Subtype satisfiability and entailment

Subtype constraints were introduced in advanced programming language research for designing subtype systems and program analysis algorithms. Two logical problems arise in this context: subtype satisfiability and subtype entailment. Subtype satisfiability underlies subtype inference; subtype entailment is for simplifying subtyping constraints in the same application. In this thesis, we investigate both problems systematically for a number of dialects of subtyping constraint languages that may vary in the following dimensions: types may be simple (finite) or recursive (infinite), type constants may be ordered in lattices or in general partially ordered sets, subtyping can be structural or non-structural, depending on whether least and greatest types are permitted. We use and develop new formal reasoning techniques based on automata, unification, and modal logic. Subtype satisfiability is well understood for all dialects with constants ordered in a lattice. Although cubic time algorithms are given by Palsberg and O\u27Keefe (1995), Pottier (1996), and Palsberg, Wand, and O\u27Keefe (1997), little is known about dialects where constants belong to arbitrary partially ordered sets. We present a uniform treatment to determine the complexities of all these classes. As a consequence, we settle a problem left open by Tiuryn and Wand in 1993 and also subsume complexity bounds given by Wand and Tiuryn (1993), Tiuryn (1992), and Frey (2002). Our results are based on a new connection between modal logic and subtype constraints that we present. Subtype entailment is known to be hard even for simple subtype constraint languages. Rehof and Henglein determined the complexity of structural subtype entailment with type constants ordered in a lattice. They proved coNP-completeness for simple types (1997) and PSPACE-completeness for recursive types (1998). Furthermore, they showed that non-structural subtype entailment is PSPACE-hard and is conjectured PSPACE-complete for the case with only two type constants for the least and greatest types respectively (1998). Yet the problem still remains open today. We argue that the difficulty occurs due to e ects linked to non-regular word languages. In order to do so, we precisely characterize subtype entailment by finite word automata with word equations. This characterization induces new results on non-structural subtype entailment, constituting a promising starting point for future investigation on decidability.Diese Arbeit untersucht zwei logische Probleme der programmiersprachlichen Typinferenz: Erfüllbarkeit und Subsumption von Teiltyp-Constraints. Wir untersuchen diese Probleme systematisch für eine Reihe von Constraintsprachen. Dabei greifen wir auf Methoden der computationalen Logik, Unifikations- und Automatentheorie zurück. Teiltyp-Erfüllbarkeit ist für den Fall wohl verstanden, dass die Typkonstanten in einem Verband angeordnet sind (Palsberg und O\u27Keefe (1995), Pottier (1996), Palsberg, Wand und O\u27Keefe (1997)). Der allgemeinere Fall mit beliebig angeordneten Konstanten wurde bislang weniger untersucht. Wir stellen einen ersten universellen Ansatz vor, indem wir erstmals einen Zusammenhang zwischen Teiltyp-Constraints und Modallogik aufzeigen. Dadurch lösen wir unter Anderem ein seit 1993 offenes Komplexitätsproblem von Wand und Tiuryn. Teiltyp-Subsumption ist selbst für einfachste Constraintsprachen von hoher Komplexität. Rehof und Henglein zeigten dies für den strukturellen Verbandsfall (mit zwei Typkonstanten 1997, 1998), ließen jedoch den nicht-strukturellen Fall offen. In dieser Arbeit betrachten wir den einfachsten nicht-strukturellen Fall. Hier zeigen wir, dass versteckte Wortgleichungen neue Schwierigkeiten verursachen. Hierzu charakterisieren wir Teiltyp-Subsumption durch spezielle endliche Automaten mit Wortgleichungen. Unsere Charakterisierung liefert partielle Entscheidbarkeitsresulte zur nichtstrukturellen Teiltyp-Subsumption und kann als Grundlage für künftige Untersuchungen dienen

Type Inference with Bounded Quantification

In this thesis we study some of the problems which occur when type inference is used in a type system with subtyping. An underlying poset of atomic types is used as a basis for our subtyping systems. We argue that the class of Helly posets is of significant interest, as it includes lattices and trees, and is closed under type formation not only with structural constructors such as function space and list, but also records, tagged variants, Abadi-Cardelli object constructors, top and bottom. We develop a general theory relating consistency, solvability, and solution of sets of constraints between regular types built over Helly posets with these constructors, and introduce semantic notions of simplification and entailment for sets of constraints over Helly posets of base types. We extend Helly posets with inequalities of the form a <= tau, where tau is not necessarily atomic, and show how this enables us to deal with bounded quantification. Using bounded quantification we define a subtyping system which combines structural subtype polymorphism and predicative parametric polymorphism, and use this to extend with subtyping the type system of Laufer and Odersky for ML with type annotations. We define a complete algorithm which infers minimal types for our extension, using factorisations, solutions of subtyping problems analogous to principal unifiers for unification problems. We give some examples of typings computed by a prototype implementation

