How to say greedy in fork algebras

Because of their expressive power, binary relations are widely used in program specification and development within formal calculi. The existence of a finite equational axiomatization for algebras of binary relations with a fork operation guarantees that the heuristic power coming from binary relations is captured inside an abstract equational calculus. In this paper we show how to express the greedy program design strategy into the first order theory of fork algebras.Eje: TeoríaRed de Universidades con Carreras en Informática (RedUNCI

How to say greedy in fork algebras

Because of their expressive power, binary relations are widely used in program specification and development within formal calculi. The existence of a finite equational axiomatization for algebras of binary relations with a fork operation guarantees that the heuristic power coming from binary relations is captured inside an abstract equational calculus. In this paper we show how to express the greedy program design strategy into the first order theory of fork algebras.Eje: TeoríaRed de Universidades con Carreras en Informática (RedUNCI

On the construction of explosive relation algebras

Fork algebras are an extension of relation algebras obtained by extending the set of logical symbols with a binary operator called fork. This class of algebras was introduced by Haeberer and Veloso in the early 90's aiming at enriching relation algebra, an already successful language for program specification, with the capability of expressing some form of parallel computation. The further study of this class of algebras led to many meaningful results linked to interesting properties of relation algebras such as representability and finite axiomatizability, among others. Also in the 90's, Veloso introduced a subclass of relation algebras that are expansible to fork algebras, admitting a large number of non-isomorphic expansions, referred to as explosive relation algebras. In this work we discuss some general techniques for constructing algebras of this type

Tarski's Q-relation algebras and Thompson's groups

The connections between Tarski's Q-relation algebras and Thompson's groups F, T, V, and monoid M are reviewed here, along with Jonsson-Tarski algebras, fork algebras, true pairing algebras, and tabular relation algebras. All of these are related to the finitization problem and Tarski's formalization of set theory without variables. Most of the technical details occur in the variety of J-algebras, which is obtained from relation algebras by omitting union and complementation and adopting a set of axioms created by Jonsson. Every relation algebra or J-algebra that contains a pair of conjugated quasiprojections satisfying the Domain and Unicity conditions, such as those that arise from J\'onsson-Tarski algebras or fork algebras, will also contain homomorphic images of F, T, V, and M. The representability of tabular relation algebras is extended here to J-algebras, using a notion of tabularity equivalent among relation algebras to the original definition.Comment: 64 pages, 4 figures, 1 tabl

From Specifications to Programs: A Fork-Algebraic Approach to Bridge the Gap

The development of programs from first-order specifications has as its main difficulty that of dealing with universal quantifiers. This work is focused in that point, i.e., in the construction of programs whose specifications involve universal quantifiers. This task is performed within a relational calculus based on fork algebras. The fact that first-order theories can be translated into equational theories in abstract fork algebras suggests that such work can be accomplished in a satisfactory way. Furthermore, the fact that these abstract algebras are representable guarantees that all properties valid in the standard models are captured by the axiomatization given for them, allowing the reasoning formalism to be shifted back and forth between any model and the abstract algebra. In order to cope with universal quantifiers, a new algebraic operation — relational implication — is introduced. This operation is shown to have deep significance in the relational statement of first-order expressions involving universal quantifiers. Several algebraic properties of the relational implication are stated showing its usefulness in program calculation. Finally, a non-trivial example of derivation is given to asses the merits of the relational implication as an specification tool, and also in calculation steps, where its algebraic properties are clearly appropriate as transformation rules.Laboratorio de Investigación y Formación en Informática Avanzad

Translating fork specifications into logic programs

In this work a compiler from fork specifications into logic programs is presented. The technique implemented by the compiler consists of transforming a set of fork equations (with some restrictions) into normal logic programs in such a way that the semantics of the fork equations is preserved. After translating a fork specification, it can be executed by consulting the generated logic program. The fork compiler, a tool for the translation, is also introduced.Facultad de Informátic

A proof of the interpretability of P/PML in a relational setting

In [1] we presented the logic P=PML, a formalism suitable for the speci cation and construction of Real-Time systems. The main algebraic result, namely, the interpretability of P/PML into an equa- tional calculus based on w-closure fork algebras (which allows to reason about Real-Time systems in an equational calculus) was stated but not proved because of the lack of space. In this paper we present a detailed proof of the interpretability theorem, as well as the proof of the representation theorem for w-closure fork alge- bras which provides a very natural semantics based on binary relations for the equational calculus.Eje: TeoríaRed de Universidades con Carreras en Informática (RedUNCI

A proof of the interpretability of P/PML in a relational setting

In [1] we presented the logic P=PML, a formalism suitable for the speci cation and construction of Real-Time systems. The main algebraic result, namely, the interpretability of P/PML into an equa- tional calculus based on w-closure fork algebras (which allows to reason about Real-Time systems in an equational calculus) was stated but not proved because of the lack of space. In this paper we present a detailed proof of the interpretability theorem, as well as the proof of the representation theorem for w-closure fork alge- bras which provides a very natural semantics based on binary relations for the equational calculus.Eje: TeoríaRed de Universidades con Carreras en Informática (RedUNCI

Noncommutative localization in noncommutative geometry

The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of ``spaces'', locally described by noncommutative rings and their categories of one-sided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical techniques are studied as well. We also describe a counterexample for a folklore test principle. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with emphasis on descent formalism, flatness, abelian categories of quasicoherent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Watts, Deligne and Rosenberg. The Cohn universal localization does not have good flatness properties, but it is determined by the localization map already at the ring level. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but with few smaller new result

"Forks without philosophers" o de cómo la cuantificación universal perdió una batalla y de las ventajas que ello reportó

En sentido general, este trabajo trata sobre la construcción formal de programas. La necesidad de construir los programas formalmente ha sido ampliamente discutida a lo largo de los últimos años, y ha adquirido una importancia cada vez mayor. En los comienzos, la programación se realizaba de una forma intuitiva, casi se podría decir artística; con el correr de los años se comprobó que los métodos utilizados eran inadecuados, por lo que se desarrollaron nuevas técnicas, mediante las cuales un programa debía ser diseñado al mismo tiempo que la prueba de su corrección. Quizás la mejor metáfora para describir la necesidad de formalidad es la que D’Argenio describe en el prefacio de su tesis de grado, donde compara a los programas con moscas, y a los métodos formales con una máquina mata-moscas; si los programadores utilizasen sus zapatos para aplastar las moscas (no utilizando los métodos formales, sino la intuición), sucedería lo que se cita en el epígrafe. Es por ello que se hace tanto hincapié en la investigación de métodos que permitan un tratamiento formal del proceso de desarrollo de software, o bien para la construcción, o bien para la especificación y verificación. En esta tesis utilizaremos un método de construcción que consiste en realizar transformaciones sobre una especificación hasta obtener un programa que la satisfaga.Tesis digitalizada en SEDICI gracias a la colaboración de la Biblioteca de la Facultad de Informática.Facultad de Ciencias Exacta