in which the sum of coordinates of each cell is less than or equal to n -1 an r-dimensional word of length n. We denote the set of all r-dimensional words over the alphabet X by W r X ( ). An r-dimensional word of length n is recognized by the automaton if, during the work with the word, it turns out to be in a final state after reading out the cell whose sum of coordinates equals n -1.We call the filled portion of an r-dimensional tape in which the sum of coordinates of each cell is equal to k a k-word diagonal and denote it by d k . We denote the first (last) i i k ( ) 0 < £ symbols of such a diagonal d k by d i d i k k ( _ )( (_ )) . The length of a word coincides with the number of its diagonals. Let a signature S m m = { } ( , ), ( ...