Subtyping constraints in quasi-lattices

In this report, we show the decidability and NP-completeness of the satisfiability problem for non-structural subtyping constraints in quasi-lattices. This problem, first introduced by Smolka in 1989, is important for the typing of logic and functional languages. The decidability result is obtained by generalizing Trifonov and Smith's algorithm over lattices, to the case of quasi-lattices. Similarly, we extend Pottier's algorithm for computing explicit solutions to the case of quasi-lattices. Finally we evoke some applications of these results to type inference in constraint logic programming and functional programming languages

Classification-based phrase structure grammar: an extended revised version of HPSG

This thesis is concerned with a presentation of Classification -based Phrase Structure Grammar (or cPSG), a grammatical theory that has grown out of extensive revisions of, and extensions to, HPSG. The fundamental difference between this theory and HPSG concerns the central role that classification plays in the grammar: the grammar classifies strings, according to their feature structure descriptions, as being of various types. Apart from the role of classification, the theory bears a close resemblance to HPSG, though it is by no means a direct translation, including numerous revisions and extensions. A central goal in the development of the theory has been its computational implementation, which is included in the thesis.The presentation may be divided into four parts. In the first, chapters 1 and 2, we present the grammatical formalism within which the theory is stated. This consists of a development of the notion of a classificatory system (chapter 1), and the incorporation of hierarchality into that notion (chapter 2).The second part concerns syntactic issues. Chapter 3 revises the HPSG treatment of specifiers, complements and adjuncts, incorporating ideas that specifiers and complements should be distinguished and presenting a treatment of adjuncts whereby the head is selected for by the adjunct. Chapter 4 presents several options for an account of unbounded dependencies. The accounts are based loosely on that of GPSG, and a reconstruction of GPSG's Foot Feature Principle is presented which does not involve a notion of default. Chapter 5 discusses coordination, employing an extension of Rounds- Kasper logic to allow a treatment of cross -categorial coordination.In the third part, chapters 6, 7 and 8, we turn to semantic issues. We begin (Chapter 6) with a discussion of Situation Theory, the background semantic theory, attempting to establish a precise and coherent version of the theory within which to work. Chapter 7 presents the bulk of the treatment of semantics, and can be seen as an extensive revision of the HPSG treatment of semantics. The aim is to provide a semantic treatment which is faithful to the version of Situation Theory presented in Chapter 6. Chapter 8 deals with quantification, discussing the nature of quantification in Situation Theory before presenting a treatment of quantification in CPSG. Some residual questions about the semantics of coordinated noun phrases are also addressed in this chapter.The final part, Chapter 9, concerns the actual computational implementation of the theory. A parsing algorithm based on hierarchical classification is presented, along with four strategies that might be adopted given that algorithm. Also discussed are some implementation details. A concluding chapter summarises the arguments of the thesis and outlines some avenues for future research

Adaptive Constraint Solving for Information Flow Analysis

In program analysis, unknown properties for terms are typically represented symbolically as variables. Bound constraints on these variables can then specify multiple optimisation goals for computer programs and nd application in areas such as type theory, security, alias analysis and resource reasoning. Resolution of bound constraints is a problem steeped in graph theory; interdependencies between the variables is represented as a constraint graph. Additionally, constants are introduced into the system as concrete bounds over these variables and constants themselves are ordered over a lattice which is, once again, represented as a graph. Despite graph algorithms being central to bound constraint solving, most approaches to program optimisation that use bound constraint solving have treated their graph theoretic foundations as a black box. Little has been done to investigate the computational costs or design e cient graph algorithms for constraint resolution. Emerging examples of these lattices and bound constraint graphs, particularly from the domain of language-based security, are showing that these graphs and lattices are structurally diverse and could be arbitrarily large. Therefore, there is a pressing need to investigate the graph theoretic foundations of bound constraint solving. In this thesis, we investigate the computational costs of bound constraint solving from a graph theoretic perspective for Information Flow Analysis (IFA); IFA is a sub- eld of language-based security which veri es whether con dentiality and integrity of classified information is preserved as it is manipulated by a program. We present a novel framework based on graph decomposition for solving the (atomic) bound constraint problem for IFA. Our approach enables us to abstract away from connections between individual vertices to those between sets of vertices in both the constraint graph and an accompanying security lattice which defines ordering over constants. Thereby, we are able to achieve significant speedups compared to state-of-the-art graph algorithms applied to bound constraint solving. More importantly, our algorithms are highly adaptive in nature and seamlessly adapt to the structure of the constraint graph and the lattice. The computational costs of our approach is a function of the latent scope of decomposition in the constraint graph and the lattice; therefore, we enjoy the fastest runtime for every point in the structure-spectrum of these graphs and lattices. While the techniques in this dissertation are developed with IFA in mind, they can be extended to other application of the bound constraints problem, such as type inference and program analysis frameworks which use annotated type systems, where constants are ordered over a lattice

