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Deciding type equivalence in a language with singleton kinds

Christopher A. Stone, Robert Harper
2000 Proceedings of the 27th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '00  
Work on the TILT compiler for Standard ML led us to study a language with singleton kinds: SA is the kind of all types provably equivalent to the type A.  ...  enriched with singleton kinds, dependent product and function kinds and , and a sub-kinding relation.  ...  with the following kind structure recall that kinds classify type constructors: A kind T classifying ordinary types; Singleton kinds SA classifying all types of kind T provably equivalent t o A ; Dependent  ... 
doi:10.1145/325694.325724 dblp:conf/popl/StoneH00 fatcat:npf5psmdfzdc3lva3jmv5rm3ua

Sound and Complete Elimination of Singleton Kinds [chapter]

Karl Crary
2001 Lecture Notes in Computer Science  
I present a translation from a language with singleton kinds to one without, and prove that translation to be sound and complete.  ...  The proof of soundness and completeness is done by normalizing type equivalence derivations using Stone and Harper's type equivalence decision procedure.  ...  The second says that equivalent types are also considered equivalent as members of their singleton kind.  ... 
doi:10.1007/3-540-45332-6_1 fatcat:sflnjrrumre47kvaaacbaz7x2e

Sound and complete elimination of singleton kinds

Karl Crary
2007 ACM Transactions on Computational Logic  
I present a translation from a language with singleton kinds to one without, and prove that translation to be sound and complete.  ...  The proof of soundness and completeness is done by normalizing type equivalence derivations using Stone and Harper's type equivalence decision procedure.  ...  The second says that equivalent types are also considered equivalent as members of their singleton kind.  ... 
doi:10.1145/1227839.1227840 fatcat:5xbtokvj5be27h3od6twql6hrq

A syntactic account of singleton types via hereditary substitution

Karl Crary
2009 Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages Theory and Practice - LFMTP '09  
We give a syntactic proof of decidability and consistency of equivalence for the singleton type calculus, which lies at the foundation of modern module systems such as that of ML.  ...  The proof works by translation of the singleton type calculus into a canonical presentation, adapted from work on logical frameworks, in which equivalent terms are written identically.  ...  In this paper, we give a syntactic proof of the metatheoretic results (decidable type equivalence and consistency) in the presence of singleton kinds.  ... 
doi:10.1145/1577824.1577829 dblp:conf/lfmtp/Crary09 fatcat:q2n5ephm3rf7pcrhbtyjsrvz3y

Extensional equivalence and singleton types

Christopher A. Stone, Robert Harper
2006 ACM Transactions on Computational Logic  
We study the λ S ≤ calculus, which contains singleton types S(M ) classifying terms of base type provably equivalent to the term M .  ...  The decidability of λ S ≤ term equivalence allows us to show decidability of type checking for TILT's intermediate language.  ...  This permitted a more straightforward rewriting-based approach to deciding equivalence. Crary has made use of singleton types and singleton kinds in several contexts.  ... 
doi:10.1145/1166109.1166112 fatcat:wklyh5yuvvau7onwbhctjnyihu

Extensional equivalence and singleton types

Christopher A. Stone, Robert Harper
2006 ACM Transactions on Computational Logic  
We study the λ S ≤ calculus, which contains singleton types S(M ) classifying terms of base type provably equivalent to the term M .  ...  The decidability of λ S ≤ term equivalence allows us to show decidability of type checking for TILT's intermediate language.  ...  This permitted a more straightforward rewriting-based approach to deciding equivalence. Crary has made use of singleton types and singleton kinds in several contexts.  ... 
doi:10.1145/1183278.1183281 fatcat:tpyi53ceujel7mvp7u5qp52fie

Dynamic Translucency with Abstraction Kinds and Higher-Order Coercions

Andreas Rossberg
2008 Electronical Notes in Theoretical Computer Science  
As an idealisation of such a module language, we present a foundational calculus that combines higher-order type generation, modelling type abstraction, with singleton kinds, modelling translucency.  ...  These are analogous to singletons, but instead of inducing equivalence they induce an isomorphism that is witnessed by explicit type coercions on the term level.  ...  The main difficulty is posed by type equivalence with singleton kinds.  ... 
doi:10.1016/j.entcs.2008.10.019 fatcat:kmxa6xutjzfvbad4ravek55xpy

A syntactic approach to eta equality in type theory

Healfdene Goguen
2005 Proceedings of the 32nd ACM SIGPLAN-SIGACT sysposium on Principles of programming languages - POPL '05  
The approach extends to other inductive types with a single constructor, and is demonstrated for singleton types and Σ-types.  ...  This paper outlines a new approach to showing the decidability of type checking for type theories with βηequality, relevant to foundations for modules systems and type theory-based proof systems.  ...  Acknowledgments I would like to thank Mariangiola Dezani for her kind invitation to visit the Università di Torino, Robert Harper for suggesting that I study the metatheory of singletons, Gilles Barthe  ... 
doi:10.1145/1040305.1040312 dblp:conf/popl/Goguen05 fatcat:ioyyd4nulzbjjc642hx74z6gta

