Results 11  20
of
58
Representations of First Order Function Types as Terminal Coalgebras
 In Typed Lambda Calculi and Applications, TLCA 2001, number 2044 in Lecture Notes in Computer Science
, 2001
"... terminal coalgebras ..."
Iteration and Coiteration Schemes for HigherOrder and Nested Datatypes
, 2004
"... This article studies the implementation of inductive and coinductive constructors of higher kinds (higherorder nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several wellbehaved e ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
This article studies the implementation of inductive and coinductive constructors of higher kinds (higherorder nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several wellbehaved extensions of System with some form of iteration and coiteration uniform in all kinds. In what we call Mendlerstyle systems, the iterator and coiterator have a computational behavior similar to the general recursor, but their types guarantee termination. In conventionalstyle systems, monotonicity witnesses are used for a notion of monotonicity de ned uniformly for all kinds. Our most expressive systems GMIt of generalized Mendler resp. conventional (co)iteration encompass Martin, Gibbons and Bailey's ecient folds for rank2 inductive types. Strong normalization of all systems considered is proved by providing an embedding of the basic Mendlerstyle system MIt into System F .
Some domain theory and denotational semantics in Coq
, 2009
"... Abstract. We present a Coq formalization of constructive ωcpos (extending earlier work by PaulinMohring) up to and including the inverselimit construction of solutions to mixedvariance recursive domain equations, and the existence of invariant relations on those solutions. We then define operatio ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Abstract. We present a Coq formalization of constructive ωcpos (extending earlier work by PaulinMohring) up to and including the inverselimit construction of solutions to mixedvariance recursive domain equations, and the existence of invariant relations on those solutions. We then define operational and denotational semantics for both a simplytyped CBV language with recursion and an untyped CBV language, and establish soundness and adequacy results in each case. 1
Monotone Inductive and Coinductive Constructors of Rank 2
 Proceedings of CSL 2001
, 2001
"... A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and nal coalgebras in a functor category and the CurryHowardcorrespondence. The denition of the system as ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and nal coalgebras in a functor category and the CurryHowardcorrespondence. The denition of the system as a calculus requires an appropriate denition of monotonicity to overcome subtle problems, most notably to ensure that the (co)inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes. 1
Strongly Typed Term Representations in Coq
 J AUTOM REASONING
"... There are two approaches to formalizing the syntax of typed object languages in a proof assistant or programming language. The extrinsic approach is to first define a type that encodes untyped object expressions and then make a separate definition of typing judgements over the untyped terms. The int ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
There are two approaches to formalizing the syntax of typed object languages in a proof assistant or programming language. The extrinsic approach is to first define a type that encodes untyped object expressions and then make a separate definition of typing judgements over the untyped terms. The intrinsic approach is to make a single definition that captures welltyped object expressions, so illtyped expressions cannot even be expressed. Intrinsic encodings are attractive and naturally enforce the requirement that metalanguage operations on object expressions, such as substitution, respect object types. The price is that the metalanguage types of intrinsic encodings and operations involve nontrivial dependency, adding significant complexity. This paper describes intrinsicstyle formalizations of both simplytyped and polymorphic languages, and basic syntactic operations thereon, in the Coq proof assistant. The Coq types encoding objectlevel variables (de Bruijn indices) and terms are indexed by both type and typing environment. One key construction is the bootstrapping of definitions and lemmas about the action of substitutions in terms of similar ones for a simpler notion of renamings. In the simplytyped case, this yields definitions that are free of any use of type equality coercions. In the polymorphic case, some substitution operations do still require type coercions, which we at least partially tame by uniform use of heterogeneous equality.
Semicontinuous sized types and termination
 In Zoltán Ésik, editor, Computer Science Logic, 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL
"... Abstract. Some typebased approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is onl ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
Abstract. Some typebased approaches to termination use sized types: an ordinal bound for the size of a data structure is stored in its type. A recursive function over a sized type is accepted if it is visible in the type system that recursive calls occur just at a smaller size. This approach is only sound if the type of the recursive function is admissible, i.e., depends on the size index in a certain way. To explore the space of admissible functions in the presence of higherkinded data types and impredicative polymorphism, a semantics is developed where sized types are interpreted as functions from ordinals into sets of strongly normalizing terms. It is shown that upper semicontinuity of such functions is a sufficient semantic criterion for admissibility. To provide a syntactical criterion, a calculus for semicontinuous functions is developed. 1.
A functional specification of effects
, 2009
"... This dissertation is about effects and type theory. Functional programming languages such as Haskell demonstrate that monads can safely encapsulated side effects. If you want to reason about effectful code, however, it is not enough to only encapsulate effects: you must explain their meaning. In thi ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
This dissertation is about effects and type theory. Functional programming languages such as Haskell demonstrate that monads can safely encapsulated side effects. If you want to reason about effectful code, however, it is not enough to only encapsulate effects: you must explain their meaning. In this dissertation I provide pure specifications of effects in type theory. After initial investigations using Haskell, I show how such specifications can be made total in a language with dependent types. Hoare Type Theory takes an alternative approach to incorporating effects in a dependently typed language. Instead of giving functional specifications, they follow Haskell’s lead and postulate the existence of primitive functions. This dissertation shows how some of these primitives may be implemented in a programming language with dependent types. I believe that functional specifications may provide the foundations on top of which richer logics may be constructed. The results presented in this dissertation may be used to write and verify
Functional pearl: I am not a number–I am a free variable
 In Proc. Haskell workshop
"... In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [5] for bound variables. By doing so, we retain the advantages of both representations: naming supports easy, arithmeticfree ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
In this paper, we show how to manipulate syntax with binding using a mixed representation of names for free variables (with respect to the task in hand) and de Bruijn indices [5] for bound variables. By doing so, we retain the advantages of both representations: naming supports easy, arithmeticfree manipulation of terms; de Bruijn indices eliminate the need for αconversion. Further, we have ensured that not only the user but also the implementation need never deal with de Bruijn indices, except within key basic operations. Moreover, we give a hierarchical representation for names which naturally reflects the structure of the operations we implement. Name choice is safe and straightforward. Our technology combines easily with an approach to syntax manipulation inspired by Huet’s ‘zippers’[10]. Without the ideas in this paper, we would have struggled to implement EPIGRAM [19]. Our example—constructing inductive elimination operators for datatype families—is but one of many where it proves invaluable.
Monads Need Not Be Endofunctors
"... Abstract. We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finitedimensional vector spaces, untyped and typed λcalculus syntax and indexed containers. We show that the Kleisli and EilenbergMoore constr ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Abstract. We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finitedimensional vector spaces, untyped and typed λcalculus syntax and indexed containers. We show that the Kleisli and EilenbergMoore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between monads and relative monads. Arrows are also an instance of relative monads. 1
A predicative strong normalisation proof for a λcalculus with interleaving inductive types
 TYPES FOR PROOF AND PROGRAMS, INTER40 A. ABEL AND T. ALTENKIRCH NATIONAL WORKSHOP, TYPES '99, SELECTED PAPERS. LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metaleve ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone