Armando Martin Haeberer

A Finite Axiomatization for Fork Algebras

Proper fork algebras are algebras of binary relations over a structured set. The underlying set has changed from a set of pairs to a set closed under an injective function. In this paper we present a representation theorem for their abstract counterpart, that entails that proper fork algebras — whose underlying set is closed under an injective function — constitute a finitely based variety.1

Comparing Two Different Approaches to Products in Abstract Relation Algebra

During the development of relation algebra as a formal programming tool, the need of some form of “categorical product” of relations became apparent, whether as a type or as an operation. Two approaches arose in the late 70’s and the early 80’s which will be referred here as the “Munich approach” (see, e.g., [18, 7]) and the “Rio approach” (see, e.g., [13, 12, 22]).

On the Smooth Calculation of Relational Recursive Expressions out of First-Order Non-Constructive Specifications Involving Quantifiers

The work presented here has its focus on the formal construction of programs out of non-constructive specifications involving quantifiers. This is accomplished by means of an extended abstract algebra of relations whose expressive power is shown to encompass that of first-order logic. Our extension was devised for tackling the classic issue of lack of expressiveness of abstract relational algebras first stated by Tarski and later formally treated by Maddux, Nemeti, etc. First we compare our extension with classic approaches to expressiveness and our axiomatization with modern approaches to products. Then, we introduce some non-fundamental operations. One of them, the relational implication, is shown to have heavy heuristic significance both in the statement of Galois connections for expressing relational counterparts for universally quantified sentences and for dealing with them. In the last sections we present two smooth program derivations based on the theoretical framework introduced previously.

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.

Representability and program construction within fork algebras

The representation theorem for fork algebras was always misunderstood regarding its applications in program construction. Its application was always described as “the portability of properties of the problem domain into the abstract calculus of fork algebras”. In this paper we show that the results provided by the representation theorem are by far more important. We show that not only the heuristic power coming from concrete binary relations is captured inside the abstract calculus, but also design strategies for program development can be successfully expressed. This result makes fork algebras a programming calculus by far more powerful than it was previously thought.

Adding Design Strategies to Fork Algebras

The representation theorem for fork algebras was always misunderstood regarding its applications in program construction. Its application was always described as “the portability of properties of the problem domain into the abstract calculus of fork algebras”. In this paper we show that the results provided by the representation theorem are by far more important. Here we show that not only the heuristic power coming from concrete binary relations is captured inside the abstract calculus, but also design strategies for program development can be successfully expressed. This result makes fork algebras a programming calculus by far more powerful than it was previously thought.

A calculus for program construction based on fork algebras, design strategies and generic algorithms

At the end of Chapter 4 of the RelMiCS book [11] an application of fork algebras as the basis for a calculus for program construction is outlined. In this paper we make a detailed presentation of the calculus as well as present some examples. We present a methodology for program construction based on the first-order theory of fork algebras. In this theory we will describe program design strategies, for instance case analysis, trivialization, divide-and-conquer and others. Using these strategies, from generic specifications (i.e., parameterized specifications) we will derive parametric algorithms. We will also provide conditions that will help in finding the parameters of the generic algorithms from the parameters in the specifications. We assume the reader is acquainted with the terminology and notation for relation and fork algebras, as well as with their basic properties as they were presented in the RelMiCS book [11].