A high fidelity, surface-based method of nucleic acid analysis has been developed based on DNA polymerase extension of primer-template complexes on DNA microchips. The ability of the method to discriminate against mismatches and provide an almost "digital" signal recommended it for molecular computation. A DNA computer with the capability of solving nondeterministic polynomial time (NP)-complete problems (those whose time-complexity function rises exponentially with the problem size) in polynomial time using this Arrayed Primer EXtension (APEX) method was experimentally demonstrated. An algorithm involving extension of surface-bound primer-template complexes, representing solutions and clauses of a Boolean formula, is described for the solution of two-, three-, and four-variable satisfiability (SAT) problems, including a 3SAT, exploiting the theoretical concepts of Lipton. A discussion of the principles of nondeterministic computing with APEX is also provided.