Proof-Relevant Resolution for Elaboration of Programming Languages

Proof-relevant resolution is a new variant of resolution in Horn-clause logic and its extensions. We propose proof-relevant resolution as a systematic approach to elaboration in programming languages that is close to formal specification and hence allows for analysis of semantics of the language. We demonstrate the approach on two case studies; we describe a novel, proof-relevant approach to type inference and term synthesis in dependently types languages and we show how proof-relevant resolution allows for analysis of inductive and coinductive soundness of type class resolution. We conclude by a discussion of overall contributions of our current work

Elaboration in Dependent Type Theory

To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary mathematical text, and resolving ambiguities in mathematical expressions. We refer to the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one as elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean's elaborator supports higher-order unification, type class inference, ad hoc overloading, insertion of coercions, the use of tactics, and the computational reduction of terms. The interactions between these components are subtle and complex, and the elaboration algorithm has been carefully designed to balance efficiency and usability. We describe the central design goals, and the means by which they are achieved

Trust, but Verify: Two-Phase Typing for Dynamic Languages

A key challenge when statically typing so-called dynamic languages is the ubiquity of value-based overloading, where a given function can dynamically reflect upon and behave according to the types of its arguments. Thus, to establish basic types, the analysis must reason precisely about values, but in the presence of higher-order functions and polymorphism, this reasoning itself can require basic types. In this paper we address this chicken-and-egg problem by introducing the framework of two-phased typing. The first "trust" phase performs classical, i.e. flow-, path- and value-insensitive type checking to assign basic types to various program expressions. When the check inevitably runs into "errors" due to value-insensitivity, it wraps problematic expressions with DEAD-casts, which explicate the trust obligations that must be discharged by the second phase. The second phase uses refinement typing, a flow- and path-sensitive analysis, that decorates the first phase's types with logical predicates to track value relationships and thereby verify the casts and establish other correctness properties for dynamically typed languages

Dependent Types for Pragmatics

This paper proposes the use of dependent types for pragmatic phenomena such as pronoun binding and presupposition resolution as a type-theoretic alternative to formalisms such as Discourse Representation Theory and Dynamic Semantics.Comment: This version updates the paper for publication in LEU

Formal logic: Classical problems and proofs

Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational

Proof Relevant Corecursive Resolution

Resolution lies at the foundation of both logic programming and type class context reduction in functional languages. Terminating derivations by resolution have well-defined inductive meaning, whereas some non-terminating derivations can be understood coinductively. Cycle detection is a popular method to capture a small subset of such derivations. We show that in fact cycle detection is a restricted form of coinductive proof, in which the atomic formula forming the cycle plays the role of coinductive hypothesis. This paper introduces a heuristic method for obtaining richer coinductive hypotheses in the form of Horn formulas. Our approach subsumes cycle detection and gives coinductive meaning to a larger class of derivations. For this purpose we extend resolution with Horn formula resolvents and corecursive evidence generation. We illustrate our method on non-terminating type class resolution problems.Comment: 23 pages, with appendices in FLOPS 201

A practical approach to goal modelling for time-constrained projects

Goal modelling is a well known rigorous method for analysing problem rationale and developing requirements. Under the pressures typical of time-constrained projects its benefits are not accessible. This is because of the effort and time needed to create the graph and because reading the results can be difficult owing to the effects of crosscutting concerns. Here we introduce an adaptation of KAOS to meet the needs of rapid turn around and clarity. The main aim is to help the stakeholders gain an insight into the larger issues that might be overlooked if they make a premature start into implementation. The method emphasises the use of obstacles, accepts under-refined goals and has new methods for managing crosscutting concerns and strategic decision making. It is expected to be of value to agile as well as traditional processes