Proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science 42nd International Symposium on Mathematical Foundations of Computer Science Leibniz International Proceedings in Informatics (2017)
Ghani, Neil (Principal Investigator) McBride, Conor (Co-investigator)
"The cost of software failure is truly staggering. Well known individual cases include the Mars Climate Orbiter failure (£80 million), Ariane Rocket disaster (£350 million), Pentium Chip Division failure (£300 million), and more recently the heartbleed bug (est. £400 million). There are many, many more examples. Even worse, failures such as one in the Patriot Missile System and another in the Therac-25 radiation system have cost lives. More generally, a 2008 study by the US government estimated that faulty software costs the US economy £100 billion annually.
There are many successful approaches to software verification (testing, model checking etc). One approach is to find mathematical proofs that guarantees of software correctness. However, the complexity of modern software means that hand-written mathematical proofs can be untrustworthy and this has led to a growing desire for computer-checked proofs of software correctness. Programming languages and interactive proof systems like Coq, Agda, NuPRL and Idris have been developed based on a formal system called Martin Type Theory. In these systems, we can not only write programs, but we can also express properties of programs using types, and write programs to express proofs that our programs are correct. In this way, both large mathematical theorems such as the Four Colour Theorem, and large software systems such as the CompCert C compiler have been formally verified. However, in such large projects, the issue of scalability arises: how can we use these systems to build large libraries of verified software in an effective way?
This is related to the problem of reusability and modularity: a component in a software system should be replaceable by another which behaves the same way even though it may be constructed in a completely different way. That is, we need an extensional equality which is computationally well behaved (that is, we want to run programs using this equality). Finding such an ty is a fundamental and difficult problem which has remained unresolved for over 40 years.
But now it looks like we might have a solution! Fields medallist Vladimir Voevodsky has come up with a completely different take on the problem by thinking of equalities as paths such as those which occur in one of the most abstract branches of mathematics, namely homotopy theory, leading to Homotopy Type Theory (HoTT). In HoTT, two objects are completely interchangeable if they behave the same way. However, most presentations of HoTT involve axioms which lack computational justification and, as a result, we do not have programming languages or verification systems based upon HoTT. The goal of our project is to fix that, thereby develop the first of a new breed of HoTT-based programming languages and verification systems, and develop case studies which demonstrate the power of HoTT to programmers and those interested in formal verification.
We are an ideal team to undertake this research because i) we have unique skills and ideas ranging from the foundations of HoTT to the implementation and deployment of programming language and verification tools; and ii) the active collaboration of the most important figures in the area (including Voevodsky) as well as industrial participation to ensure that we keep in mind our ultimate goal -- usable programming language and verification tools."