arXiv subjects

Frédéric Blanqui

Publications and source records attributed to Frédéric Blanqui.

47 records · Page 3Linked to original sources

arXiv2006

Inductive types in the Calculus of Algebraic Constructions

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 higher-order 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 non-strictly positive types and inductive-recursive types together with non-free constructors and pattern-matching on defined symbols.

cs.LO↗

arXiv2006

Definitions by rewriting in the Calculus of Constructions

This paper presents general syntactic conditions ensuring the strong normalization and the logical consistency of the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. On the one hand, the Calculus of Constructions is a powerful type system in which one can formalize the propositions and natural deduction proofs of higher-order logic. On the other hand, rewriting is a simple and powerful computation paradigm. The combination of both allows, among other things, to develop formal proofs with a reduced size and more automation compared with more traditional proof assistants. The main novelty is to consider a general form of rewriting at the predicate-level which generalizes the strong elimination of the Calculus of Inductive Constructions.

cs.LO↗

arXiv2006

Termination and Confluence of Higher-Order Rewrite Systems

In the last twenty years, several approaches to higher-order rewriting have been proposed, among which Klop's Combinatory Rewrite Systems (CRSs), Nipkow's Higher-order Rewrite Systems (HRSs) and Jouannaud and Okada's higher-order algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higher-order pattern-matching mechanism, resulting in simply-typed CRSs. Then, we show how the termination criterion developed for IDTSs with first-order pattern-matching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higher-order pattern-matching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow's higher-order critical pair analysis technique for proving local confluence can be applied to IDTSs.

cs.LO↗

arXiv2006

Definitions by Rewriting in the Calculus of Constructions

The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the beta-rule and the user-defined rules under some general syntactic conditions including confluence. As examples, we show that two important systems satisfy these conditions: a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq, and the Natural Deduction Modulo a large class of equational theories.

cs.LO↗

arXiv2006

An Isabelle formalization of protocol-independent secrecy with an application to e-commerce

A protocol-independent secrecy theorem is established and applied to several non-trivial protocols. In particular, it is applied to protocols proposed for protecting the computation results of free-roaming mobile agents doing comparison shopping. All the results presented here have been formally proved in Isabelle by building on Larry Paulson's inductive approach. This therefore provides a library of general theorems that can be applied to other protocols.

cs.LO↗

arXiv2006

Inductive types in the Calculus of Algebraic Constructions

In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), which is 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 higher-order rewrite rules. In this paper, we not only prove that CIC as a whole can be seen as a CAC, but also that it can be extended with non-free constructors, pattern-matching on defined symbols, non-strictly positive types and inductive-recursive types.

cs.LO↗

arXiv2006

Rewriting modulo in Deduction modulo

We study the termination of rewriting modulo a set of equations in the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. In a previous work, we defined general syntactic conditions based on the notion of computable closure for ensuring the termination of the combination of rewriting and beta-reduction. Here, we show that this result is preserved when considering rewriting modulo a set of equations if the equivalence classes generated by these equations are finite, the equations are linear and satisfy general syntactic conditions also based on the notion of computable closure. This includes equations like associativity and commutativity, and provides an original treatment of termination modulo equations.

cs.LO↗

arXiv2006

Combining typing and size constraints for checking the termination of higher-order conditional rewrite systems

In a previous work, the first author extended to higher-order rewriting and dependent types the use of size annotations in types, a termination proof technique called type or size based termination and initially developed for ML-like programs. Here, we go one step further by considering conditional rewriting and explicit quantifications and constraints on size annotations. This allows to describe more precisely how the size of the output of a function depends on the size of its inputs. Hence, we can check the termination of more functions. We first give a general type-checking algorithm based on constraint solving. Then, we give a termination criterion with constraints in Presburger arithmetic. To our knowledge, this is the first termination criterion for higher-order conditional rewriting taking into account the conditions in termination.

cs.LO↗

arXiv2006

Decidability of Type-checking in the Calculus of Algebraic Constructions with Size Annotations

Since Val Tannen's pioneer work on the combination of simply-typed lambda-calculus and first-order rewriting (LICS'88), many authors have contributed to this subject by extending it to richer typed lambda-calculi and rewriting paradigms, culminating in calculi like the Calculus of Algebraic Constructions. These works provide theoretical foundations for type-theoretic proof assistants where functions and predicates are defined by oriented higher-order equations. This kind of definitions subsumes induction-based definitions, is easier to write and provides more automation. On the other hand, checking that user-defined rewrite rules are strongly normalizing and confluent, and preserve the decidability of type-checking when combined with beta-reduction, is more difficult. Most termination criteria rely on the term structure. In a previous work, we extended to dependent types and higher-order rewriting, the notion of ``sized types'' studied by several authors in the simpler framework of ML-like languages, and proved that it preserves strong normalization. The main contribution of the present paper is twofold. First, we prove that, in the Calculus of Algebraic Constructions with size annotations, the problems of type inference and type-checking are decidable, provided that the sets of constraints generated by size annotations are satisfiable and admit most general solutions. Second, we prove the later properties for a size algebra rich enough for capturing usual induction-based definitions and much more.

cs.LO↗

arXiv2006

On the confluence of lambda-calculus with conditional rewriting

The confluence of untyped lambda-calculus with unconditional rewriting has already been studied in various directions. In this paper, we investigate the confluence of lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Muller and Dougherty for unconditional rewriting. Two cases are considered, whether beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing beta-reduction to weakly normalizing beta-reduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using an extended notion of orthogonality that takes advantage of the conditional part of rewrite rules.

cs.LO↗

arXiv2006

(HO)RPO Revisited

The notion of computability closure has been introduced for proving the termination of the combination of higher-order rewriting and beta-reduction. It is also used for strengthening the higher-order recursive path ordering. In the present paper, we study in more details the relations between the computability closure and the (higher-order) recursive path ordering. We show that the first-order recursive path ordering is equal to an ordering naturally defined from the computability closure. In the higher-order case, we get an ordering containing the higher-order recursive path ordering whose well-foundedness relies on the correctness of the computability closure. This provides a simple way to extend the higher-order recursive path ordering to richer type systems.

cs.LO↗