Object oriented institutions to specify symbolic computation systems

Abstract

The specification of the data structures used in EAT, a software system for symbolic computation in algebraic topology, is based on an operation that defines a link among different specification frameworks like hidden algebras and coalgebras. In this paper, this operation is extended using the notion of institution, giving rise to three institution encodings. These morphisms define a commutative diagram which shows three possible views of the same construction, placing it in an equational algebraic institution, in a hidden institution or in a coalgebraic institution. Moreover, these morphisms can be used to obtain a new description of the final objects of the categories of algebras in these frameworks, which are suitable abstract models for the EAT data structures. Thus, our main contribution is a formalization allowing us to encode a family of data structures by means of a single algebra (which can be described as a coproduct on the image of the institution morphisms). With this aim, new particular definitions of hidden and coalgebraic institutions are presented.Mathematics Subject Classification. 68Q65, 68Q60.

Formalization of a normalization theorem in simplicial topology

“…Then, some concrete outputs of the program cannot be tested, that is, compared with any expected value. This is the reason why a project to apply formal methods to the study of Kenzo as a software system was launched some years ago [6,12]. Eventually, this research line arrived to the formalization of some parts of Algebraic Topology and Homological Algebra by using proof assistants as Isabelle/HOL [2,3] or Coq [7].…”

A Parameterization Process: From a Functorial Point of View

“…In view of the obtained results, some years ago, the first author of this paper began the formal study of the programs, in order to reach a good understanding on the internal calculation processes of these software systems. In particular, our study of the data types used in EAT and Kenzo [13,6,8] shows that there are two different layers of data structures in the systems. In the first layer, one finds the usual abstract data types, like the type of integers.…”

A case-study in algebraic manipulation using mechanized reasoning tools

“…Previous works have been devoted to the algebraic specification of EAT [15] and Kenzo [9][10][11]; afterwards, we focused on formally proving some basic but significant properties directly extracted from Kenzo. Some relevant properties of the Kenzo system have been formally proved in the proof assistant Isabelle [2][3][4].…”