13
votes
Why/when do we ever need transfinite loop variants?
You are correct when you observe that for any particular terminating loop $L$ we may simply define the invariant "we're getting one step closer to termination". But proving that this is indeed a valid ...
- 28.3k
12
votes
Formal semantics of OCaml in Coq
Have you seen Arthur Charguéraud's PhD thesis, Characteristic Formulae for Mechanized Program Verification?
Rather than building the type system and small-step semantics as inductive relations, he ...
- 32.4k
9
votes
Why/when do we ever need transfinite loop variants?
I would like to add the following to Andrej's response (not enough rep for a comment).
Indeed, we cannot avoid ordinals but we may hide them. One approach is to use some modal logic that takes ...
- 326
9
votes
Accepted
Why is it impossible to prove software to be correct?
Whatever you heard is false. Proving programs correct is not only possible, it is also done in practice on large scale. There is an entire branch of computer science, namely formal methods which ...
- 28.3k
8
votes
Formal semantics of OCaml in Coq
You could be interested in Jacques Garrigue's A Certified Implementation of ML with Structural Polymorphism and Recursive Types, which establishes the soundness of static and dynamic semantics and ...
- 2,040
6
votes
Verified type checkers
Here are some results of a simple Google search:
Certification of a Type Inference Tool for ML: Damas–Milner within Coq by Catherine Dubois and Valérie Ménissier-Morain
Formalization of a Polymorphic ...
- 28.3k
6
votes
Accepted
Lee's algorithm for synthesis of ranking functions in size-change termination proofs
The complexity is acceptable in current verifiers, and has been implemented in at least the AProVE termination analysis tool for term rewrite systems.
They describe their implementation in Lazy ...
- 13.7k
6
votes
How to determine whether a proof requires "higher-order reasoning techniques"?
Briefly, every theorem stated in first-order logic has a first-order proof.
In his book "An Introduction to Mathematical Logic and Type Theory", Peter B. Andrews develops both first-order logic and a ...
- 161
5
votes
Accepted
Complexity of a graph-rewriting problem
I don't know if it has been studied before, but after a quick look I think it should be PSPACE complete.
We can build a reduction using the Nondeterministic Constraint Logic model of computation (NCL)...
- 22.6k
4
votes
Accepted
Ordering sequences containing bitvectors for size-change termination
Surely, you want $s_1 < s_2$ if there is a $t$ such that $s_1.t < s_2.t$ and $s_1.u \leq s_2.u$ for every other field $u$. That, at least, gives you a well-founded order.
But there are many ...
- 13.7k
4
votes
What's the difference between an invariant and an "inductive" invariant?
Just to expand on Jukka's excellent answer, the question of finding invariants usually arises when trying to establish safety properties. A safety property $S$ is a property that we wish for the ...
- 181
4
votes
Accepted
Proof that the theory of rationals is convex
The key idea here is that for any conjunction of equations $F\equiv u_1=v_1\wedge\ldots\wedge u_k=v_k$, the set $S_F$ is convex in the geometric sense, i.e. for any two points $p,q\in S_F$, all points ...
- 2,500
4
votes
Accepted
Is it possible to verify a typechecker for a total dependently-typed language in that language's logic?
As often in these matters, the encodings are important: you could have some silly encoding of Turing machines or Gallina terms or both where an input string $\langle n\rangle$ represents "the $n$-th ...
- 13.7k
3
votes
Accepted
Practical example: how to formally verify "file name" implementation from a spec?
In general, the technique used is known as "fuzzing". Not all errors are equally likely. Let's consider two hypothetical errors:
System A incorrectly rejects a filename if it contains an ...
- 334
3
votes
Accepted
State of the Art for the Monadic Class?
I found signs that such a decision procedure was implemented in the (general purpose) theorem prover SPASS.
In particular see the thesis of Ann-Christin Knoll, On Resolution
Decision Procedures
for ...
- 13.7k
3
votes
State of the Art for the Monadic Class?
In a 1993 LICS paper, Bachmair, Ganzinger and Waldmann showed that set constraints are equivalent to monadic FOL, in Set Constraints are the Monadic Class. If memory serves, set constraints are ...
- 32.4k
3
votes
Why/when do we ever need transfinite loop variants?
If I may add something, too, I'd suggest that some of the ways ordinals are presented tend to make them sound more 'suspicious' than they actually are. At least, I think there are other ways of ...
- 921
3
votes
Nominal Tree Languages i.e. with Binders and Infinite Symbols?
I think section 3.1 (titled "Nominal tree automata") of this paper: https://drops.dagstuhl.de/opus/volltexte/2019/11434/ may be what you're looking for, they use λ-terms as their example.
In ...
2
votes
How are safety/liveness languages defined on the set of finite or infinite words?
The answer is yes to all questions, so it is enough to answer 2 and 4, as the definitions work in particular for languages of finite words:
A language $P\subseteq \Sigma^*\cup\Sigma^\omega$ is safety ...
- 8,598
2
votes
Practical example: how to formally verify "file name" implementation from a spec?
For the concrete case of a specification of a regular language, there is the Java String Analyzer which roughly is able to compute a finite state automaton (i.e. regular expression) of the set of ...
- 393
1
vote
Why is it impossible to prove software to be correct?
I'm wondering if you've misheard or heard a distorted version of the Donald Knuth Quote:
Beware of bugs in the above code; I have only proved it correct, not tried it.'
There is definitely some sort ...
- 2,830
1
vote
Confusions about the technique for verifying implementations of linearizable objects in [Herlihy and Wing, 1990]
Herlihy and Wing write on p. 477:
In conclusion, the rep invariant $\mathbf{I}$ must be continually satisfied and the abstraction function continually defined, not only between abstract operations,...
- 845
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