Dominance constraints are logical descriptions of trees that are widely used in computational linguistics. Their general satisfiability problem is known to be NP-complete. Here we identify normal dominance constraints and present an efficient graph algorithm for testing their satisfiablity in deterministic polynomial time. Previously, no polynomial time algorithm was known.

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, 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 determine the complexity of the entailment problem of FT with existential quantification to be PSPACE-complete, by proving its equivalence to the inclusion problem of non-deterministic finite automata. Our reduction from the entailment problem to the inclusion problem is based on a new alogrithm that, given an existential formula of FT , computes a finite automaton which accepts all its logic consequences.

Extending a subtyping-constraint-based type inference framework with conditional constraints and rows yields a powerful type inference engine. We illustrate this claim by proposing solutions to three delicate type inference problems: \accurate" pattern matchings, record concatenation, and rst-class messages. Until now, known solutions required signicantly different techniques; our theoretical contribution is in using only a single set of tools. On the practical side, this allows all three problems to bene t from a common set of constraint simplication techniques, a formal description of which is given in an appendix. 1 Introduction Type inference is the task of examining a program which lacks some (or even all) type annotations, and recovering enough type information to make it acceptable by a type checker. Its original, and most obvious, application is to free the programmer from the burden of manually providing these annotations, thus making static typing a less dreary d...

The language FT of ordering constraints over feature trees has been introduced as an extension of the system FT of equality constraints over feature trees. While the first-order theory of FT is well understood, only few decidability results are known for the first-order theory of FT . We introduce a new method for proving the decidability of fragments of the first-order theory of FT . This method is based on reduction to second order monadic logic that is decidable according to Rabin's famous tree theorem. The method applies to any fragment of the first-order theory of FT for which one can change the model towards sufficiently labeled feature trees -- a class of trees that we introduce. As we show, the first order-theory of ordering constraints over sufficiently labeled feature trees is equivalent to second-order monadic logic (S2S for infinite and WS2S for finite feature trees). We apply our method for proving that entailment of FT with existential quantifiers j 1 j=9x 1 : : :9x n j 2 is decidable. Previous results were restricted to entailment without existential quantifiers which can be solved in cubic time. Meanwhile, entailment with existential quantifiers has been shown PSPACE-complete (for finite and infinite feature trees respectively).

We study relations on trees defined by first-order constraints over a vocabulary that includes the tree extension relation T T , holding if and only if every branch of T extends to a branch of T , unary node-tests, and a binary relation checking if the domains of two trees are equal. We show that from such a formula one can generate a tree automaton that accepts the set of tuples of trees defined by the formula, and conversely that every automaton over tree-tuples is captured by such a formula. We look at the fragment with only extension inequalities and leaf tests, and show that it corresponds to a new class of automata on tree tuples, which is strictly weaker then general tree-tuple automata. We use the automata representations to show separation and expressibility results for formulae in the logic. We then turn to relational calculi over the logic defined here: that is, from constraints we extend to queries that have second-order parameters for a finite set of tree tuples. We give normal forms for queries, and use these to get bounds on the data complexity of query evaluation, showing that while general query evaluation is unbounded within the polynomial hierarchy, generic query evaluation has very low complexity, giving strong bounds on the expressive power of relational calculi with tree extension constraints. We also give normal forms for safe queries in the calculus.

Entailment of subtype constraints was introduced for constraint simplification in subtype inference systems. Designing an efficient algorithm for subtype entailment turned out to be surprisingly difficult. The situation was clarified by Rehof and Henglein who proved entailment of structural subtype constraints to be coNP-complete for simple types and PSPACE-complete for recursive types. For entailment of non-structural subtype constraints of both simple and recursive types they proved PSPACE-hardness and conjectured PSPACE-completeness but failed in finding a complete algorithm. In this paper, we investigate the source of complications and isolate a natural subproblem of non-structural subtype entailment that we prove PSPACE-complete. We conjecture (but this is left open) that the presented approach can be extended to the general case.

We study relations on trees defined by first-order constraints over a vocabulary that includes the tree extension relation T ≺ T ′ , holding if and only if every branch of T extends to a branch of T ′, unary node-tests, and a binary relation checking if the domains of two trees are equal. We consider both ranked and unranked trees. These are trees with and without a restriction on the number of children of nodes. We adopt the model-theoretic approach to tree relations and study relations definable over the structure consisting of the set of all trees and the above predicates. We relate definability of sets and relations of trees to computability by tree automata. We show that some natural restrictions correspond to familiar logics in the more classical setting, where every tree is a structure over a fixed vocabulary, and to logics studied in the context of XML pattern languages. We then look at relational calculi over collections of trees, and obtain quantifier-restriction results that give us bounds on the expressive power and complexity. As unrestricted relational calculi can express problems complete for each level of the polynomial hierarchy, we look at their restrictions, corresponding to the restricted logics over the family of all unranked trees, and find several calculi with low (NC 1) data complexity, while still expressing properties important for database and

We present a constraint system OF of feature trees that is appropriate to specify and implement type inference for first-class messages. OF extends traditional systems of feature constraints by a selection constraint xhyiz "by first-class feature tree" y, in contrast to the standard selection constraint x[ f ]y "by fixed feature" f . We investigate the satisfiability problem of OF and show that it can be solved in polynomial time, and even in quadratic time in an important special case. We compare OF with Treinen's constraint system EF of feature constraints with first-class features, which has an NP-complete satisfiability problem. This comparison yields that the satisfiability problem for OF with negation is NP-hard. Based on OF we give a simple account of type inference for first-class messages in the spirit of Nishimura's recent proposal, and we show that it has polynomial time complexity: We also highlight an immediate extension that is desirable but makes type inference NP-hard.

. During the last decade, Higher-Order Unification (HOU) has become a popular tool for constructing the semantic representation of natural language expressions. But there is a well-known problem with this approach: it over-generates that is, it produces solutions which although they are mathematically valid, are linguistically incorrect because they do not represent possible meanings of the expression being analysed. In this paper, we argue that Higher-Order Colored Unification (HOCU) can help prevent over-generation and we describe the linguistic, logical and computational aspects of an HOCU--based approach to semantic construction. MOTS-CLS : Unification d'ordre suprieur, Structures de traits, Smantique de la langue naturelle KEY WORDS : Higher-Order Unification, Feature Trees, Natural Language Semantics 1. Introduction Higher-Order Unification (HOU) has become increasingly popular in computational linguistics as a tool for constructing the semantic representation of constructs Te...

The Higher-Order Unification approach to ellipsis presented in [DSP91] lacks an interface with linguistic modules other than semantics. As a result, it cannot handle such phenomena as the many-pronouns puzzle or the distinction between full and pronominal NPs. In this paper, we extend [GK96]'s Higher-Order Coloured Unification approach to handle those cases. 1 Introduction [DSP91, SPD96] (henceforth, DSP) present what is today one of the most influential theories of ellipsis. The basic idea underlying this theory is very simple: ellipses are represented by free variables whose values are then determined using Higher-Order Unification (HOU). For instance, the semantic representation of Jon likes Mary and Peter does too is like(j; m)R(p) and the value of R (the semantic representation of the ellipsis) is determined by the equation like(j; m) = R(j). The process of solving such equations is called Higher-Order Unification and can be stated as follows: given an equation M = N , find a sub...