Type-based termination of recursive definitions

This paper introduces "lambda-hat", a simply typed lambda calculus supporting inductive types and recursive function definitions with termination ensured by types. The system is shown to enjoy subject reduction, strong normalisation of typable terms and to be stronger than a related system "lambda-G" in which termination is ensured by a syntactic guard condition. The system can, at will, be extended to also support coinductive types and corecursive function definitions.Information Society Technologies (IST) - Fifth Framework Programm (FP5) - TYPES.Fundação para a Ciência e a Tecnologia (FCT) – PRAXIS XXI/C/EEI/14172/98.INRIA-ICCTI.Estonian Science Foundation (ETF) - grant no. 4155

Type-based termination of recursive definitions and constructor subtyping in typed lambda calculi

Tese de doutoramento em Informática, ramo de Fundamentos da Computação.In type systems, a combination of subtyping and overloading is a way to achieve more precise typings. This thesis explores how to use these mechanisms in two directions: (i) as a way to ensure termination of recursive functions; (ii) as a way to capture in a type-theoretic context the use of subtyping as inclusion between inductively defined sets. The first part of the thesis presents a mechanism that ensures termination through types and defines a system that incorporates it. More precisely, we formalize the notion of type-based termination using a restricted form of type dependency (also known as indexed types). Every datatype is replaced by a family of approximations indexed over a set of stages; then being in a certain approximation means that a term can be seen as having a certain bound on constructor usage. We introduce λˆ, a simply typed λ-calculus à la Curry, supporting parametric inductive datatypes, case-expressions and letrec-expressions with termination ensured by types. We show that λˆ enjoys important meta-theoretical properties, including confluence, subject reduction and strong normalization. We also show that the calculus is powerful enough to encode many recursive definitions rejected by existing type systems, and give some examples. We prove that this system encompasses in a strict way Giménez' λς, a system in which termination of typable expressions is ensured by a syntactical condition constraining the uses of recursive calls in the body of definitions. The second part of the thesis studies properties of a type system featuring constructor subtyping. Constructor subtyping is a form of subtyping in which an inductive type σ is viewed as a subtype of another inductive type τ if each constructor c of σ is also a constructor of τ (but τ may have more constructors), and whenever c : θ'→σ is a declaration for τ, then c : θ'→τ is a declaration for τ with θ'→≤θ'. In this thesis we allow for this form of subtyping in the system λcs, which is a simply typed λ-calculus à la Curry, supporting mutually recursive parametric datatypes, case-expressions and letrec-expressions. We establish the properties of confluence, subject reduction and decidability of type checking for this calculus. As the system features general recursion, the reduction calculus is obviously non-terminating. However, we sketch two ways of achieving strong normalization. One way is to constrain the system to guard-by-destructors recursion, following what is done for λς. The other way is to enrich the type system with stages (following the ideas presented for λˆ) and enforcing termination through typing. Potential uses of constructor subtyping include proof assistants and functional programming languages. In particular, constructor subtyping provides a suitable foundation for extensible datatypes, and is specially adequate to re-usability. The combination of subtyping between datatypes and overloading of constructors allows the definition of new datatypes by restricting or by expanding the set of constructors of an already defined datatype. This flexibility in the definition of datatypes induces a convenient form of code reuse for recursive functions, allowing the definition of new functions by restricting or by expanding already defined ones. We enrich a calculus featuring constructor subtyping with a mechanism to define extensible overloaded recursive functions by pattern-matching, obtaining the system λcs+fun. We formalize the concept of well-formed environment of function declarations and establish that under such environments the properties of confluence, subject reduction and decidability of type-checking hold. Moreover, we prove that the requirements imposed for the well-formed environments are decidable and show how standard techniques can still be used for compiling pattern-matching into case-expressions.Em sistemas de tipos, a combinação de mecanismos de subtipagem e de sobrecarga de construtores permite alcançar tipagens mais precisas para os termos. Esta tese investiga a utilização destes mecanismos, quer como forma de assegurar a terminação de funções recursivas, quer como forma de captar subtipagem através de inclusão de conjuntos num sistema com tipos indutivos. A primeira parte da tese apresenta um sistema de tipos capaz de assegurar a terminação de funções recursivas, unicamente por tipagem. Mais concretamente, a noção de terminação baseada em tipos é formalizada utilizando uma forma restrita de dependência de tipos, também conhecida por tipos indexados. Cada tipo de dados é visto como uma família de aproximações, indexada por um conjunto de níveis, fornecendo tais níveis indicações sobre o uso de construtores na formação de termos. Esta forma de garantir terminação por tipos encontra-se formalizada no que é um cálculo lambda simplesmente tipado à la Curry, com tipos indutivos paramétricos, com expressões de ponto fixo e de análise de casos. Demonstra-se que λˆ é um cálculo bem comportado, satisfazendo as propriedades de confluência, preservação de tipos ao longo da cadeia de redução, e normalização forte. O sistema λˆ permite codificar muitas definições recursivas que são rejeitadas por outros sistemas com preocupações semelhantes de garantia de terminação. Em particular, prova-se que este cálculo engloba de modo estrito o sistema λς de Giménez, um sistema em que a terminação das expressões tipáveis é assegurada por uma condição sintáctica que restringe as chamadas recursivas de funções. Na segunda parte da tese, apresenta-se um sistema de tipos com subtipagem por construtores e estudam-se as suas propriadades. A subtipagem por construtores é uma forma de subtipagem na qual um tipo indutivo σ é visto como um subtipo de um outro tipo indutivo τ, se τ tiver mais construtores do que σ. Neste trabalho, a subtipagem por construtores está presente no sistema λcs, um cálculo lambda simplesmente tipado, à la Curry, com tipos inductivos paramétricos e mutuamente recursivos, com expressões de ponto fixo e de análise de casos. Demonstra-se que este cálculo é confluente, a tipagem é decidível e a redução preserva tipos. Para garantir a normalização forte, são propostas duas abordagens: satisfação de uma condição sintáctica nas definições recursivas (à semelhança de λς), ou enriquecimento do sistema de tipos com níveis (à semelhança de λˆ) de forma a garantir terminação por tipagem. Esta forma de subtipagem encontra aplicações nos sistemas de prova assistida e nas linguagens funcionais de programação. Em particular, a subtipagem por construtores revela-se adequada para o tratamento de tipos de dados extensíveis. A combinação da subtipagem com a sobrecarga de construtores permite que a definição de novos tipos de dados possa ser feita por restrição ou expansão do conjunto de construtores de um tipo de dados jà definido. Esta flexibilidade na definição de tipos de dados induz uma forma de re-utilização de código adequada às funções recursivas, permitindo que a definição de novas funções se possa fazer também por restrição ou expansão de funções já definidas. Estes mecanismos são estudados no âmbito do sistema λcs+fun, um cálculo lambda com subtipagem por construtores e com definições recursivas sobrecarregadas e extensíveis, definidas por concordância de padrões num ambiente global. Define-se, para este cálculo, o conceito de ambiente bem formado de funções, e demonstra-se que, para estes ambientes, as propriedades de confluência, decidibilidade de tipagem e preservação de tipos são válidas. Também se demonstra que os vários requisitos impostos para garantir a boa formação do ambiente global de funções correspondem a propriedades decidíveis. Finalmente, descreve-se um algoritmo de compilação das funções definidas por concordância de padrões para expressões com análise de casos

Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types

Type systems certify program properties in a compositional way. From a bigger program one can abstract out a part and certify the properties of the resulting abstract program by just using the type of the part that was abstracted away. Termination and productivity are non-trivial yet desired program properties, and several type systems have been put forward that guarantee termination, compositionally. These type systems are intimately connected to the definition of least and greatest fixed-points by ordinal iteration. While most type systems use conventional iteration, we consider inflationary iteration in this article. We demonstrate how this leads to a more principled type system, with recursion based on well-founded induction. The type system has a prototypical implementation, MiniAgda, and we show in particular how it certifies productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317

Beating the Productivity Checker Using Embedded Languages

Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problem-specific language as a data type, writing the program in the problem-specific language, and writing a guarded interpreter for this language.Comment: In Proceedings PAR 2010, arXiv:1012.455

Inductive and Coinductive Components of Corecursive Functions in Coq

In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008

A Typed Language for Truthful One-Dimensional Mechanism Design

We first introduce a very simple typed language for expressing allocation algorithms that allows automatic verification that an algorithm is monotonic and therefore truthful. The analysis of truthfulness is accomplished using a syntax-directed transformation which constructs a proof of monotonicity based on an exhaustive critical-value analysis of the algorithm. We then define a more high-level, general-purpose programming language with typical constructs, such as those for defining recursive functions, along with primitives that match allocation algorithm combinators found in the work of Mu'alem and Nisan [10]. We demonstrate how this language can be used to combine both primitive and user-defined combinators, allowing it to capture a collection of basic truthful allocation algorithms. In addition to demonstrating the value of programming language design techniques in application to a specific domain, this work suggests a blueprint for interactive tools that can be used to teach the simple principles of truthful mechanism desig

Refinement Types as Higher Order Dependency Pairs

Refinement types are a well-studied manner of performing in-depth analysis on functional programs. The dependency pair method is a very powerful method used to prove termination of rewrite systems; however its extension to higher order rewrite systems is still the object of active research. We observe that a variant of refinement types allow us to express a form of higher-order dependency pair criterion that only uses information at the type level, and we prove the correctness of this criterion

General Recursion via Coinductive Types

A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations, implementation of operational semantics, formalization of domain theory, and inductive definition of domain predicates. Here, a different solution is proposed: exploiting coinductive types to model infinite computations. To every type A we associate a type of partial elements Partial(A), coinductively generated by two constructors: the first, return(a) just returns an element a:A; the second, step(x), adds a computation step to a recursive element x:Partial(A). We show how this simple device is sufficient to formalize all recursive functions between two given types. It allows the definition of fixed points of finitary, that is, continuous, operators. We will compare this approach to different ones from the literature. Finally, we mention that the formalization, with appropriate structural maps, defines a strong monad.Comment: 28 page

Un système X, Raisonner formellement sur les programmes ML

Nous proposons dans cet article un système de types dit "renforcé" permettant de garantir la terminaison de programmes exprimés dans le noyau fonctionnel pur de ML. Le caractère "renforcé" du système proposé vient de l'ajout aux règles de typage traditionnelles de ML d'un principe d'induction structurelle sur les types de données qui apporte la garantie de terminaison. La preuve de normalisation des termes ainsi typés suit la tradition des preuves par réductibilité au sens de Tait