A certified module of Simplicial Complexes for the Kenzo system

In the field of Intelligent Information Processing, mechanized reasoning systems provide a chance of increasing the reliability of software systems, namely Computer Algebra Systems. This paper is devoted to a concrete case of this topic. The notion of simplicial complex, see [7], is the most elementary method to settle a connection between common “general” topology and homological algebra. The notion of topological space is too “abstract” in order to perform computations. A triangulation, by means of simplicial complexes, can be provided for “sensible” spaces, so every topological space can be considered as a simplicial complex, making the computations easier. Nevertheless, many common constructions in topology are difficult to make explicit in the framework of simplicial complexes. It soon became clear in the forties the notion of simplicial set is much better. The reference [7] remains the basic reference in this subject. The Kenzo system [2] is a Common Lisp program which works with the main mathematical structures used in Simplicial Algebraic Topology, namely it is able to work with simplicial sets. However the notion of simplicial complex is not included in the Kenzo system. Kenzo was written mainly as a research tool and has got relevant results which have not been confirmed nor refuted by any other means. Then, the question of Kenzo reliability (beyond testing) arose in a natural way. Several works (see [1] and [6]) have focussed on studying the correctness of first order fragments of Kenzo with the ACL2 theorem prover [5]. We have undertaken two tasks: on the one hand, the development of a new Kenzo module which integrates the notion of simplicial complex. On the other hand, certifying the correctness of this module using the ACL2 theorem prover. The complete development can be found in [3].