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

Applied Measure Theory for Probabilistic Modeling

Probabilistic programming and statistical computing are vibrant areas in the development of the Julia programming language, but the underlying infrastructure dramatically predates recent developments. The goal of MeasureTheory.jl is to provide Julia with the right vocabulary and tools for these tasks. In the package we introduce a well-chosen set of notions from the foundations of probability together with powerful combinators and transforms, giving a gentle introduction to the concepts in this article. The task is foremost achieved by recognizing measure as the central object. This enables us to develop a proper concept of densities as objects relating measures with each others. As densities provide local perspective on measures, they are the key to efficient implementations. The need to preserve this computationally so important locality leads to the new notion of locally-dominated measure solving the so-called base measure problem and making work with densities and distributions in Julia easier and more flexible

Proof-relevant Horn clauses for dependent type inference and term synthesis

This work has been supported by the EPSRC grant “Coalgebraic Logic Programming for Type Inference” EP/K031864/1-2, EU Horizon 2020 grant “RePhrase: Refactoring Parallel Heterogeneous Resource Aware Applications - a Software Engineering Approach” (ICT-644235), and by COST Action IC1202 (TACLe), supported by COST (European Cooperation in Science and Technology).First-order resolution has been used for type inference for many years, including in Hindley-Milner type inference, type-classes, and constrained data types. Dependent types are a new trend in functional languages. In this paper, we show that proof-relevant first-order resolution can play an important role in automating type inference and term synthesis for dependently typed languages. We propose a calculus that translates type inference and term synthesis problems in a dependently typed language to a logic program and a goal in the proof-relevant first-order Horn clause logic. The computed answer substitution and proof term then provide a solution to the given type inference and term synthesis problem. We prove the decidability and soundness of our method.PostprintPeer reviewe

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

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs