Determinization and Minimization of Automata for Nested Words Revisited

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 (SHAs) that generalize naturally on both (stepwise) tree automata and on finite word automata. We then show how to determinize SHAs, yielding reasonably small deterministic automata for the NREs from the XPath benchmark. The size of deterministic SHAs automata can be reduced further by a novel minimization algorithm for a subclass of SHAs. In order to understand why the new approach to determinization and minimization works so nicely, we investigate the relationship between NWAs and SHAs further. Clearly, deterministic SHAs can be compiled to deterministic NWAs in linear time, and conversely NWAs can be compiled to nondeterministic SHAs in polynomial time. Therefore, we can use SHAs as intermediates for determinizing NWAs, while avoiding the huge size increase with the usual determinization algorithm for NWAs. Notably, the NWAs obtained from the SHAs 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 SHAs 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 SHAs 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 SHAs, 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.

Model-Checking Structured Context-Free Languages

The problem of model checking procedural programs has fostered much research towards the definition of temporal logics for reasoning on context-free structures. The most notable of such results are temporal logics on Nested Words, such as CaRet and NWTL. Recently, the logic OPTL was introduced, based on the class of Operator Precedence Languages (OPL), more powerful than Nested Words. We define the new OPL-based logic POTL, and provide a model checking procedure for it. POTL improves on NWTL by enabling the formulation of requirements involving pre/post-conditions, stack inspection, and others in the presence of exception-like constructs. It improves on OPTL by being FO-complete, and by expressing more easily stack inspection and function-local properties. We developed a model checking tool for POTL, which we experimentally evaluate on some interesting use-cases.

Visibly Pushdown Transducers with Well-Nested Outputs

Visibly pushdown transducers (VPTs) are visibly pushdown automata extended with outputs. They have been introduced oto model transformations of nested words, i.e. words with a call/return structure. When outputs are also structured and well nested words, VPTs are a natural formalism to express tree transformations evaluated in streaming. We prove the class of VPTs with well-nested outputs to be decidable in PTIME. Moreover, we show that this class is closed under composition and that its type-checking against visibly pushdown languages is decidable.

WEIGHTED NESTED WORD AUTOMATA AND LOGICS OVER STRONG BIMONOIDS

Nested words have been introduced by Alur and Madhusudan as a model for e.g. recursive programs or XML documents and have received much recent interest. In this paper, we investigate a quantitative automaton model and a quantitative logic for nested words. The behavior resp. the semantics map nested words to weights which are taken from a strong bimonoid. Strong bimonoids can be viewed as semirings without requiring the distributivity assumption which was essential in the classical theory of formal power series; strong bimonoids include e.g. all bounded lattices and many other structures from multi-valued logics. Our main results show that weighted nested word automata and suitable weighted MSO logics are expressively equivalent. This extends the classical Büchi-Elgot result from words to a weighted setting for nested words.