Modern development tools cannot eliminate the fundamental limits dictated by mathematical theory, leaving developers unable to computationally resolve all program behaviors. This dilemma traces back to David Hilbert's failed quest to prove that formal systems could be entirely consistent, complete, and decidable. Kurt Gödel, Alonzo Church, and Alan Turing ultimately dismantled this vision by mathematically demonstrating inherent computational limitations. Today, these historical boundaries continue to restrict modern verification methods.
Key points
Achieving total program correctness comes with inescapable costs governed by the fundamen…: Achieving total program correctness comes with inescapable costs governed by the fundamental laws of computation.
Automated verification tools face permanent mathematical limits that prevent the resoluti…: Automated verification tools face permanent mathematical limits that prevent the resolution of certain critical software behaviors.
Mathematical theory demonstrates that complete program understanding and computation are…: Mathematical theory demonstrates that complete program understanding and computation are inherently at odds with one another.
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… and why math (alone) won’t help us
I was very happy to speak at Curry On once again. Last year, following our work with Quasar, I spoke on continuations vs. monads. This year I gave a talk about computational complexity results in software verification, a subject I became interested in while formally specifying and verifying our upcoming revamped version of Galaxy.
What follows is an extended version of the talk (embedded above), containing some additional proofs and results I didn’t have time to cover, complete with footnotes, for those wishing to study the topics further. Reading time: ~45 mins.
In this article, we try to understand the relationship between programs and correctness, and in particular, why writing correct programs is hard. We will review results from computability theory and complexity theory, and see that programs and complete understanding – which is required for correctness – are fundamentally at odds. The main contribution of this text is aggregating in one place some results (as well as some references) pertaining to the essential difficulty of producing correct software. My motivation was…
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