Electronic Communication of Mathematics and the Interaction of Computer Algebra Systems and Proof Assistants

Abstract

Present day computer algebra systems (CASs) and proof assistants (PAs) are specialized programs that help humans with mathematical computations and deductions. Although several such systems are impressive, they all have certain limitations. In most CASs side conditions that are essential for the truth of an equality are not formulated; moreover there are bugs. The PAs have a limited power for computing and hence also for assistance with proofs. Almost all examples of both categories are stand alone special purpose systems and therefore they cannot communicate with each other. We will argue that the present state of the art in logic is such that there is a natural formal language, independent of the special purpose application in question, by which these systems can communicate mathematical statements. In this way their individual power will be enhanced. Statements received at one particular location from other sites fall into two categories: with or without the qualification “evidently impeccable", a notion that is methodologically precise and sound. For statements having this quality assessment the evidence may come from the other site or from the local site itself, but in both cases it is verified locally. In cases where there is no evidence of impeccability one has to rely on cross checking. There is a trade-off between these two kinds of statements: for impeccability one has to pay the price of obtaining less power. Some examples of communication forms are given that show how the participants benefit.