Determinization and Minimization of Automata for Nested Words Revisited

Resumen

We consider the problem of determinizing and minimizing automata for nested words in practice. For this we compile the nested regular expressions (???????? NREs ) from the usual XPath benchmark to nested word automata (???????? NWAs ). The determinization of these ???????? NWAs , however, fails to produce reasonably small automata. In the best case, huge deterministic ???????? NWAs are produced after few hours, even for relatively small ???????? NREs of the benchmark. We propose a different approach to the determinization of automata for nested words. For this, we introduce stepwise hedge automata (?????? SHA s) that generalize naturally on both (stepwise) tree automata and on finite word automata. We then show how to determinize ?????? SHA s, yielding reasonably small deterministic automata for the ???????? NREs from the XPath benchmark. The size of deterministic ?????? SHA s automata can be reduced further by a novel minimization algorithm for a subclass of ?????? SHA s. In order to understand why the new approach to determinization and minimization works so nicely, we investigate the relationship between ???????? NWAs and ?????? SHA s further. Clearly, deterministic ?????? SHA s can be compiled to deterministic ???????? NWAs in linear time, and conversely ???????? NWAs can be compiled to nondeterministic ?????? SHA s in polynomial time. Therefore, we can use ?????? SHA s as intermediates for determinizing ???????? NWAs , while avoiding the huge size increase with the usual determinization algorithm for ???????? NWAs . Notably, the ???????? NWAs obtained from the ?????? SHA s perform bottom-up and left-to-right computations only, but no top-down computations. This ?????? NWA behavior can be distinguished syntactically by the (weak) single-entry property, suggesting a close relationship between ?????? SHA s and single-entry ???????? NWAs . In particular, it turns out that the usual determinization algorithm for ???????? NWAs behaves well for single-entry ???????? NWAs , while it quickly explodes without the single-entry property. Furthermore, it is known that the class of deterministic multi-module single-entry ???????? NWAs enjoys unique minimization. The subclass of deterministic ?????? SHA s to which our novel minimization algorithm applies is different though, in that we do not impose multiple modules. As further optimizations for reducing the sizes of the constructed ?????? SHA s, we propose schema-based cleaning and symbolic representations based on apply-else rules that can be maintained by determinization. We implemented the optimizations and report the experimental results for the automata constructed for the XPathMark benchmark.