Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Decreasing diagrams for confluence and commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a wellfounded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf ’s theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Point-free program calculation

Tese de doutoramento em Informática, ramo de Fundamentos da ComputaçãoDue to referential transparency, functional programming is particularly appropriate for equational reasoning. In this thesis we reason about functional programs by calculation, using a program calculus built upon two basic ingredients. The first is a set of recursion patterns that allow us to define recursive functions implicitly. These are encoded as higher-order operators that encapsulate typical forms of recursion, such as the well-known foldr operator on lists. The second is a point-free style of programming in which programs are expressed as combinations of simpler functions, without ever mentioning their arguments. The basic combinators are derived from standard categorical constructions, and are characterized by a rich set of equational laws. In order to be able to apply this calculational methodology to real lazy functional programming languages, a concrete category of partial functions and elements is used. While recursion patterns are already well accepted and a lot of research has been carried out on this topic, the same cannot be said about point-free programming. This thesis addresses precisely this component of the calculus. One of the contributions is a mechanism to translate classic pointwise code into the point-free style. This mechanism can be applied to a λ-calculus rich enough to represent the core functionality of a real functional programming language. A library that enables programming in a pure pointfree style within Haskell is also presented. This library is useful for understanding the expressions resulting from the above translation, since it allows their direct execution and, where applicable, the graphical visualization of recursion trees. Another contribution of the thesis is a framework for performing point-free calculations with higher-order functions. This framework is based on the internalization of some basic combinators, and considerably shortens calculations in this setting. In order to assess the viability of mechanizing calculations, several issues are discussed concerning the decidability of equality and the implementation of fusion laws.Devido à transparência referencial, o paradigma funcional é particularmente adequado ao raciocínio equacional. Nesta tese a manipulação de programas funcionais será feita por cálculo, sendo o cálculo de programas constituído por dois ingredientes fundamentais. O primeiro é um conjunto de padrões de recursividade que nos permite definir funções recursivas implicitamente. Estes padrões são codificados como operadores de ordem superior que ecapsulam formas de recursão típicas, tal como o bem conhecido operador foldr para listas. O segundo ingrediente é um estilo de programação “pointfree”, no qual os programas são definidos por combinação de funções mais simples sem nunca mencionar explicitamente os seus argumentos. Os combinadores fundamentais são derivados de construções categoriais padrão, e são caracterizados por um conjunto expressivo de leis equacionais. Para ser possível aplicar este método de cálculo a linguagens de programação funcional “lazy”, foi usada uma categoria concreta onde as funções e os elementos podem ser parciais. Ao contrário dos padrões de recursividade, que já são bem aceites e sobre os quais já se fez muita investigação, o mesmo não se pode dizer sobre a programação “point-free”. Esta tese aborda precisamente este componente do cálculo. Uma das contribuições é um mecanismo que permite traduzir código “pointwise” clássico para o estilo “point-free”. Este mecanismo pode ser aplicado a um λ-calculus suficientemente expressivo para representar a funcionalidade básica de uma linguagem de programação funcional real. Também se apresenta uma biblioteca que permite programar num estilo “point-free” puro dentro da linguagem Hankell. Esta biblioteca é útil para compreender as expressões que resultam da tradução acima referida, pois permite a sua execução directa e, quando aplicável, a visualização gráfica de árvores de recursividade. Outra contribuição da tese consiste numa metodologia para realizar cálculos “point-free” sobre funções de ordem superior. Esta metodologia é baseada na internalização de alguns combinadores fundamentais, e permite encurtar significativamente os cálculos. Para estabelecer a viabilidade de mecanização, também se discutem várias questões relacionadas com a decidibilidade da igualdade e a implementação de leis de fusão