Abstract
The system FT< of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. We investigate the first-order theory of FT< and its fragments in detail, both over finite trees and over possibly infinite trees. We prove that the first-order theory of FT< is undecidable, in contrast to the first-order theory of FT which is well-known to be decidable. We show that the entailment problem of FT< with existential quantification is PSPACE-complete. So far, this problem has been shown decidable, coNP-hard in case of finite trees, PSPACE-hard in case of arbitrary trees, and cubic time when restricted to quantifier-free entailment judgments. To show PSPACE-completeness, we show that the entailment problem of FT< with existential quantification is equivalent to the inclusion problem of non-deterministic finite automata. Available at http://www.ps.uni-saarland.de/Publications/documents/FTSubTheory_98.pdf
Highlights
4 Undecidability Results 1365–8050 c 2001 Maison de l’Informatique et des Mathématiques Discrètes (MIMD), Paris, FranceMartin Müller and Joachim Niehren and Ralf TreinenFeature constraints have been used for describing records in constraint programming [1, 30, 31, 36] and record-like structures in computational linguistics [14, 12, 23, 26]
Following [2, 4, 3], we consider feature constraints as predicate logic formulas interpreted in the structure of feature trees
There, the non-satisfiability of a weak subsumption constraint φ was equivalent to the fact that two labeling path constraints ax π℄μ and bx π℄μ for different label symbols a and b are entailed by φ, which could be checked by inspection of the automaton that describes all the labeling path constraints entailed by φ
Summary
1365–8050 c 2001 Maison de l’Informatique et des Mathématiques Discrètes (MIMD), Paris, France. The constraints of FT are interpreted in the structure of feature trees with the weak subsumption ordering. Once the undecidability of the first-order theory of FT is settled, it remains to distinguish decidable fragments and their complexity The application domains of ordering constraints over feature trees are quite diverse. They have been used to describe so-called coordination phenomena in natural language [7] and for the analysis of concurrent constraint programming languages [20]. We assume an infinite set V of variables ranged over by x y z, a set F of at least two features ranged over by f g and a set L of labels ranged over by a b
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