Strong Normalization Property Research Articles

Extensional proofs in a propositional logic modulo isomorphisms

System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as A∧B and B∧A, or A⇒(B∧C) and (A⇒B)∧(A⇒C) are made equal. System I enjoys the strong normalization property. This is sufficient to prove the existence of empty types, but not to prove the introduction property (every closed term in normal form is an introduction). Moreover, a severe restriction had to be made on the types of the variables in order to obtain the existence of empty types. We show here that adding η-expansion rules to System I permits to drop this restriction, and yields a strongly normalizing calculus which enjoys the full introduction property.

Normalization in the simply typed -calculus

AbstractIn this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the $\lambda \mu$ -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual $\beta$ -, $\mu$ -, and $\mu '$ -reductions, we consider the $\lambda \mu$ -calculus augmented with the reduction rules $\rho$ , $\theta$ and $\varepsilon$ . We show that we need all of these rules for this purpose. Then we prove that, with the syntax of Parigot, the calculus enjoys the strong normalization property even when we add the rules $\rho$ , $\theta$ , and $\epsilon$ , while the $\lambda \mu$ -calculus presented with the more flexible de Groote-style syntax, in contrast, has only the weak normalization property. In particular, we present a normalization algorithm for the $\beta \mu \mu '\rho \theta \varepsilon$ -reduction in the de Groote-style calculus.

Propositional quantifiers in labelled natural deduction for normal modal logic

Abstract This article concerns the treatment of propositional quantification in a framework of labelled natural deduction for modal logic developed by Basin, Matthews and Viganò. We provide a detailed analysis of a basic calculus that can be used for a proof-theoretic rendering of minimal normal multimodal systems with quantification over stable domains of propositions. Furthermore, we consider variations of the basic calculus obtained via relational theories and domain theories allowing for quantification over possibly unstable domains of propositions. The main result of the article is that fragments of the labelled calculi not exploiting reductio ad absurdum enjoy the Church–Rosser property and the strong normalization property; such result is obtained by combining Girard’s method of reducibility candidates and labelled languages of lambda calculus codifying the structure of modal proofs.

On Strong Normalization in Proof-Graphs for Propositional Logic

Traditional proof theory of Propositional Logic deals with proofs whose size can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The use of proof-graphs, instead of trees or lists, for representing proofs is getting popular among proof-theoreticians. Proof-graphs serve as a way to study complexity of propositional proofs and to provide more efficient theorem provers, concerning size of propositional proofs.Fpl-graphs were initially developed for minimal implicational logic representing proofs through references rather than copy. Thus, formulas and sub-deductions preserved in the graph structure, can be shared deleting unnecessary sub-deductions resulting in the reduced proof. In this work, we consider full minimal propositional logic and show how to reduce (eliminating maximal formulas) these representations such that strong normalization theorem can be proved by simply counting the number of maximal formulas in the original derivation. In proof-graphs, the main reason for obtaining the strong normalization property using such simple complexity measure is a direct consequence of the fact that each formula occurs only once in the proof-graph and the case of the hidden maximum formula that usually occurs in the tree-form derivation is already represented in the fpl-graph.

New Proofs of Important Theorems of Untyped Extensional λ Calculus

This paper contains new proofs of the following two theorems of the untyped extensional ?-calculus: the Curry theorem stating that any ?-term has a β?-normal form if and only if it has a β-normal form and the normalization theorem for β?-reduction. The proposed approach is based on the following well-known results: the postponement theorem of ?-reduction and the strong normalization property of ?-reduction that allow one to naturally extend some propositions from the usual ?-calculus to the extensional case.

Atomic polymorphism

AbstractIt has been known for six years that the restriction of Girard's polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait's method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the full intuitionistic propositional calculus translates into one or more βη-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to η-conversions.

Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach

We present an extension of the second-order logic AF2 with iso-style inductive and coinductive definitions specifically designed to extract programs from proofs a la Krivine-Parigot by means of primitive (co)recursion principles. Our logic includes primitive constructors of least and greatest fixed points of predicate transformers, but contrary to the common approach, we do not restrict ourselves to positive operators to ensure monotonicity, instead we use the Mendler-style, motivated here by the concept of monotonization of an arbitrary operator on a complete lattice. We prove an adequacy theorem with respect to a realizability semantics based on saturated sets and saturated-valued functions and as a consequence we obtain the strong normalization property for the proof-term reduction, an important feature which is absent in previous related work.

