Equivalence of two-dimensional multitape automata

Abstract

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 = { } ( , ), ( ...

The equivalence problem of multidimensional multitape automata

“…The special case of the equivalence problem of multidimensional multitape automata, where the dimensions of the tapes are less than or equal to 2, was solved in [4]. Here we extend the technique introduced in [4] and prove that the equivalence problem of multidimensional multitape automata can be reduced to the equivalence problem of multitape automata [5].…”

“…Here we extend the technique introduced in [4] and prove that the equivalence problem of multidimensional multitape automata can be reduced to the equivalence problem of multitape automata [5].…”

“…An example of such coding for c = 2 is shown in [4]. An example for c = 3 is adduced below: a 3-dimensional word of length n + 1 (Fig.…”