A syntactic approach to eta equality in type theory

Healfdene Goguen
2005 SIGPLAN notices  
The approach extends to other inductive types with a single constructor, and is demonstrated for singleton types and Σ-types.  ...  This paper outlines a new approach to showing the decidability of type checking for type theories with βηequality, relevant to foundations for modules systems and type theory-based proof systems.  ...  Acknowledgments I would like to thank Mariangiola Dezani for her kind invitation to visit the Università di Torino, Robert Harper for suggesting that I study the metatheory of singletons, Gilles Barthe  ... 
doi:10.1145/1047659.1040312 fatcat:svzyjzsgrjfw7n2x7r55cxbrna

Relating Church-Style and Curry-Style Subtyping

Adriana Compagnoni, Healfdene Goguen
2011 Electronic Proceedings in Theoretical Computer Science  
Type theories with higher-order subtyping or singleton types are examples of systems where computation rules for variables are affected by type information in the context.  ...  The algorithm for subtyping in F-Omega-Sub can then be given on types without context or kind information.  ...  theory with singleton types, Σ and Π types, and all of the η rules.  ... 
doi:10.4204/eptcs.45.1 fatcat:e75aonjtu5fi5gkfbioujqglbi

Constructive Analysis of S1S and Büchi Automata [article]

Moritz Lichter, Gert Smolka
2018 arXiv   pre-print
We study S1S and B\"uchi automata in the constructive type theory of the Coq proof assistant.  ...  For UP semantics (ultimately periodic sequences), we verify B\"uchi's translation of formulas to automata and thereby establish decidability of S1S constructively.  ...  Notationally, we will identify the pair (x, y) with the sequence xy ω . A discrete type is a type X together with a boolean function deciding equality on X.  ... 
arXiv:1804.04967v1 fatcat:vy464lvv7ffipez6uglpgc7chq

Singleton

Simon Winwood, Manuel Chakravarty
2011 Proceedings of the 7th ACM SIGPLAN workshop on Types in language design and implementation - TLDI '11  
In addition to the primitive singleton types of other languages, these generalised singleton types allow the values from arbitrary inductive types to be associated with the contents of registers and memory  ...  Along with Singleton's facility for term and proof abstraction, generalised singleton types allow strong statements to be made about the functional behaviour of Singleton programs.  ...  Both typing and term equivalence are decidable. rs using p IS ::= I ; IS | case r d , rs ,L | br L | jump r | halt [τ ] Word values wv ::= ?  ... 
doi:10.1145/1929553.1929557 dblp:conf/tldi/WinwoodC11 fatcat:fjjyn6yan5bzjme562levnydjq

Well-Definedness and Semantic Type-Checking in the Nested Relational Calculus and XQuery [article]

Jan Van den Bussche, Dirk Van Gucht, Stijn Vansummeren
2004 arXiv   pre-print
Nevertheless, for a "pure" variant of XQuery, in which no identification is made between an item and the singleton containing that item, the problems become decidable.  ...  We show that well-definedness and semantic type-checking are undecidable, even in the positive-existential case.  ...  If the well-definedness or semantic type-checking problem is decidable for PENRC[kind] , then it is also decidable for pure PERX.  ... 
arXiv:cs/0406060v1 fatcat:ujnvhkcw7zecdgoeni5qxtxzca

Introduction to Univalent Foundations of Mathematics with Agda [article]

Martín Hötzel Escardó
2020 arXiv   pre-print
We introduce Voevodsky's univalent foundations and univalent mathematics, and explain how to develop them with the computer system Agda, which is based on Martin-L\"of type theory.  ...  Compared to most expositions of the subject, we work with explicit universe levels.  ...  '-center x , singleton-type'-centered x Voevodsky's notion of type equivalence The main notions of univalent mathematics conceived by Voevodsky, with formulations in MLTT, are those of singleton type  ... 
arXiv:1911.00580v3 fatcat:vb5sesz4wvfbtlhxsxmkltchay

A type system for higher-order modules

Derek Dreyer, Karl Crary, Robert Harper
2003 Proceedings of the 30th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '03  
Our type system harmonizes design elements from previous work, resulting in a simple, economical account of modular programming.  ...  We present a type theory for higher-order modules that accounts for many central issues in module system design, including translucency, applicativity, generativity, and modules as first-class values.  ...  D An Algorithm for Deciding Equivalence In this section we define an algorithm for deciding type, signature, and module equivalence, and prove it sound with respect to declarative equivalence.  ... 
doi:10.1145/604131.604151 fatcat:i6vqfn4ncbcrnf45rwcbmopnwy
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