Intersection Types from a Proof-theoretic Perspective

In this work we present a proof-theoretical justification for the intersection type assignment system IT by means of the logical system Intersection Synchronous Logic ISL. ISL builds classes of equivalent deductions of the implicative and conjunctive fragment of the intuitionistic logic NJ. ISL results from decomposing intuitionistic conjunction into two connectives: a synchronous conjunction, that can be used only among equivalent deductions of NJ, and an asynchronous one, that can be applied among any sets of deductions of NJ. A term decoration of ISL exists so that it matches both: the IT assignment system, when only the synchronous conjunction is used, and the simple types assignment with pairs and projections, when the asynchronous conjunction is used. Moreover, the proof of strong normalization property for ISL is a simple consequence of the same property in NJ and hence strong normalization for IT comes for free.

Strong normalization property for second order linear logic

The paper contains the first complete proof of strong normalization (SN) for full second order linear logic (LL): Girard’s original proof uses a standardization theorem which is not proven. We introduce sliced pure structures (sps), a very general version of Girard’s proof-nets, and we apply to sps Gandy’s method to infer SN from weak normalization (WN). We prove a standardization theorem for sps: if WN without erasing steps holds for an sps, then it enjoys SN. A key step in our proof of standardization is a confluence theorem for sps obtained by using only a very weak form of correctness, namely acyclicity slice by slice. We conclude by showing how standardization for sps allows to prove SN of LL, using as usual Girard’s reducibility candidates.

Church-Rosser property and intersection types

We give a proof via reducibility of the Church-Rosser property for the system D of λ-calculus with intersection types. As a consequence we can get the confluence property for developments directly, without making use of the strong normalization property for developments, by using only the typability in D and a suitable embedding of developments in this system. As an application we get a proof of the Church-Rosser theorem for the untyped λ-calculus.

Polymorphic higher-order recursive path orderings

This article extends the termination proof techniques based on reduction orderings to a higher-order setting, by defining a family of recursive path orderings for terms of a typed lambda-calculus generated by a signature of polymorphic higher-order function symbols. These relations can be generated from two given well-founded orderings, on the function symbols and on the type constructors. The obtained orderings on terms are well founded, monotonic, stable under substitution and include β-reductions. They can be used to prove the strong normalization property of higher-order calculi in which constants can be defined by higher-order rewrite rules using first-order pattern matching. For example, the polymorphic version of Gödel's recursor for the natural numbers is easily oriented. And indeed, our ordering is polymorphic, in the sense that a single comparison allows to prove the termination property of all monomorphic instances of a polymorphic rewrite rule. Many nontrivial examples are given that exemplify the expressive power of these orderings. All have been checked by our implementation. This article is an extended and improved version of Jouannaud and Rubio [1999]. Polymorphic algebras have been made more expressive than in our previous framework. The intuitive notion of a polymorphic higher-order ordering has now been made precise. The higher-order recursive path ordering itself has been made much more powerful by replacing the congruence on types used there by an ordering on types satisfying some abstract properties. Besides, using a restriction of Dershowitz's recursive path ordering for comparing types, we can integrate both orderings into a single one operating uniformly on both terms and types.

Investigations on the Dual Calculus

The Dual Calculus, proposed recently by Wadler, is the outcome of two distinct lines of research in theoretical computer science: (A) Efforts to extend the Curry–Howard isomorphism, established between the simply-typed lambda calculus and intuitionistic logic, to classical logic. (B) Efforts to establish the tacit conjecture that call-by-value (CBV) reduction in lambda calculus is dual to call-by-name (CBN) reduction. This paper initially investigates relations of the Dual Calculus to other calculi, namely the simply-typed lambda calculus and the Symmetric lambda calculus. Moreover, Church–Rosser and Strong Normalization properties are proven for the calculus’ CBV reduction relation. Finally, extensions of the calculus to second-order types are briefly introduced.

Recursion over objects of functional type

This paper presents an extension of the simply typed λ-calculus that allows iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax. This calculus aims at being the kernel for a type theory in which the user will be able to formalize logics or formal systems using the LF methodology, while taking advantage of new induction and recursion principles, extending the principles available in a calculus such as the Calculus of Inductive Constructions. The key idea of our system is the use of modal logic S4. We present here the system, its typing rules and reduction rules. The system enjoys the decidability of typability, soundness of typed reduction with respect to the typing rules, the Church–Rosser and strong normalization properties and it is a conservative extension over the simply typed λ-calculus. These properties entail the preservation of the adequacy of encodings.

How to normalize the Jay

In this note we give an elementary proof of the strong normalization property of the J combinator by providing an explicit bound for the maximal length of the reduction paths of a term. This result shows clearly that in the theorem of Toyama, Klop and Barendregt on completeness of unions of left-linear term rewriting systems, disjointness is essential.

