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We show that a one-way two-dimensional iterative array of finite-state machines (2-DIA) can recognize and parse strings of any context-free language in linear time. What makes this result interesting and rather surprising is the fact that each processor of the array holds only a fixed amount of information (independent of the size of the input) and communicates with its neighbors in only one direction. This makes for a simple VLSI implementation. Although it is known that recognition can be done on a 2-DIA, previous parsing algorithms require the processors to have unbounded memory, even when the communication is two-way. We also consider the problem of finding approximate patterns in strings, the string-to-string correction problem, and the longest common subsequence problem, and show that they can be solved in linear time on a 2-DIA.