Advanced flow-based type systems for object-oriented languages

Ph.DDOCTOR OF PHILOSOPH

Constraint Satisfaction Techniques for Combinatorial Problems

The last two decades have seen extraordinary advances in tools and techniques for constraint satisfaction. These advances have in turn created great interest in their industrial applications. As a result, tools and techniques are often tailored to meet the needs of industrial applications out of the box. We claim that in the case of abstract combinatorial problems in discrete mathematics, the standard tools and techniques require special considerations in order to be applied effectively. The main objective of this thesis is to help researchers in discrete mathematics weave through the landscape of constraint satisfaction techniques in order to pick the right tool for the job. We consider constraint satisfaction paradigms like satisfiability of Boolean formulas and answer set programming, and techniques like symmetry breaking. Our contributions range from theoretical results to practical issues regarding tool applications to combinatorial problems. We prove search-versus-decision complexity results for problems about backbones and backdoors of Boolean formulas. We consider applications of constraint satisfaction techniques to problems in graph arrowing (specifically in Ramsey and Folkman theory) and computational social choice. Our contributions show how applying constraint satisfaction techniques to abstract combinatorial problems poses additional challenges. We show how these challenges can be addressed. Additionally, we consider the issue of trusting the results of applying constraint satisfaction techniques to combinatorial problems by relying on verified computations

Efficient Content-based Routing, Mobility-aware Topologies, and Temporal Subspace Matching

Event-based systems are seen as good candidates for supporting distributed applications in dynamic and ubiquitous environments because they support decoupled and asynchronous many-to-many information dissemination. Event systems are widely used, because asynchronous messaging provides a flexible alternative to RPC (Remote Procedure Call). They are typically implemented using an overlay network of routers. A content-based router forwards event messages based on filters that are installed by subscribers and other routers. The filters are organized into a routing table in order to forward incoming events to proper subscribers and neighbouring routers. This thesis addresses the optimization of content-based routing tables organized using the covering relation and presents novel data structures and configurations for improving local and distributed operation. Data structures are needed for organizing filters into a routing table that supports efficient matching and runtime operation. We present novel results on dynamic filter merging and the integration of filter merging with content-based routing tables. In addition, the thesis examines the cost of client mobility using different protocols and routing topologies. We also present a new matching technique called temporal subspace matching. The technique combines two new features. The first feature, temporal operation, supports notifications, or content profiles, that persist in time. The second feature, subspace matching, allows more expressive semantics, because notifications may contain intervals and be defined as subspaces of the content space. We also present an application of temporal subspace matching pertaining to metadata-based continuous collection and object tracking.Tapahtumapohjaiset järjestelmät nähdään hyvänä tapana tukea ja kehittää hajautettuja sovelluksia dynaamisissa ympäristöissä. Nämä järjestelmät tukevat asynkronista viestien välitystä. Tapahtumapohjaisia järjestelmiä käytetään, koska asynkroninen viestintä mahdollistaa etäproseduurikutsuja vapaammat sidokset sovellusten välille. Tapahtumapohjaiset järjestelmät toteutetaan tyypillisesti ns. "overlay" verkkoina sovelluskerroksella. Sisältöpohjainen reititin välittää tapahtumaviestejä tilaajien asettamien suotimien (eng. filter) perusteella. Tiedon tilaajat ja tuottajat kytketään suotimien avulla niin että tuottajien tuottamat tapahtumat välittyvät aktiivisille tilaajille. Suodin valikoi viestivirrasta halutut viestit erityisten sääntöjen avulla. Suotimet järjestetään reititystauluksi, jonka perusteella päätetään kenelle reititin ohjaa viestejä. Väitöskirja käsittelee suodinpohjaisten reititystaulujen optimointia ja esittää uusia tietorakenteita ja konfiguraatioita paikalliseen sekä hajautettuun toimintaan. Työssä esitetään yleinen formaali suotimien yhdistämismalli, joka integroidaan esitettyjen tietorakenteiden kanssa. Lisäksi työssä tutkitaan liikkuvien tilaajien ja tuottajien aiheuttamia kustannuksia. Työssä esitetään myös uusi tekniikka aikaan kytketyn tiedon välitykseen. Tekniikka yhdistää kaksi uutta piirrettä. Ensimmäinen piirre on temporaalinen toiminta, jossa välitettävä tieto on määritetty olemaan voimassa tietyn ajanjakson. Toinen ominaisuus mahdollistaa sekä kyselyiden että datan, johon kyselyt kohdistuvat, määrittelyn moniulotteisten suotimien avulla. Työssä esitetään tekniikasta esimerkkisovellus, joka käsittelee joukkojen ja olioiden seurantaa