The inclusion of Regular Expressions (REs) is the kernel of any type-checking

algorithm for XML manipulation languages. XML applications would

benefit from the extension of REs with interleaving and counting, but this is

not feasible in general, since inclusion is EXPSPACE-complete for such extended REs.

In previous works we introduced a notion of ``conflict-free REs'', which are extended REs with excellent complexity behavior, including a polynomial inclusion algorithm and linear membership.

Conflict-free REs have interleaving and counting, but the complexity is

tamed by the ``conflict-free'' limitations, which have been found to be

satisfied by the vast majority of the content models published on

the Web.

However, a type-checking algorithm needs to compare machine-generated

subtypes against human-defined supertypes. The conflict-free restriction,

while quite harmless for the human-defined supertype, is far too restrictive

for the subtype. We show here that the PTIME inclusion algorithm can be actually extended to deal with totally unrestricted REs with counting and interleaving in the subtype position,

provided that the supertype is conflict-free.

This is exactly the expressive power that we need in order to use subtyping

inside type-checking

algorithms, and the cost of this generalized algorithm is only quadratic,

which is as good as the best algorithm we have for the

symmetric case. The result is extremely surprising, since we had previously found that

symmetric inclusion becomes NP-hard as soon as the candidate subtype

is enriched with binary intersection, a generalization that looked much

more innocent than what we achieve here.

**Keyword**
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Databases °

Type Theory
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Databases
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