arXiv subjects

Brigitte Pientka

Publications and source records attributed to Brigitte Pientka.

28 records · Page 2Linked to original sources

arXiv2022

A Category Theoretic View of Contextual Types: from Simple Types to Dependent Types

We describe the categorical semantics for a simply typed variant and a simplified dependently typed variant of Cocon, a contextual modal type theory where the box modality mediates between the weak function space that is used to represent higher-order abstract syntax (HOAS) trees and the strong function space that describes (recursive) computations about them. What makes Cocon different from standard type theories is the presence of first-class contexts and contextual objects to describe syntax trees that are closed with respect to a given context of assumptions. Following M. Hofmann's work, we use a presheaf model to characterise HOAS trees. Surprisingly, this model already provides the necessary structure to also model Cocon. In particular, we can capture the contextual objects of Cocon using a comonad $\flat$ that restricts presheaves to their closed elements. This gives a simple semantic characterisation of the invariants of contextual types (e.g. substitution invariance) and identifies Cocon as a type-theoretic syntax of presheaf models. We further extend this characterisation to dependent types using categories with families and show that we can model a fragment of Cocon without recursor in the Fitch-style dependent modal type theory presented by Birkedal et. al..

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arXiv2021

Moebius: Metaprogramming using Contextual Types -- The stage where System F can pattern match on itself (Long Version)

We describe the foundation of the metaprogramming language, Moebius, which supports the generation of polymorphic code and, more importantly the analysis of polymorphic code via pattern matching. Moebius has two main ingredients: 1) we exploit contextual modal types to describe open code together with the context in which it is meaningful. In Moebius, open code can depend on type and term variables (level 0) whose values are supplied at a later stage, as well as code variables (level 1) that stand for code templates supplied at a later stage. This leads to a multi-level modal lambda-calculus that supports System-F style polymorphism and forms the basis for polymorphic code generation. 2) we extend the multi-level modal lambda-calculus to support pattern matching on code. As pattern matching on polymorphic code may refine polymorphic type variables, we extend our type-theoretic foundation to generate and track typing constraints that arise. We also give an operational semantics and prove type preservation. Our multi-level modal foundation for Moebius provides the appropriate abstractions for both generating and pattern matching on open code without committing to a concrete representation of variable binding and contexts. Hence, our work is a step towards building a general type-theoretic foundation for multi-staged metaprogramming that, on the one hand, enforces strong type guarantees and, on the other hand, makes it easy to generate and manipulate code. This will allow us to exploit the full potential of metaprogramming without sacrificing the reliability of and trust in the code we are producing and running.

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arXiv2019

A Type Theory for Defining Logics and Proofs

We describe a Martin-Löf-style dependent type theory, called Cocon, that allows us to mix the intensional function space that is used to represent higher-order abstract syntax (HOAS) trees with the extensional function space that describes (recursive) computations. We mediate between HOAS representations and computations using contextual modal types. Our type theory also supports an infinite hierarchy of universes and hence supports type-level computation thereby providing metaprogramming and (small-scale) reflection. Our main contribution is the development of a Kripke-style model for Cocon that allows us to prove normalization. From the normalization proof, we derive subject reduction and consistency. Our work lays the foundation to incorporate the methodology of logical frameworks into systems such as Agda and bridges the longstanding gap between these two worlds.

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arXiv2019

Cocon: Computation in Contextual Type Theory

We describe a Martin-Löf style dependent type theory, called Cocon, that allows us to mix the intensional function space that is used to represent higher-order abstract syntax (HOAS) trees with the extensional function space that describes (recursive) computations. We mediate between HOAS representations and computations using contextual modal types. Our type theory also supports an infinite hierarchy of universes and hence supports type-level computation -- thereby providing metaprogramming and (small-scale) reflection. Our main contribution is the development of a Kripke-style model for Cocon that allows us to prove normalization. From the normalization proof, we derive subject reduction and consistency. Our work lays the foundation to incorporate the methodology of logical frameworks into systems such as Agda and bridges the longstanding gap between these two worlds.

