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36
Automated Termination Analysis for Haskell: From Term Rewriting to Programming Languages
 In Proc. RTA ’06, LNCS
, 2006
"... Abstract. There are many powerful techniques for automated termination analysis of term rewriting. However, up to now they have hardly been used for real programming languages. We present a new approach which permits the application of existing techniques from term rewriting in order to prove termin ..."
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Cited by 40 (12 self)
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Abstract. There are many powerful techniques for automated termination analysis of term rewriting. However, up to now they have hardly been used for real programming languages. We present a new approach which permits the application of existing techniques from term rewriting in order to prove termination of programs in the functional language Haskell. In particular, we show how termination techniques for ordinary rewriting can be used to handle those features of Haskell which are missing in term rewriting (e.g., lazy evaluation, polymorphic types, and higherorder functions). We implemented our results in the termination prover AProVE and successfully evaluated them on existing Haskelllibraries. 1
The Calculus of Algebraic Constructions
 In Proc. of the 10th Int. Conf. on Rewriting Techniques and Applications, LNCS 1631
, 1999
"... Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by hi ..."
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Cited by 33 (11 self)
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Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols. 1.
The Computability Path Ordering: the End of a Quest
"... Abstract. In this paper, we first briefly survey automated termination proof methods for higherorder calculi. We then concentrate on the higherorder recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture ..."
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Cited by 22 (2 self)
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Abstract. In this paper, we first briefly survey automated termination proof methods for higherorder calculi. We then concentrate on the higherorder recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability arguments à la Tait and Girard, therefore explaining the name of the improved ordering. 1
Inductive types in the calculus of algebraic constructions
 FUNDAMENTA INFORMATICAE 65(12) (2005) 61–86 JOURNAL VERSION OF TLCA’03
, 2005
"... In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrit ..."
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Cited by 19 (4 self)
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In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrite rules. In this paper, we prove that CIC as a whole can be seen as a CAC, and that it can be extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols.
A Unifying Approach to Recursive and Corecursive Definitions
 IN [5
, 2002
"... In type theory based logical frameworks, recursive and corecursive definitions are subject to syntactic restrictions that ensure their termination and productivity. These restrictions however greately decrease the expressive power of the language. In this work we propose a general approach for s ..."
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Cited by 18 (1 self)
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In type theory based logical frameworks, recursive and corecursive definitions are subject to syntactic restrictions that ensure their termination and productivity. These restrictions however greately decrease the expressive power of the language. In this work we propose a general approach for systematically defining fixed points for a broad class of well given recursive definition. This approach unifies the ones based on wellfounded order to the ones based on complete metrics and contractive functions, thus allowing for mixed recursive/corecursive definitions.
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 ..."
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Cited by 16 (6 self)
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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.
Implementing a Normalizer Using Sized Heterogeneous Types
 Journal of Functional Programming, MSFP’06 special issue
"... In the simplytyped lambdacalculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus, giving rise to a structurally recursive normalize ..."
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Cited by 13 (2 self)
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In the simplytyped lambdacalculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus, giving rise to a structurally recursive normalizer for the simplytyped lambdacalculus. We generalize this scheme to simultaneous substitutions, preserving its simple termination argument. We further implement hereditary simultaneous substitutions in a functional programming language with sized heterogeneous inductive types, Fωb, arriving at an interpreter whose termination can be tracked by the type system of its host programming language.
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 .
On the stability by union of reducibility candidates
 of Lecture Notes in Computer Science
, 2007
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 6 (1 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Programming errors in traversal programs over structured data
"... Traversal strategies provide an established means of describing automated queries, analyses, transformations, and other nontrivial computations on deeply structured data (including, most notably, data representations of software artifacts such as programs). The resulting traversal programs are pron ..."
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Cited by 6 (3 self)
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Traversal strategies provide an established means of describing automated queries, analyses, transformations, and other nontrivial computations on deeply structured data (including, most notably, data representations of software artifacts such as programs). The resulting traversal programs are prone to programming errors. We are specifically concerned with errors that go beyond classic type errors, in particular: (i) divergence of traversal, (ii) unintentional extent of traversal into data, (iii) trivial traversal results, (iv) inapplicability of the constituents of a traversal program along traversal. We deliver a taxonomy of programming errors, and start attacking some of them by refinements of traversal programming.