Verifying the bridge between simplicial topology and algebra: the Eilenberg-Zilber algorithm

Abstract

The Eilenberg-Zilber algorithm is one of the central components of the computer algebra system called Kenzo, devoted to computing in Algebraic Topology. In this article we report on a complete formal proof of the underlying Eilenberg-Zilber theorem, using the ACL2 theorem prover. As our formalization is executable, we are able to compare the results of the certified programme with those of Kenzo on some universal examples. Since the results coincide, the reliability of Kenzo is reinforced. This is a new step in our long-term project towards certified programming for Algebraic Topology.

The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?

“…The commercial computer algebra systems are black boxes and their algorithms are opaque to the users (of course, also the source code), and certainly this does not contribute to avoid errors. It makes difficult to apply modern techniques of software verification to this kind of systems (as an example of verification in the context of an open source computer algebra systems, see [5]). Moreover, known bugs of computer algebra systems should be available to the users; this is usual in free software, but an anathema for commercial packages.…”

Modelling algebraic structures and morphisms in ACL2

“…We are mainly interested in the Kenzo system, where the methodologies and tools presented in this paper can reduce the formalisation effort in works like [44,47].…”

“…loops were replaced by tail-recursive functions). The work presented in [2,35,36,50] did not concern algebraic structures, but ACL2 has also been used to formalise constructions (the Normalisation theorem and the Eilenberg-Zilberg theorem [59]) involving algebraic structures implemented in Kenzo, see [44,47]. In contrast to the work in Isabelle and Coq, the algebraic structures involved in the ACL2 formalisations were developed from scratch instead of using, as a basis, a previously developed algebraic hierarchy.…”

“…The formalisations presented in [44,47] show that it is possible to reason about algebraic concepts in ACL2; however, the first-order quantifier-free logic of ACL2 stands in the way of a widespread use of this ITP to work with algebraic structures. In this paper, we show how this weakness can be overcome thanks to two of ACL2's great strengths: programmable extensibility and proof automation.…”

“…In the next section, we present a brief introduction to the ACL2 system and the tools employed in our development. Section 3 presents how algebraic structures can be modelled from scratch in ACL2 (this is the approach followed in [44,47]), and the difficulties that are associated with this process. To overcome those problems, we present a methodology (by means of examples) to implement a set of tools for algebraic structures in Sect.…”