Data Abstraction and General Recursion

Existing approaches to semantics of algebraically specified data types such as Initial Algebra Semantics and Final Algebra Semantics do not take into account the possibility of general recursion and hence non-termination in the ambient programming language. Any technical development of this problem needs to be in the setting of domain theory. In this paper we present extensions of initial and final algebra semantics to algebras with an underlying domain structure. Four possibilities for specification methodologies arise: two each in the Initial and Final algebra paradigms. We demonstrate that the initial/final objects (as appropriate) exist in all four situations. The final part of the paper attempts to explicate the notion of abstractness of ADT\u27s by defining a notion of operational semantics for ADT\u27s, and then studying the relationship between the various algebraic-semantics proposed and the operational semantics

Resumptions, Weak Bisimilarity and Big-Step Semantics for While with Interactive I/O: An Exercise in Mixed Induction-Coinduction

We look at the operational semantics of languages with interactive I/O through the glasses of constructive type theory. Following on from our earlier work on coinductive trace-based semantics for While, we define several big-step semantics for While with interactive I/O, based on resumptions and termination-sensitive weak bisimilarity. These require nesting inductive definitions in coinductive definitions, which is interesting both mathematically and from the point-of-view of implementation in a proof assistant. After first defining a basic semantics of statements in terms of resumptions with explicit internal actions (delays), we introduce a semantics in terms of delay-free resumptions that essentially removes finite sequences of delays on the fly from those resumptions that are responsive. Finally, we also look at a semantics in terms of delay-free resumptions supplemented with a silent divergence option. This semantics hinges on decisions between convergence and divergence and is only equivalent to the basic one classically. We have fully formalized our development in Coq.Comment: In Proceedings SOS 2010, arXiv:1008.190

Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201

Types for X10 Clocks

X10 is a modern language built from the ground up to handle future parallel systems, from multicore machines to cluster configurations. We take a closer look at a pair of synchronisation mechanisms: finish and clocks. The former waits for the termination of parallel computations, the latter allow multiple concurrent activities to wait for each other at certain points in time. In order to better understand these concepts we study a type system for a stripped down version of X10. The main result assures that well typed programs do not run into the errors identified in the X10 language reference, namely the ClockUseException. The study will open, we hope, doors to a more flexible utilisation of clocks in the X10 language.Comment: In Proceedings PLACES 2010, arXiv:1110.385

Termination Casts: A Flexible Approach to Termination with General Recursion

This paper proposes a type-and-effect system called Teqt, which distinguishes terminating terms and total functions from possibly diverging terms and partial functions, for a lambda calculus with general recursion and equality types. The central idea is to include a primitive type-form "Terminates t", expressing that term t is terminating; and then allow terms t to be coerced from possibly diverging to total, using a proof of Terminates t. We call such coercions termination casts, and show how to implement terminating recursion using them. For the meta-theory of the system, we describe a translation from Teqt to a logical theory of termination for general recursive, simply typed functions. Every typing judgment of Teqt is translated to a theorem expressing the appropriate termination property of the computational part of the Teqt term.Comment: In Proceedings PAR 2010, arXiv:1012.455

Constraint Handling Rules with Binders, Patterns and Generic Quantification

Constraint Handling Rules provide descriptions for constraint solvers. However, they fall short when those constraints specify some binding structure, like higher-rank types in a constraint-based type inference algorithm. In this paper, the term syntax of constraints is replaced by $\lambda$-tree syntax, in which binding is explicit; and a new $\nabla$ generic quantifier is introduced, which is used to create new fresh constants.Comment: Paper presented at the 33nd International Conference on Logic Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017 16 pages, LaTeX, no PDF figure