On the undecidability of splicing systems

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Abstract
The notion of splicing system has been used to abstract the process of DNA digestion by restriction enzymes and subsequent religation. A splicing system language is the formal language of all DNA strings producible by such a process. The membership problem is to devise an algorithm (if possible) to answer the question of whether or not a given DNA string belongs to a splicing system language given by initial strings and enzymes. In this paper the concept of a sequential splicing system is introduced. A sequential splicing system differs from a splicing system in that the latter allows arbitrarily many copies of any string in the initial set whereas the sequential splicing system may restrict the initial number of copies of some strings. The main result is that there exist sequential splicing systems with recursively unsolvable membership problem. The technique of the proof is to embed Turing machine computations in the languages.

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