Alejandro Ríos

Relating the λσ- and λs-styles of explicit substitutions

The aim of this article is to compare two styles of Explicit Substitutions: the and s-styles. We start by introducing a criterion of adequacy to simulate -reduction in calculi of explicit substitutions and we apply it to several calculi: , * , , s, t and u. The latter is presented here for the first time and may be considered as an adequate variant of s. By doing so, we establish that calculi à la s are usually more adequate at simulating -reduction than calculi in the -style. In fact, we prove that t is more adequate than and that u is more adequate than , * and s. We also give counterexamples to show that all other comparisons are impossible according to our criterion. Our next step consists in presenting the ! and !e calculi, the two-sorted (term and substitution) versions of the s and se calculi, respectively. We establish an isomorphism between the s-calculus and the term restriction of the !-calculus, which extends to an isomorphism between se and the term restriction of !e. Since the ! and !e calculi are given in the style of the -calculus they are bridge calculi between s and and between se and and thus we are able to better understand one calculus in terms of the other. Finally, we present typed versions of all the calculi and check that the above mentioned isomorphism preserves types. As a consequence, the !-calculus is a calculus in the -style that has the following properties: (a) ! simulates one step -reduction, (b) ! is confluent (on closed terms), (c) ! preserves strong normalization, (d) !’s associated calculus of substitutions is SN, (e) the simply typed ! calculus is SN, (f) the !-calculus possesses an extension !e that is confluent on open terms (terms with eventual metavariables of sort term only), and (g) the simply typed !e calculus is weakly normalizing (on open term). As far as we know, the !-calculus is the first calculus in the -style that has all the properties (a)–(g). However, the open problem of the SN of the associated calculus of substitution of !e remains unsolved and like in the case of , and se, !e does not have PSN.

Strong Normalization of Substitutions

Acr-calculus is an extended A-calculus where substitutions are handled explicitly. It is similar to, and inspired by, Categorical Combinatory logic (CCL). The strong normalization of a, the subcalculus which computes substitutions, may be inferred from the strong normalization of the similar subsystem SUBST of CCL. We present here an independent proof of the termination of several substitution calculi, including a and SUBST.

A de Bruijn Notation for Higher-Order Rewriting

We propose a formalism for higher-order rewriting in de Bruijn notation. This notation not only is used for terms (as usually done in the literature) but also for metaterms, which are the syntactical objects used to express general higher-order rewrite systems. We give formal translations from higher-order rewriting with names to higher-order rewriting with de Bruijn indices, and vice-versa. These translations can be viewed as an interface in programming languages based on higher-order rewrite systems, and they are also used to show some properties, namely, that both formalisms are operationally equivalent, and that confluence is preserved when translating one formalism into the other.

From Higher-Order to First-Order Rewriting

We show how higher-order rewriting may be encoded into first-order rewriting modulo an equational theory Ɛ. We obtain a characterization of the class of higher-order rewriting systems which can be encoded by first-order rewriting modulo an empty theory (that is, Ɛ = θ). This class includes of course the λ-calculus. Our technique does not rely on a particular substitution calculus but on a set of abstract properties to be verified by the substitution calculus used in the translation.

Pure type systems with de Bruijn indices

Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics there are many different type systems. They serve several purposes and are formulated in various ways. A general framework called Pure Type Systems (PTSs) has been introduced independently by Terlouw and Berardi in order to provide a unified formalism in which many type systems can be represented. In particular, PTSs allow the representation of the simple theory of types, the polymophic theory of types, the dependent theory of types and various other well-known type systems such as the Edinburgh Logical Frameworks and the Automath system. PTSs are usually presented using variable names. In this article, we present a formulation of PTSs with de Bruijn indices. De Bruijn indices avoid the problems caused by variable names during the implementation of type systems. We show that PTSs with variable names and PTSs with de Bruijn indices are isomorphic. This isomorphism enables us to answer questions about PTSs with de Bruijn indices including confluence, termination (strong normalization) and safety (subject reduction).

Proof Terms for Infinitary Rewriting

We generalize the notion of proof term to the realm of transfinite reduction. Proof terms represent reductions in the first-order term format, thereby facilitating their formal analysis. Transfinite reductions can be faithfully represented as infinitary proof terms, unique up to infinitary associativity. We use proof terms to define equivalence of transfinite reductions on the basis of permutation equations. A proof of the compression property via proof terms is presented, which establishes permutation equivalence between the original and the compressed reductions.

Normalisation for Dynamic Pattern Calculi

The Pure Pattern Calculus (PPC) [10, 11] extends the -calculus, as well as the family of algebraic pattern calculi [20, 6, 12], with first-class patterns i.e. patterns can be passed as arguments, evaluated and returned as results. The notion of matching failure of PPC in [11] not only provides a mechanism to define functions by pattern matching on cases but also supplies PPC with parallelor-like, non-sequential behaviour. Therefore, devising normalising strategies for PPC to obtain well-behaved implementations turns out to be challenging. This paper focuses on normalising reduction strategies for PPC. We define a (multistep) strategy and show that it is normalising. The strategy generalises the leftmost-outermost strategy for -calculus and is strictly finer than parallel-outermost. The normalisation proof is based on the notion of necessary set of redexes, a generalisation of the notion of needed redex encompassing non-sequential reduction systems.

Call-by-Need, Neededness and All That.

We show that call-by-need is observationally equivalent to weak-head needed reduction. The proof of this result uses a semantical argument based on a (non-idempotent) intersection type system called

Relating Higher-order and First-order Rewriting

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de Bruijn Indices for Metaterms

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