Why Writing Correct Software Is Hard

Writing correct software is an incredibly difficult task. As you read through the story of theoretical computer science, you'll discover the deep-rooted challenges that make achieving software correctness a daunting endeavor.

Hilbert's program, proposed over a century ago, aimed to formalize all of mathematics. However, Gödel's incompleteness theorems shattered this ideal, proving that it's impossible to prove the consistency and completeness of a formal system. Moreover, Church and Turing shattered the third pillar of Hilbert's program – the decidability of mathematical statements – by proving the halting theorem. This theorem states that no algorithm can determine if a given program will ever halt, or terminate.

The halting theorem has far-reaching implications for software verification. It is closely tied to the problem of software correctness. Specifically, there is no algorithm that can determine whether a program will reach a certain state, given an initial state. This has immediate corollaries for software verification. One such corollary is Rice's theorem, which states that no algorithm can decide any non-trivial property of a program's partial functions.

This means that verifying that a program possesses certain desirable properties, such as ensuring termination or avoiding certain behaviors, is fundamentally difficult.

In conclusion, the pursuit of software correctness is a hard challenge due to the inherent limitations of computation. Despite the existence of various tools and techniques that can assist in ensuring correctness, the fundamental cost of correctness lies in the natural laws of computation. These limitations are deeply rooted in the history and development of theoretical computer science, and they continue to shape our understanding of the inherent difficulty in producing correct software.