I highly recommend people interested in Coq start with Pierce's Logical Foundations. It is by far the most accessible introduction to Coq I've found. I'm working through Chlipala's books next. CPDT is a great deep-dive into more advanced use cases of Coq as well as its theoretical underpinnings. I'll be taking a class this semester based on FRAP, my understanding is it hits those formal methods and language topics without specifically emphasizing Coq, except for the exercises.
As an aside, since Coq is the only theorem proving language I know well, I am curious what the pros/cons are as compared to Isabelle/HOL, F*, Lean, etc. Coq has been around for a while, but I keep seeing new theorem provers pop up. What are the alleged weaknesses of Coq that other languages are trying to fix?
The main issue is proof automation. Curry-Howard based provers (Coq, Agda, Lean etc) are nice for teaching but if you want to get work done (e.g. you want to verify an OS kernel), you need to automate as much as possible.
Isabelle/HOL currently has by far the most powerful automation. This is partly because, like Coq, it is an old system (unlike Lean and Agda) and lots of automation has been built in the past, but also because a lot of automation is much easier to adapt to provers based on classical logic (as opposed to the constructive logics that Curry-Howard needs (yes, I know that classical reasoning can be done within constructive logic)). Adapting proof automation that has been available for a while for Isabelle/HOL to Curry-Howard is at least in parts an open research problem. Lean has been created in parts because of the urgency of given Curry-Howard systems much better proof automation.
There is an additional dimension which is strength of the logic at hand. The logics Coq, Lean et al are based on are more expressive in various dimensions. Whether that's relevant to your use case depends on what kind of maths you are trying to formalise. If it is higher category theory, it matters a great deal, for program verification it tends not to be important b/c most program verification can be done in rather weak logics.
How does being classical help with automation? I'm more familiar with ACL2, which seems to gain a lot more on the automation front from being first-order than from anything else (though it has sort of a weird take on the excluded middle).
That's a subtle and deep question. One example are nominal techniques. Consider Nominal Isabelle. It is known that nominal techniques are contructive. However, when adding nominal techniques to a classical logic, it is natural to express some of this using double negation, leading to only a small blowup in formula size. Existing proof automation can easily handle this blowuip. Doing the same in a simple minded constructive way, leads to a massive formula size increase, defeating (at the time) existing proof automation.
This was a while back, maybe the problem has been solved now, I have not followed this field. But as far as I'm aware, the 'hammering' approach to proof automation that has been working well with Isabelle/HOL has been successfully ported to Coq/Agda etc.
I know. There have been a lot of impressive verification work in Coq. Huawei is using Coq to do OS verification. Having less automation to hand, in comparison with Isabelle/HOL makes this a bit harder.
Coq has less built-in automation, but with Ltac it's possible to achieve impressive levels of automation through explicitly-crafted decision procedures, as demonstrated in Chlipala's CPDT and many of his papers.
Yes, Coq has a
tactics language. So does Isabelle. What I have in mind are primarily
"Hammers", i.e. hooking in fully automatic theorem prover (e.g. MizAR for
Mizar, Sledgehammer for Isabelle/HOL, HOL(y)Hammer for HOL Light and
HOL4). As far as I'm aware, CoqHammer for Coq is
work-in-progress and does not yet provide the level of automation that you get from Sledgehammer in Isabelle/HOL for years. But admittedly, it's been 4-5 years since I last
used Coq. At the time, two core issues were the following:
- Existing automatic theorem provers often produced non-constructive
proofs.
- Existing automatic theorem provers are mostly for FOL, so the gap to
the much more powerful logics of Curry-Howard based provers is
bigger than with weaker systems.
Have both problems been solved comprehensively as of January 2020?
They have not been solved. My point was that automation via Coq's tactic language can serve as a decent substitute for "push-button" automation like Hammers, at least if one's continually working in the same domain and so can build up a library of tactics.
This is more programming language theory, or, rather the PL-theory portion of formal methods, which is a small portion of formal methods research. Most formal methods research these days focuses on sound static analysis, model checking and concolic testing, techniques that are showing better scalability than the deductive methods favored by PL theorists.
Alloy is a model checker that has a GUI for exploring models. It's more focused on exploring structures. Their site links to a book, Software Abstractions, that will teach you a lot of concepts without needing a strong, mathematical background.
If focus is programming (esp low-level), SPARK Ada lets programmers annotate code with the expected behavior before that's fed into provers that confirm or refute the code operates that way in all situations. Barnes and Barnes wrote a nice book teaching both it and Praxis' Correct-by-Construction method for software development:
Many people don't make it through the stuff teaching formal proof in Coq and so on. It's really hard with few gains along the way. TLA+, SPIN, Alloy, and SPARK Ada are much more accessible with a higher work-to-reward ratio. Those easier results might justify further investment in learning the hard stuff.
Those wondering about how formal methods are used in a lot of situations might like our list on Lobsters. We have a number of professionals and enthusiasts there who stay posting both historically significant and cutting-edge work.
And with SPARK if you want to dive into manual proof of a property (usually because SPARK & Why3 and all the SMT solvers can't manage and you don't have some appropriate lemmas) with Coq or Isabelle.
Not the best way to learn manual proof, but I've seen it used very efficiently.
But then if course you're faced with the difficulty of increasing automation : the more you automate, the more it is difficult to have the 'muscle memory' (and lower cognitive load) that you get when flexing the manual proof often...
Even that is getting automation as I long ago predicted. I think the SPARK-type proofs will be easier to automate this way, too, since they're more mechanical.
* https://softwarefoundations.cis.upenn.edu/
* http://www.concrete-semantics.org/
* http://adam.chlipala.net/frap/
Even the Little series covers types and proofs now:
* https://mitpress.mit.edu/books/little-typer
* https://mitpress.mit.edu/books/little-prover
Plus some topic-specific classics:
* https://mitpress.mit.edu/books/principles-model-checking
* https://www.springer.com/gp/book/9783540654100
* https://www.cis.upenn.edu/~bcpierce/tapl/
And some constructive foundations:
* https://www.cambridge.org/core/books/conceptual-mathematics/...
* https://www.cambridge.org/core/books/sets-for-mathematics/E8...
* https://www.paultaylor.eu/~pt/prafm/
* https://www.cs.cmu.edu/~rwh/pfpl/