Weak convergence and uniform normalization in infinitary rewriting

We study infinitary term rewriting systems containing finitely many rules. For these, we show that if a weakly convergent reduction is not strongly convergent, it contains a term that reduces to itself in one step (but the step itself need not be part of the reduction). Using this result, we prove the starkly surprising result that for any orthogonal system with finitely many rules, the system is weakly normalizing under weak convergence if{f} it is strongly normalizing under weak convergence if{f} it is weakly normalizing under strong convergence if{f} it is strongly normalizing under strong convergence. As further corollaries, we derive a number of new results for weakly convergent rewriting: Systems with finitely many rules enjoy unique normal forms, and acyclic orthogonal systems are confluent. Our results suggest that it may be possible to recover some of the positive results for strongly convergent rewriting in the setting of weak convergence, if systems with finitely many rules are considered. Finally, we give a number of counterexamples showing failure of most of the results when infinite sets of rules are allowed

Vicious circles in orthogonal term rewriting systems

In this paper we first study the difference between Weak Normalization (WN) and Strong Normalization (SN), in the framework of first order orthogonal rewriting systems. With the help of the Erasure Lemma we establish a Pumping Lemma, yielding information about exceptional terms, defined as terms that are WN but not SN. A corollary is that if an orthogonal TRS is WN, there are no cyclic reductions in finite reduction graphs. This is a stepping stone towards the insight that orthogonal TRSs with the property WN, do not admit cyclic reductions at all

Reescritura de términos y sustituciones explícitas

La operación de sustitución constituye un engranaje básico en los fundamentos de la teoría de lenguajes de programación. Juega un rol central en el lambda cálculo (por ende, en lenguajes de programación funcional), en unificación de primer orden y de orden superior (por ende, en lenguajes de programación basados en el paradigma lógico), en modalidades de pasaje de parámetros (por ende, en lenguajes de programación imperativos), etc. Recientemente, investigadores en informática se han interesado en el pasaje de la noción usual de la sustitución, atómica, y de gruesa granularidad, hacia una noción más refinada, de más fina granularidad. La noción de sustitución es transportada del metalenguaje (nuestro lenguaje de discurso) al lenguaje objeto (nuestro lenguaje de estudio). Como consecuencia de ello se obtienen los llamados cálculos de sustituciones explícitas. Estos son de sumo interés a la hora de estudiar la interpretación operacional de los formalismos en cuestión y constituyen los objetos de interés de esta tesis. Se desarrollan los siguientes tres ejes de estudio: Primero, se consideran estrategias de reescritura perpetuas en lambda cálculos con sustituciones explícitas. Estas son estrategias de reescritura que preservan la posibilidad de reducciones infinitas. Se propone una caracterización inductiva del conjunto de términos que no poseen reducciones infinitas (los llamados fuertemente normalizantes). Un lambda cálculo polimórfico con sustituciones explícitas también es analizado, incluyendo propiedades tales como subject reduction y normalización fuerte. Segundo, colocamos el ς-cálculo de M. Abadi and L. Cardelli enriquecido con sustituciones explícitas bajo el microscopio. Este cálculo se encuentra en un nivel semejante de abstracción al lambda cálculo pero se basa en objetos en lugar de funciones. Propiedades tales como simulación del lambda cálculo, confluencia y preservación de la normalización fuerte (aquellos términos que son fuertemente normalizantes en ς también lo son en ς con sustituciones explícitas) son consideradas. Finalmente, dirigimos nuestra atención a la tarea de relacionar la reescritura de orden superior con aquella de primer orden. Fijamos una variante de los ERS (apodados SERSdb) de Z. Khasidashvili como nuestro formalismo de orden superior de partida y definimos un proceso de conversión que permite codificar cualquier SERSdb como un sistema de reescritura de primer orden. En este último, cada paso de reescritura se lleva a cabo módulo una teoría ecuacional determinada por un cálculo de sustituciones explícitas. La misma se formula de manera genérica a través de una presentación de cálculos de sustituciones explícitas basada en macros y axiomas sobre estas macros, parametrizando de esta manera al procedimiento de conversión sobre cualquier cálculo de sustituciones explícitas que obedece la presentación basada en macros. El procedimiento de conversión se encarga de codificar pattern matching de orden superior y sustitución en el entorno de reescritura de primer orden. Asimismo, propiedades que relacionan la noción de reescritura en el orden superior con aquella de primer orden son analizadas en detalle. Se identifica una clase de SERSdb para los cuales el sistema de primer orden resultante de su conversión no requiere una teoría ecuacional para implementar pattern matching de orden superior, bastando para ello matching sintáctico. También se argumenta que esta clase de sistemas de orden superior es apropiada para transferir resultados del entorno de reescritura de orden superior a aquella de primer orden. A modo de ejemplo no-trivial de ello, estudiamos la transferencia del teorema de standarización (fuerte).Substitution spans many areas in programming language theory. It plays a central role in the lambda calculus (hence functional programming), in first and higher-order unikation (hence logic programming), parameter passing methods (hence imperative programming), etc. Recently researchers became interested in shifting from the usual atomic, coarse grained view of substitution to a more refined, fine grained one. Substitution is promoted from the metalevel (our language of discourse) to the object-level (our language of study). This is interesting when studying the operational interpretation of the formalisms in question. Calculi of object-level or explicit substitution is the concern of this thesis. The following three study axes are developed. First we consider perpetual rewrite strategies in lambda calculi of explicit substitutions. These are rewrite strategies that preserve the possibility of inhite derivations. Also, we study how to characterize inductively the set of terms that do not possess infinite derivations (the strongly normalizing terms). Polymorphic lambda calculus with explicit substitutions shall receive our attention too, including properties such as subject reduction and strong normalization. Secondly, we put the ς-calculus of M.Abadi and L.Cardelli augmented with explicit substitutions under the microscope. This calculus is at the level of the lambda calculus but is based on objects instead of functions. Properties such as simulation of the lambda calculus, confluence and preservation of strong normalization (terms which are strongly normalizing in ς are also strongly normalizii in ς with explicit substitutions) are considered. Finally, we address the task of reducing higher-order rewriting to first-order rewriting. We fix a variant of Z-Khasidashvili's ERS (dubbed SERSdb) as our departing formalism and provide a conversion procedure to encode any ERS as a first-order rewrite system in which a rewrite step takes place modulo an equational theory determined by a calculus of explicit substitutions. The latter is achieved with the aid of a macro-based presentation of calculi of explicit substitutions, thus parametrizing the conversion procedure over any calculus of explicit substitutions in compliance with the aforementioned presentation. The conversion procedure is in charge of encoding higherorder pattern matching and substitution in the first-order framework. Properties relating the rewrite relation in the higher-order framework and that of the resulting first-order system are studied in detail. We then identify a class of SERSdb for which the resulting first-order system does not require the equational theory to implement higher-order pattern matching, thus contenting itself with syntactic matching. It is argued that this class of systems is appropriate for transferring results from the first-order framework to the higher-order one. As a non-trivial example we study the transfer of the (strong) standardization theorem.Fil:Bonelli, Eduardo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina

