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Generalized Iteration and Coiteration for HigherOrder Nested Datatypes
 PROC. OF FOSSACS 2003
, 2003
"... We solve the problem of extending Bird and Paterson's generalized folds for nested datatypes and its dual to inductive and coinductive constructors of arbitrarily high ranks by appropriately generalizing Mendlerstyle (co)iteration. Characteristically to Mendlerstyle schemes of disciplined (co ..."
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Cited by 8 (5 self)
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We solve the problem of extending Bird and Paterson's generalized folds for nested datatypes and its dual to inductive and coinductive constructors of arbitrarily high ranks by appropriately generalizing Mendlerstyle (co)iteration. Characteristically to Mendlerstyle schemes of disciplined (co)recursion, the schemes we propose do not rest on notions like positivity or monotonicity of a constructor and facilitate programming in a natural and elegant style close to programming with the customary letrec construct, where the typings of the schemes, however, guarantee termination. For rank 2, a smoothened version of Bird and Paterson's generalized folds and its dual are achieved; for rank 1, the schemes instantiate to Mendler's original (re)formulation of iteration and coiteration. Several examples demonstrate the power of the approach. Strong normalization of our proposed extension of system F of higherorder parametric polymorphism is proven by a reductionpreserving embedding into pure F .
(Co)iteration for higherorder nested datatypes
 POSTCONF. PROC. OF IST WG TYPES 2ND ANN. MEETING, TYPES'02, LECT. NOTES IN COMPUT. SCI
, 2003
"... The problem of defining iteration for higherorder nested datatypes of arbitrary (finite) rank is solved within the framework of System F ω of higherorder parametric polymorphism. The proposed solution heavily relies on a general notion of monotonicity as opposed to a syntactic criterion on the sh ..."
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Cited by 8 (5 self)
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The problem of defining iteration for higherorder nested datatypes of arbitrary (finite) rank is solved within the framework of System F ω of higherorder parametric polymorphism. The proposed solution heavily relies on a general notion of monotonicity as opposed to a syntactic criterion on the shape of the type constructors such as positivity or even being polynomial. Its use is demonstrated for some rank2 heterogeneous/nested datatypes such as powerlists and de Bruijn terms with explicit substitutions. An important feature is the availability of an iterative definition of the mapping operation (the functoriality) for those rank1 type transformers (i. e., functions from types to types) arising as least fixedpoints of monotone rank2 type transformers. Strong normalization is shown by an embedding into F ω. The results dualize to greatest fixedpoints, hence to coinductive constructors with coiteration.
Fixed points of type constructors and primitive recursion
 Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 2024, 2004, Proceedings, volume 3210 of Lecture Notes in Computer Science
, 2004
"... Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable decla ..."
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Cited by 7 (3 self)
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Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable declarations without variance information. Hence, the second rule is too liberal; we must insist on p = ◦. The corrected set of rules is then: ⋄ cxt ∆ cxt ∆, X ◦κ cxt ∆ cxt ∆ ⊢ A: ∗ ∆, x:A cxt Definition of welltyped terms (immediately following). Since wellformed typing contexts ∆ contain no variance information, hence ◦ ∆ = ∆, we might drop the “◦ ” in the instantiation rule (fifth rule). The new set of rules is consequently, (x:A) ∈ ∆ ∆ cxt ∆ ⊢ x: A ∆, X ◦κ ⊢ t: A ∆ ⊢ t: ∀X κ. A ∆, x:A ⊢ t: B ∆ ⊢ λx.t: A → B ∆ ⊢ t: ∀X κ. A ∆ ⊢ F: κ
Continuous semantics for strong normalization
 In CiE’05, volume 3526 of LNCS
, 2005
"... Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G"odel's system T, but also to various forms of bar ..."
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Cited by 4 (2 self)
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Abstract. We prove a general strong normalization theorem for higher type rewrite systems based on Tait's strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of G&quot;odel's system T, but also to various forms of bar recursion for which strong normalization was hitherto unknown.
Realisability for induction and coinduction with applications to constructive analysis
 J. Univers. Comput. Sci
, 2010
"... Abstract: We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in th ..."
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Cited by 1 (1 self)
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Abstract: We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation and hint at further nontrivial applications in computable analysis.
Program extraction via typed realisability for induction and coinduction
, 2009
"... and coinduction ..."
Realisability and adequacy for (co)induction
"... Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in th ..."
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Cited by 1 (1 self)
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Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation, and hint at further nontrivial applications in computable analysis. 1
Substitution in nonwellfounded . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the nonwellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.
Under consideration for publication in Math. Struct. in Comp. Science Continuous semantics for strong normalisation
, 2005
"... We prove a general strong normalisation theorem for higher type rewrite systems based on Tait’s strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of Gödel’s system T, but also to various forms of barrecursion for which strong norm ..."
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We prove a general strong normalisation theorem for higher type rewrite systems based on Tait’s strong computability predicates and a strictly continuous domaintheoretic semantics. The theorem applies to extensions of Gödel’s system T, but also to various forms of barrecursion for which strong normalisation was hitherto unknown.