The simply typed rewriting calculus

The rewriting calculus is a rule construction and application framework. As such it embeds in a uniform way term rewriting and lambda-calculus. Since rule application is an explicit object of the calculus, it allows us also to handle the set of results explicitly.We present a simply typed version of the rewriting calculus. With a good choice of the type system, we show that the calculus is type preserving and terminating, i.e. verifies the subject reduction and strong normalization properties.

Explicit substitution on the edge of strong normalization

We use the recursive path ordering (RPO) technique of semantic labelling to show the preservation of strong normalization (PSN) property for several calculi of explicit substitution. PSN states that if a term M is strongly normalizing under ordinary β-reduction (using ‘global’ substitutions), then it is strongly normalizing if the substitution is made explicit (‘local’). There are different ways of making global substitution explicit and PSN is a quite natural and desirable property for the explicit substitution calculus. Our method for proving PSN is very general and applies to several known systems of explicit substitutions, both with named variables and with De Bruijn indices: λν of Lescanne et al., λs of Kamareddine and Rios and λx of Rose and Bloo. We also look at two small extensions of the explicit substitution calculus that allow to permute substitutions. For one of these extensions PSN fails (using the counterexample in Melliès 1995). For the other we can prove PSN using our method, thus showing the subtlety of the subject and the generality of our method. One of the key ideas behind our proof is that, for λx the set of terms of the explicit substitution calculus, we look at the set λx <∞, consisting of the terms A such that the substitution-normalform of each subterm of A is β-SN. This is a kind of ‘induction loading’: if we prove that λx-reduction is SN on the set λx <∞, then we have proved PSN for λx. To prove λx-SN on the set λx <∞, we define the β- size of a term A ϵ λx <∞ as the maximum length of a β-reduction path from the substitution-normal-form of A. Using this β-size, we define a translation from λx <∞ to some well-founded order > rpo on labelled terms, such that any infinite λx-reduction path starting from an A ϵ λx <∞ translates to an infinite > rpo -descending sequence. The well-founded order > rpo is defined by using a technique similar to semantic labelling.

Termination of SystemF-bounded: A Complete Proof

System F -bounded is a second-order typed λ -calculus with subtyping which has been defined to carry out foundational studies about the type systems of object-oriented languages. The almost recursive nature of the essential feature of this system makes one wonder whether it retains the strong normalization property, with respect to first- and second-order βϵ reduction of system F ⩽ . We prove that this is the case. The proof is carried out to the last detail to allow the reader to be convinced of its correctness.

A Symmetric Lambda Calculus for Classical Program Extraction

We introduce aλ-calculus with symmetric reduction rules and “classical” types, i.e., types corresponding to formulas of classical propositional logic. The strong normalization property is proved to hold for such a calculus, as well as for its extension to a system equivalent to Peano arithmetic. A theorem on the shape of terms in normal form is also proved, making it possible to get recursive functions out of proofs ofΠ02formulas, i.e., those corresponding to program specifications.

Modular proofs for completeness of hierarchical term rewriting systems

In this paper, we study modular aspects of hierarchical combinations of term rewriting systems. A combination R 0 ∪ R 1 is hierarchical if the defined symbols of the two subsystems R 0 and R 1 are disjoint, some of the defined symbols of R 0 are constructors in R 1 and the defined symbols of R 1 do not occur in R 0. It is shown that in hierarchical combinations, a reduction can increase the rank of a term. Therefore, techniques employed in proving the modularity results for direct sums and constructor sharing systems are not applicable for hierarchical combinations. We propose a set of sufficient conditions for the modularity of completeness of hierarchical combinations. The sufficient conditions are syntactic ones (about recursion) and can be easily tested for finite systems. First, the modularity of strong innermost normalization (SIN) for a class of hierarchical combinations is established. By imposing a restriction that R 0 ∪ R 1 is an overlay system, the modularity of local confluence is established for this class. Then the modularity of completeness is obtained using a recent result relating strong innermost normalization and termination properties of locally confluent overlay systems.

Strong normalisation for the linear term calculus

AbstractWe prove a strong normalisation result for the linear term calculus of Benton, Bierman, Hyland and de Paiva. Rather than prove the result from first principles, we give a translation of linear terms into terms in the second-order polymorphic lambda calculus (λ2) which allows the result to be proved by appealing to the well-known strong normalisation property of λ2. An interesting feature of the translation is that it makes use of the λ2 coding of a coinductive datatype as the translation of the !-types (exponentials) of the linear calculus.