Abstract not found

We show that the first-order theory of structural subtyping of non-recursive types is decidable. Let Σ be a language consisting of function symbols (representing type constructors) and C a decidable structure in the relational language L containing a binary relation ≤. C represents primitive types; ≤ represents a subtype ordering. We introduce the notion of Σ-term-power of C, which generalizes the structure arising in structural subtyping. The domain of the Σ-term-power of C is the set of Σ-terms over the set of elements of C. We show that the decidability of the first-order theory of C implies the decidability of the first-order theory of the Σterm-power of C. This result implies the decidability of the first-order theory of structural subtyping of non-recursive types.

Feature trees are the formal basis for algorithms manipulating record like structures in constraint programming, computational linguistics and in concrete applications like software configuration management. Feature trees model records, and constraints over feature trees yield extensible and modular record descriptions. We introduce the constraint system FT of ordering constraints interpreted over feature trees. Under the view that feature trees represent symbolic information, the relation corresponds to the information ordering ("carries less information than"). We present two algorithms in cubic time, one for the satisfiability problem and one for the entailment problem of FT . We show that FT has the independence property. We are thus able to handle negative conjuncts via entailment and obtain a cubic algorithm that decides the satisfiability of conjunctions of positive and negated ordering constraints over feature trees. Furthermore, we reduce the satisfiability problem of Dorre's weak subsumption constraints to the satisfiability problem of FT and improve the complexity bound for solving weak subsumption constraints from O(n^5) to O(n³).

Decidability of non-structural subtype entailment is a long-standing open problem in programming language theory. In this paper, we apply automata theoretic methods to characterize the problem equivalently by using regular expressions and word equations. This characterization induces new results on non-structural subtype entailment, constitutes a promising starting point for further investigations on decidability, and explains for the first time why the problem is so difficult. The difficulty is caused by implicit word equations that we make explicit.

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

Abstract Equality constraints (unification constraints) have widespread use in program analysis, most notably in static polymorphic type systems. Conditional equality constraints extend equality constraints with a weak form of subtyping to allow for more accurate analyses. We give a complete complexity characterization of the various entailment problems for conditional equality constraints and for a natural extension of conditional equality constraints.

Unranked trees, that is, trees with no restriction on the number of children of nodes, have recently attracted much attention, primarily as an abstraction of XML documents. In this paper, we study logical definability over unranked trees, as well as collections of unranked trees, that can be viewed as databases of XML documents. The traditional approach to definability is to view each tree as a structure of a fixed vocabulary, and study the expressive power of various logics on trees. A different approach, based on model theory, considers a structure whose universe is the set of all trees, and studies definable sets and relations; this approach extends smoothly to the setting of definability over collections of trees. We study the latter, model-theoretic approach. We find sets of operations on unranked trees that define regular tree languages, and show that some natural restrictions correspond to logics studied in the context of XML pattern languages. We then look at relational calculi over collections of unranked trees, and obtain quantifierrestriction 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 ) data complexity, that can express important XML properties like DTD validation and XPath evaluation.