On subsumption and semiunification in feature algebras

AbstractWe consider a generalization of term subsumption, or matching, to a class of mathematical structures which we call feature algebras. We show how these generalize both first-order terms and the feature structures used in computational linguistics. The notion of term subsumption generalizes to a natural notion of algebra homomorphism. In the setting of feature algebras, unification, corresponds naturally to solving constraints involving equalities between strings of unary function symbols, and semiunification also allows inequalities representing subsumption constraints. Our generalization allows us to show that the semiunification problem for finite feature algebras is undecidable. This implies that the corresponding problem for rational trees (cyclic terms) is also undecidable

On the expressivity of feature logics with negation, functional uncertainty, and sort equations

Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with complements and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom

On the expressivity of feature logics with negation, functional uncertainty, and sort equations

Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with complements and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom

Direct and Expressive Type Inference for the Rank 2 Fragment of System F

This thesis develops a semiunification-based type inference procedure for the rank 2 fragment of System F, with an emphasis on practical considerations for the adoption of such a procedure into existing programming languages. Current semiunification-based rank 2 inference procedures (notably that of Kfoury and Wells) are limited in several ways, which hinder their use in real-world settings. First of all, the translation from an instance of the type inference problem to an instance of the semiunification problem (SUP) is indirect; in particular, because of a series of source-level transformations that take place before translation, the translation is not syntax-directed. As a result, type errors discovered during the semiunification process cannot be cleanly translated back to specific subexpressions of the source program that caused the error. Also, because the rank 2 fragment of System F lacks a principal types property, an inference procedure cannot output a single type that encompasses all of a given term's derivable types. The procedure must therefore either rely on user assistance to produce the right type, or simply choose arbitrarily one of the given term's possible types. The algorithm of Kfoury and Wells in particular makes degenerate type assumptions in the absence of user assistance, and consequently produces types that are of no practical use. We build up our system in stages; we begin by improving the SUP translation. Whereas termination for the Kfoury-Wells rank 2 inference procedure is assured by translating terms into instances of the acyclic semiunification problem (a decidable subset of SUP, which is undecidable in general), we formulate and target the R-acyclic semiunification problem---a larger decidable subset of SUP that facilitates a more concise translation from source terms. We next eliminate the source-level transformation of terms, in order to formulate a truly syntax-directed translation from a source term to a set of SUP-like constraints. In doing so, we find that even the full SUP itself is not sufficiently equipped to support such a translation. We formulate USUP, a superset of SUP that incorporates a new class of identifier we call an unknown. We formulate decidable subsets of USUP, and then formulate a truly syntax-directed translation from source terms into USUP, with guaranteed termination. Finally, to address the principal types problem, we introduce a notation for types in which we allow a particular class of variable to stand for type constructors, rather than ordinary types (an idea based on the so-called third-order lambda-calculus). We show that, with third-order types we can not only output large sets of useful types for a given term, without programmer assistance, but the types we output also generalize over more System F types than any type within System F itself can do, and still be a valid type for the source term. Thus, our system increases opportunities for separate compilation and code reuse beyond any existing system of which we are aware. Our system is sound, though incomplete in certain well-characterized ways, despite which our system performs exactly as one would hope on a variety of examples, which we illustrate in this thesis

Relational extensions to feature logic: applications to constraint based grammars

This thesis investigates the logical and computational foundations of unification-based or more appropriately constraint based grammars. The thesis explores extensions to feature logics (which provide the basic knowledge representation services to constraint based grammars) with multi-valued or relational features. These extensions are useful for knowledge representation tasks that cannot be expressed within current feature logics.The approach bridges the gap between concept languages (such as KL-ONE), which are the mainstay of knowledge representation languages in AI, and feature logics. Va¬ rious constraints on relational attributes are considered such as existential membership, universal membership, set descriptions, transitive relations and linear precedence con¬ straints.The specific contributions of this thesis can be summarised as follows: 1. Development of an integrated feature/concept logic 2. Development of a constraint logic for so called partial set descriptions 3. Development of a constraint logic for expressing linear precedence constraints 4. The design of a constraint language CL-ONE that incorporates the central ideas provided by the above study 5. A study of the application of CL-ONE for constraint based grammarsThe thesis takes into account current insights in the areas of constraint logic programming, object-oriented languages, computational linguistics and knowledge representation

On subsumption and semiunification in feature algebras

AbstractWe consider a generalization of term subsumption, or matching, to a class of mathematical structures which we call feature algebras. We show how these generalize both first-order terms and the feature structures used in computational linguistics. The notion of term subsumption generalizes to a natural notion of algebra homomorphism. In the setting of feature algebras, unification, corresponds naturally to solving constraints involving equalities between strings of unary function symbols, and semiunification also allows inequalities representing subsumption constraints. Our generalization allows us to show that the semiunification problem for finite feature algebras is undecidable. This implies that the corresponding problem for rational trees (cyclic terms) is also undecidable