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A parenthesis string is a string of left and right parentheses. The string is wellformed when it consists of balanced pairs of left and right parentheses. This study presents a novel systolic algorithm for generating all the well-formed parenthesis strings in lexicographical order. The algorithm is cost-optimal and is run on a linear array of processors such that each well-formed parenthesis string can be generated in three time steps. The processor array is appropriate for VLSI implementation, since it has the features of modularity, regularity, and local connection.