Semantics of Types for Database Objects

This paper proposes a framework of denotational semantics of database type systems and constructs a type system for complex database objects. Starting with an abstract analysis of the relational model, we develop a mathematical theory for the structures of domains of database objects. Based on this framework, we construct a concrete database type system and its semantic domain. The type system allows arbitrarily complex structures that can be constructed using labeled records, labeled variants, finite sets and recursion. On the semantic domain, in addition to standard operations on records, variants and sets, a join and a projection are available as polymorphically typed computable functions on arbitrarily complex objects. We then show that both the type system and the semantic domain can be uniformly integrated in an ML-like programming language. This leads us to develop a database programming language that supports rich data structures and powerful operations for databases while enjoying desirable features of modern type systems of programming languages including strong static type-checking, static type inference and ML polymorphism

Abstract Interpretation of Polymorphic Higher-Order Functions

This thesis describes several abstract interpretations of polymorphic functions. In all the interpretations, information about any instance of a polymorphic function is obtained from that of the smallest, thus avoiding the computation of the instance directly. This is useful in the case of recursive functions, because it avoids the expensive computation of finding fixed points of functionals corresponding to complex instances. We define an explicitly typed polymorphic language with the Hindley-Milner type system to illustrate our ideas, and provide two semantics of polymorphism that relate separate instances of any polymorphic function. The choice of which semantics to use depends on the particular program analysis we want to study. For studying strictness analysis and binding-time analysis, we introduce a semantics based on embedding-closure pairs. We see how the abstract function of the smallest instance of a polymorphic function is used in building an approximation to that of any instance. Furthermore, we extend the language to include lists, and describe both strictness analysis and binding-time analysis of lists. Thus, this work extends previous work by others, on analyses of polymorphic first-order functions and also of monomorphic higher-order functions, to polymorphic higher-order functions. In relating distinct instances of a polymorphic function, the approximate abstract function is expressed as the greatest lower bound of a set of functions. This may not be very cheap to compute. However, there are often ways of obtaining the same result by considering a smaller set of functions. Another issue concerns how close the approximations are to the exact values. In the first-order case, it is shown that the approximate values coincide with the exact values. In general this is not the case, but experimental results on strictness analysis indicate that good approximations are obtained. Embedding-projection pairs are used to provide a semantics that is convenient for termination analysis of polymorphic functions. We show that the abstract interpretation of an instance can be approximated by the least upper bound of a set of functions that are built from that of the smallest

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later