The undecidability of the unrestricted modified edit distance

We define the unrestricted modified edit distance based on the modified edit distance defined by Galil and Giancarlo (1989) where the cost of substring deletions and insertions are context-sensitive and the cost of character substitutions are context-free. The modified edit distance is the minimum cost of converting a string X to a string Y where the sequence of edit operations has the property that all substring deletions precede all character substitutions and all character substitutions precede all substring insertions. Note that the modified edit distance does not satisfy the triangle inequality. We show that the problem of finding the unrestricted modified edit distance which is the minimum cost over all edit sequences (without these constraints) of converting X to Y is undecidable.