Mixed Inductive/Coinductive Types and Strong Normalization

Abel, Andreas

doi:10.1007/978-3-540-76637-7_19

Cited by 15 publications

(29 citation statements)

References 19 publications

(26 reference statements)

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“…There are two main theoretical issues we need to tackle: first, extension of copattern coverage to dependent types, and secondly, checking termination and productivity of functions to guarantee strong normalization. A candidate for the latter task are sized types [Hughes et al 1996;Barthe et al 2004;Abel 2007] as already implemented in MiniAgda [Abel 2012]. Further, we aim at developing a denotational model for languages with copatterns.…”

Section: Resultsmentioning

confidence: 99%

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Copatterns

Abel

Pientka

Thibodeau

et al. 2013

Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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Inductive datatypes provide mechanisms to define finite data such as finite lists and trees via constructors and allow programmers to analyze and manipulate finite data via pattern matching. In this paper, we develop a dual approach for working with infinite data structures such as streams. Infinite data inhabits coinductive datatypes which denote greatest fixpoints. Unlike finite data which is defined by constructors we define infinite data by observations. Dual to pattern matching, a tool for analyzing finite data, we develop the concept of copattern matching, which allows us to synthesize infinite data. This leads to a symmetric language design where pattern matching on finite and infinite data can be mixed.We present a core language for programming with infinite structures by observations together with its operational semantics based on (co)pattern matching and describe coverage of copatterns. Our language naturally supports both call-by-name and call-by-value interpretations and can be seamlessly integrated into existing languages like Haskell and ML. We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.

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Section: Resultsmentioning

confidence: 99%

“…In MiniAgda, one can certify termination and productivity using sized types [Hughes et al 1996;Barthe et al 2004;Abel 2007]. Copatterns provide the right syntax to decorate corecursive definitions with size variables that witness productivity.…”

Section: Extensions and Implementationsmentioning

confidence: 99%

Copatterns

Abel

Pientka

Thibodeau

et al. 2013

Proceedings of the 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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“…Termination checking requires us to identity a suitable notion of acceptable inductive datatypes, e.g. based on strict positivity as in Coq [Paulin-Mohring 1993], and we plan to adapt sized types as for example in Abel [2007Abel [ , 2008 in the future.…”

Section: Discussionmentioning

confidence: 99%

Programming with binders and indexed data-types

Cave

Pientka

2012

Proceedings of the 39th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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We show how to combine a general purpose type system for an existing language with support for programming with binders and contexts by refining the type system of ML with a restricted form of dependent types where index objects are drawn from contextual LF. This allows the user to specify formal systems within the logical framework LF and index ML types with contextual LF objects. Our language design keeps the index language generic only requiring decidability of equality of the index language providing a modular design. To illustrate the elegance and effectiveness of our language, we give programs for closure conversion and normalization by evaluation.Our three key technical contribution are: 1) We give a bidirectional type system for our core language which is centered around refinement substitutions instead of constraint solving. As a consequence, type checking is decidable and easy to trust, although constraint solving may be undecidable. 2) We give a big-step environment based operational semantics with environments which lends itself to efficient implementation. 3) We prove our language to be type safe and have mechanized our theoretical development in the proof assistant Coq using the fresh approach to binding.

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“…Instead of working around the limitations of guarded corecursion one can include language features which make it easier to explain why programs are productive. One such feature is sized types (Hughes et al 1996;Barthe et al 2004;Abel 2009), and the λ -calculi of Buchholz (2005) provide other examples. Another approach is to use cleverer algorithms for establishing productivity.…”

Section: Related Workmentioning

confidence: 99%

Beating the Productivity Checker Using Embedded Languages

Danielsson

2010

Electron. Proc. Theor. Comput. Sci.

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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 problemspecific language, and writing a guarded interpreter for this language.

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Mixed Inductive/Coinductive Types and Strong Normalization

Cited by 15 publications

References 19 publications

Copatterns

Copatterns

Programming with binders and indexed data-types

Beating the Productivity Checker Using Embedded Languages

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