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We propose and investigate a formal language operation inspired by the naturally occurring phenomenon of DNA primer extension by a DNA-template-directed DNA Polymerase enzyme. Given two DNA strings u and v, where the shorter string v (called primer) is Watson-Crick complementary and can thus bind to a substring of the longer string u (called template) the result of the primer extension is a DNA string that is complementary to a suffix of the template which starts at the binding position of the primer. The operation of DNA primer extension can be abstracted as a binary operation on two formal languages: a template language L 1 and a primer language L 2 . We call this language operation L 1 -directed extension of L 2 and study the closure properties of various language classes, including the classes in the Chomsky hierarchy, under directed extension. Furthermore, we answer the question under what conditions can a given language of target strings be generated from a given template language when the primer language is unknown. We use the canonic inverse of directed extension in order to obtain the optimal solution (the minimal primer language) to this question.

The operation of overlap assembly was defined by Csuhaj-Varju, Petre, and Vaszil as a formal model of the linear self-assembly of DNA strands: The overlap assembly of two strings, xy and yz, which share an "overlap" y, results in the string xyz. This paper continues the exploration of the properties of the overlap assembly operation by investigating closure of various language classes under iterated overlap assembly, and the decidability of the completeness of a language. It also investigates the problem of deciding whether a given string is terminal with respect to a language, and the problem of deciding if a given language can be generated by an overlap assembly operation of two given others.

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