68 research outputs found
Union of Reducibility Candidates for Orthogonal Constructor Rewriting
We revisit Girard's reducibility candidates by proposing a general of the notion of neutral terms. They are the terms which do not interact with some contexts called elimination contexts. We apply this framework to constructor rewriting, and show that for orthogonal constructor rewriting, Girard's reducibility candidates are stable by union
Toward a General Rewriting-Based Framework for Reducibility
Reducibility is a powerful proof method which applies to various properties of typed terms in different type systems. For strong normalization, different vari- ants are known, such as Girard's reducibility candidates, Tait's saturated sets and biorthogonals. They differ by the closure conditions imposed to types interpreta- tions, called here reducibility families. This paper is about the computational and observational properties underlying untyped reducibility. Our starting point is the comparison of reducibility families w.r.t. their ability to handle rewriting, for which their possible stability by union plays an important role. Indeed, usual saturated sets are generally stable by union, but with rewriting it can be difficult to define a uniform notion of saturated sets. On the other hand, rewriting is more naturally taken into account by reducibility candidates, but they are not always stable by union. It seems that for a given rewrite relation, the stability by union of reducibility candidates should imply the ability to naturally define corresponding saturated sets. In this paper, we seek to devise a general framework in which the above claim can be substantiated. In particular, this framework should be as simple as possible, while allowing the formulation of general notions of reducibility candidates and saturated sets. We present a notion of non-interaction which allows to define neutral terms and reducibility candidates in a generic way. This notion can be formulated in a very simple and general framework, based only on a rewrite relation and a set of contexts, called elimination contexts, required to satisfy some simple properties. This provides a convenient level of abstraction to prove fundamental properties of reducibility candidates, to compare them with biorthogonals, and to study their stability by union. Moreover, we propose a general form of saturated sets, issued from the stability by union of reducibility candidates
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
On the Stability by Union of Reducibility Candidates
International audienceWe investigate some aspects of proof methods for the termination of (extensions of) the second-order lambda-calculus in presence of union and existential types. We prove that Girard's reducibility candidates are stable by union iff they are exactly the non-empty sets of terminating terms which are downward-closed wrt a weak observational preorder. We show that this is the case for the Curry-style second-order lambda-calculus.As a corollary, we obtain that reducibility candidates are exactly the Tait's saturated sets that are stable by reduction. We then extend the proof to a system with product, co-product and positive iso-recursive types
Proofs and Refutations for Intuitionistic and Second-Order Logic
The ?^{PRK}-calculus is a typed ?-calculus that exploits the duality between the notions of proof and refutation to provide a computational interpretation for classical propositional logic. In this work, we extend ?^{PRK} to encompass classical second-order logic, by incorporating parametric polymorphism and existential types. The system is shown to enjoy good computational properties, such as type preservation, confluence, and strong normalization, which is established by means of a reducibility argument. We identify a syntactic restriction on proofs that characterizes exactly the intuitionistic fragment of second-order ?^{PRK}, and we study canonicity results
Strong Normalization as Safe Interaction
Abstract When enriching the λ-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the elimination rule (∨ E) of union types may also allow to type non normalizing terms (in which case we say that (∨ E) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (∨ E) amounts to the characterization, in a term, of safe interactions between some of its subterms. In this paper, we study the safety of (∨ E) for an extension of the λ-calculus with simple rewrite rules. We prove that the union and intersection type discipline without (∨ E) is complete w.r.t. strong normalization. This allows to show that (∨ E) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (∨ E), and study an alternative biorthogonality relation, based on the observation of the least reducibility candidate
Refinement Types as Higher Order Dependency Pairs
Refinement types are a well-studied manner of performing in-depth analysis on
functional programs. The dependency pair method is a very powerful method used
to prove termination of rewrite systems; however its extension to higher order
rewrite systems is still the object of active research. We observe that a
variant of refinement types allow us to express a form of higher-order
dependency pair criterion that only uses information at the type level, and we
prove the correctness of this criterion
Strong Normalization as Safe Interaction
International audienceWhen enriching the lambda-calculus with rewriting, union types may be needed to type all strongly normalizing terms. However, with rewriting, the elimination rule (UE) of union types may also allow to type non normalizing terms (in which case we say that (UE) is unsafe). This occurs in particular with non-determinism, but also with some confluent systems. It appears that studying the safety of (UE) amounts to the characterization, in a term, of safe interactions between some of its subterms. In this paper, we study the safety of (UE) for an extension of the lambda-calculus with simple rewrite rules. We prove that the union and intersection type discipline without (UE) is complete w.r.t. strong normalization. This allows to show that (UE) is safe if and only if an interpretation of types based on biorthogonals is sound for it. We also discuss two sufficient conditions for the safety of (UE), and study an alternative biorthogonality relation, based on the observation of the least reducibility candidate
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