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arXiv2018

Index-Stratified Types (Extended Version)

We present Tores, a core language for encoding metatheoretic proofs. The novel features we introduce are well-founded Mendler-style (co)recursion over indexed data types and a form of recursion over objects in the index language to build new types. The latter, which we call index-stratified types, are analogue to the concept of large elimination in dependently typed languages. These features combined allow us to encode sophisticated case studies such as normalization for lambda calculi and normalization by evaluation. We prove the soundness of Tores as a programming and proof language via the key theorems of subject reduction and termination.

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arXiv2015

An Open Challenge Problem Repository for Systems Supporting Binders

A variety of logical frameworks support the use of higher-order abstract syntax in representing formal systems; however, each system has its own set of benchmarks. Even worse, general proof assistants that provide special libraries for dealing with binders offer a very limited evaluation of such libraries, and the examples given often do not exercise and stress-test key aspects that arise in the presence of binders. In this paper we design an open repository ORBI (Open challenge problem Repository for systems supporting reasoning with BInders). We believe the field of reasoning about languages with binders has matured, and a common set of benchmarks provides an important basis for evaluation and qualitative comparison of different systems and libraries that support binders, and it will help to advance the field.

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arXiv2015

A Case Study on Logical Relations using Contextual Types

Proofs by logical relations play a key role to establish rich properties such as normalization or contextual equivalence. They are also challenging to mechanize. In this paper, we describe the completeness proof of algorithmic equality for simply typed lambda-terms by Crary where we reason about logically equivalent terms in the proof environment Beluga. There are three key aspects we rely upon: 1) we encode lambda-terms together with their operational semantics and algorithmic equality using higher-order abstract syntax 2) we directly encode the corresponding logical equivalence of well-typed lambda-terms using recursive types and higher-order functions 3) we exploit Beluga's support for contexts and the equational theory of simultaneous substitutions. This leads to a direct and compact mechanization, demonstrating Beluga's strength at formalizing logical relations proofs.

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arXiv2015

The Next 700 Challenge Problems for Reasoning with Higher-Order Abstract Syntax Representations: Part 1-A Common Infrastructure for Benchmarks

A variety of logical frameworks support the use of higher-order abstract syntax (HOAS) in representing formal systems. Although these systems seem superficially the same, they differ in a variety of ways; for example, how they handle a context of assumptions and which theorems about a given formal system can be concisely expressed and proved. Our contributions in this paper are three-fold: 1) we develop a common infrastructure for representing benchmarks for systems supporting reasoning with binders, 2) we present several concrete benchmarks, which highlight a variety of different aspects of reasoning within a context of assumptions, and 3) we design an open repository ORBI, (Open challenge problem Repository for systems supporting reasoning with BInders). Our work sets the stage for providing a basis for qualitative comparison of different systems. This allows us to review and survey the state of the art, which we do in great detail for four systems in Part 2 of this paper (Felty et al, 2015). It also allows us to outline future fundamental research questions regarding the design and implementation of meta-reasoning systems.

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arXiv2011

Multi-level Contextual Type Theory

Contextual type theory distinguishes between bound variables and meta-variables to write potentially incomplete terms in the presence of binders. It has found good use as a framework for concise explanations of higher-order unification, characterize holes in proofs, and in developing a foundation for programming with higher-order abstract syntax, as embodied by the programming and reasoning environment Beluga. However, to reason about these applications, we need to introduce meta^2-variables to characterize the dependency on meta-variables and bound variables. In other words, we must go beyond a two-level system granting only bound variables and meta-variables. In this paper we generalize contextual type theory to n levels for arbitrary n, so as to obtain a formal system offering bound variables, meta-variables and so on all the way to meta^n-variables. We obtain a uniform account by collapsing all these different kinds of variables into a single notion of variabe indexed by some level k. We give a decidable bi-directional type system which characterizes beta-eta-normal forms together with a generalized substitution operation.

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arXiv2010

Explicit Substitutions for Contextual Type Theory

In this paper, we present an explicit substitution calculus which distinguishes between ordinary bound variables and meta-variables. Its typing discipline is derived from contextual modal type theory. We first present a dependently typed lambda calculus with explicit substitutions for ordinary variables and explicit meta-substitutions for meta-variables. We then present a weak head normalization procedure which performs both substitutions lazily and in a single pass thereby combining substitution walks for the two different classes of variables. Finally, we describe a bidirectional type checking algorithm which uses weak head normalization and prove soundness.

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