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

Probabilistic Rewriting: Normalization, Termination, and Unique Normal Forms

While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. We study in this setting questions such as uniqueness of the result (unique limit distribution) and normalizing strategies (is there a strategy to find a result with greatest probability?). The goal is to have tools to analyse the operational properties of probabilistic calculi (such as probabilistic lambda-calculi) whose evaluation is also non-deterministic, in the sense that different reductions are possible

Probabilistic Rewriting: On Normalization, Termination, and Unique Normal Forms

While a mature body of work supports the study of rewriting systems, even infinitary ones, abstract tools for Probabilistic Rewriting are still limited. Here, we investigate questions such as uniqueness of the result (unique limit distribution) and we develop a set of proof techniques to analyze and compare reduction strategies. The goal is to have tools to support the operational analysis of probabilistic calculi (such as probabilistic lambda-calculi) whose evaluation is also non-deterministic, in the sense that different reductions are possible. In particular, we investigate how the behavior of different rewrite sequences starting from the same term compare w.r.t. normal forms, and propose a robust analogue of the notion of "unique normal form". Our approach is that of Abstract Rewrite Systems, i.e. we search for general properties of probabilistic rewriting, which hold independently of the specific structure of the objects.Comment: Extended version of the paper in FSCD 2019, International Conference on Formal Structures for Computation and Deductio

Encoding the Factorisation Calculus

Jay and Given-Wilson have recently introduced the Factorisation (or SF-) calculus as a minimal fundamental model of intensional computation. It is a combinatory calculus containing a special combinator, F, which is able to examine the internal structure of its first argument. The calculus is significant in that as well as being combinatorially complete it also exhibits the property of structural completeness, i.e. it is able to represent any function on terms definable using pattern matching on arbitrary normal forms. In particular, it admits a term that can decide the structural equality of any two arbitrary normal forms. Since SF-calculus is combinatorially complete, it is clearly at least as powerful as the more familiar and paradigmatic Turing-powerful computational models of Lambda Calculus and Combinatory Logic. Its relationship to these models in the converse direction is less obvious, however. Jay and Given-Wilson have suggested that SF-calculus is strictly more powerful than the aforementioned models, but a detailed study of the connections between these models is yet to be undertaken. This paper begins to bridge that gap by presenting a faithful encoding of the Factorisation Calculus into the Lambda Calculus preserving both reduction and strong normalisation. The existence of such an encoding is a new result. It also suggests that there is, in some sense, an equivalence between the former model and the latter. We discuss to what extent our result constitutes an equivalence by considering it in the context of some previously defined frameworks for comparing computational power and expressiveness

Probabilistic Rewriting and Asymptotic Behaviour: on Termination and Unique Normal Forms

While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. In this paper we study the question of uniqueness of the result (unique limit distribution), and develop a set of proof techniques to analyze and compare reduction strategies. The goal is to have tools to support the operational analysis of probabilistic calculi (such as probabilistic lambda-calculi) where evaluation allows for different reduction choices (hence different reduction paths)

Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems

We study perpetuality of reduction steps, as well as perpetuality of redexes, in orthogonal rewrite systems. A perpetual step is a reduction step which retains the possibility of infinite reductions. A perpetual redex is a redex which, when put into an arbitrary context, yields a perpetual step. We generalize and refine existing criteria for the perpetuality of reduction steps and redexes in orthogonal Term Rewriting Systems and the λ-calculus due to Bergstra and Klop and others. We first introduce Context-sensitive Conditional Expression Reduction Systems (CCERSs) and define a concept of orthogonality (which implies confluence) for them. In particular, several important λ-calculi and their extensions and restrictions can naturally be embedded into orthogonal CCERSs. We then define a perpetual reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal fully-extended CCERSs. Using the properties of the minimal perpetual strategy, we prove 1. perpetuality of any reduction step that does not erase potentially infinite arguments, which are arguments that may become, via substitution, infinite after a number of outside steps, and 2. perpetuality (in every context) of any safe redex, which is a redex whose substitution instances may discard infinite arguments only when the corresponding contracta remain infinite. We prove both these perpetuality criteria for orthogonal fully-extended CCERSs and then specialize and apply them to restricted λ-calculi, demonstrating their usefulness. In particular, we prove the equivalence of weak and strong normalization (which equivalence is here called uniform normalization) for various restricted λ-calculi, most of which cannot be derived from previously known perpetuality criteria