In Tezos, the skip-list data structure is used in several places. Verification of
skip-list
required many days of painstaking work, and this work is not yet finished.
In this post, we will talk about the techniques we used.
In particular, we describe the verification of function mem - one of the functions
operating in this data structure. More information about formal verification of skip list
can be found here.
Formal verification of the skip list of Tezos includes the work on verification of the following properties :
content and a list of back pointers),cell and pointer,equality definition,next, genesis, best_skip, mem, pp and others,back_path (the list of backpointers),express line (back_path, list of pointers, generated with
usage of best_skip function).As you may already know from the article Simulations - dependently-typed version,
in the Proto_alpha folder we keep code, generated by the coq-of-ocaml.
Many of the fixpoints here are preceded by the annotation #[bypass_check(guard)],
which means that guard checking is locally disabled. And this means that
we can not consider these functions to be total. In such cases, proof engineers
usually feel irritation, because "total" is one of their favorite words.
We know that total functions never crush and they always return a well-typed
result within a finite time.
Thanks to Alan Turing we know that the halting problem - the difficulty of determining whether a specific program terminates or not - is undecidable. Coq can't determine if a function is total in general, according to the original definition of totality. Instead, it analyzes a function's syntax and makes a conservative approximation.
Our current goal is that the doubles defined in the folder
Simulations, whose equivalence with their prototypes from Proto_alpha should
be proved, would be total (have enabled Coqs guard checking on fixpoint).
We already mentioned the Guard Checking question here.
Coq is analyzing the function's syntax to consider it to be total. One way to convince Coq that a function is total is to show that there is a decreasing argument that converges to the base case.
Let's consider the function parse_ty_aux from Proto_alpha and its clone
dep_parse_ty_aux from the Simulations folder. parse_ty_aux is having
{struct node_value} annotation. The point of the {struct ident} annotation is
to tell the Coq, which argument decreases along the recursive calls.
node_value : Alpha_context.Script.node is designed in such a way that it is
not trivial to show that the function converges. Coq can not detect that each
and every recursion call is done on the direct subterm of a given node_value
argument (and it is indeed actually not the case).