Deterministic Context-Sensitive Languages: Part II

DEFINITION. A de termin i s t i c l inear bounded a u t o m a t o n (DLBA) is a sixtuple (Q, q0, V, # 1 , # 2 , J ) where: Q is a finite set of states q o , q l .... ; q o E Q is the initial state; V is a finite vocabulary; # 1 , #2 6 V are left and right endmarkers, respectively; and J is a finite set of instructions. T h e automaton consists of two pushdown lists, called L and R, and a finite control mechanism. #1 always occurs as the bot tom entry on L, and this is its only occurrence; similarly for #2 and R. We will represent a configuration of the automaton by # 1 x l "" xnq~xn+ 1 "'" X , # 2 . This means that the finite control is in state ql , L contains #1xl "'" xn ( n / > 0), with x n topmost, and R contains xn+ 1 "" x t # 2 ( t > / n ) , with xn+ 1 topmost. T h e automaton functions as follows. For an input string Xl "'" xn E V*, it starts in the configuration # ~ q o x l . " x n # 2 . In general, if the automaton is in state q with x i and x~ at tile top of L and R respectively, it changes state and replaces x i and x~ by other symbols at the tops of L and R, under direction of an instruction of J . Alternatively, the automaton can accept the input string if the instruction so orders, or it can reject the input string if no instruction applies to the triple (xl , q, x~-). The set of strings accepted by a D L B A M is denoted T ( M ) . The instructions in J are of the form ~ --> fl, where c~ and/3 are "before" and "after" snapshots of the part of the configuration around the state qe" Each a or /3 contains one member q of Q and one or more members of