Sign In

Communications of the ACM

Review Articles

Mechanical Mathematicians

stylized mechanical gears, illustration

Credit: Martin Capek

In the 1960s, researchers dreamed of automatic theorem provers powerful enough to develop lengthy proofs of conjectures that no human was able to prove before. Some even thought these "mechanical mathematicians" would eventually replace flesh-and-blood mathematicians. However, for many decades, automatic provers remained too weak to help, let alone replace, mathematicians. Instead, mathematicians turned to computer algebra systems, which help not so much with proofs as with computations, and automatic provers turned to other application areas. These areas include hardware and software verification, either directly or via an interactive verification platform (for example, Atelier B, Dafny, F*, Frama-C, SPARK 2014, Spec#, VCC, Why3). In addition, automatic provers are used as back ends to general-purpose proof assistants (for example, ACL2, Coq, HOL, Isabelle, Lean, Mizar, PVS). As mathematicians are slowly embracing proof assistants, automatic provers are finally becoming useful to them, for discharging straight-forward but tedious proof tasks.

Back to Top

Key Insights


As a simple example from elementary mathematics, consider the formula


The logical symbols ∧ and ⇒ mean "and" and "implies," gcd is the greatest common divisor, and | is the "divides" relation between two numbers (for example, d | ab means ab is divisible by d). Let us check a special case: If a = 12, b = 35, and d = 5, we have a' = 1, b' = 5, and indeed a'b' = 5 divides d = 5 and vice versa. Mathematicians can easily see that the formula holds in general, but to write a formal proof with an acceptable level of detail for a proof assistant can easily take 15 or 30 minutes. In contrast, automatic theorem provers can prove such formulas within seconds.


No entries found

Log in to Read the Full Article

Sign In

Sign in using your ACM Web Account username and password to access premium content if you are an ACM member, Communications subscriber or Digital Library subscriber.

Need Access?

Please select one of the options below for access to premium content and features.

Create a Web Account

If you are already an ACM member, Communications subscriber, or Digital Library subscriber, please set up a web account to access premium content on this site.

Join the ACM

Become a member to take full advantage of ACM's outstanding computing information resources, networking opportunities, and other benefits.

Subscribe to Communications of the ACM Magazine

Get full access to 50+ years of CACM content and receive the print version of the magazine monthly.

Purchase the Article

Non-members can purchase this article or a copy of the magazine in which it appears.
Sign In for Full Access
» Forgot Password? » Create an ACM Web Account