Recent work has demonstrated the self-assembly of designed periodic two-dimensional arrays composed of DNA tiles, in which the intermolecular contacts are directed by 'sticky' ends. In a mathematical context, aperiodic mosaics may be formed by the self-assembly of 'Wang' tiles, a process that emulates the operation of a Turing machine. Macroscopic self-assembly has been used to perform computations; there is also a logical equivalence between DNA sticky ends and Wang tile edges. This suggests that the self-assembly of DNA-based tiles could be used to perform DNA-based computation. Algorithmic aperiodic self-assembly requires greater fidelity than periodic self-assembly, because correct tiles must compete with partially correct tiles. Here we report a one-dimensional algorithmic self-assembly of DNA triple-crossover molecules that can be used to execute four steps of a logical (cumulative XOR) operation on a string of binary bits.