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Fuel - quiet escape from the halting problem

· 4 min read
Natasha Klaus

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.

note

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).

#[bypass_check(guard)]Fixpoint parse_ty_aux {ret : Set}  (ctxt : Alpha_context.context) (stack_depth : int) (legacy : bool)  (allow_lazy_storage : bool) (allow_operation : bool) (allow_contract : bool)  (allow_ticket : bool) (ret_value : parse_ty_ret)  (node_value : Alpha_context.Script.node) {struct node_value}  : M? (ret * Alpha_context.context) :=  let parse_passable_ty_aux_with_ret {ret} := 'parse_passable_ty_aux_with_ret    ret in  let parse_any_ty_aux := 'parse_any_ty_aux in  let parse_big_map_value_ty_aux := 'parse_big_map_value_ty_aux in  let? ctxt := Alpha_context.Gas.consume ctxt Typecheck_costs.parse_type_cycle    in  if stack_depth >i 10000 then    Error_monad.error_value      (Build_extensible "Typechecking_too_many_recursive_calls" unit tt)  else ...

For dep_parse_ty_aux we change the recursive call argument annotation, we make it {struct fuel}

Fixpoint dep_parse_ty_aux   (ctxt : Alpha_context.context) (fuel : nat) (legacy : bool)  (allow_lazy_storage : bool) (allow_operation : bool) (allow_contract : bool)  (allow_ticket : bool) {ret_value : parse_ty_ret}  (node_value : Alpha_context.Script.node) {struct fuel}  : M? (ret_fam ret_value * Alpha_context.context) :=  let parse_passable_ty_aux_with_ret := 'parse_passable_ty_aux_with_ret in  let parse_any_ty_aux := 'parse_any_ty_aux in  let parse_big_map_value_ty_aux := 'parse_big_map_value_ty_aux in  let? ctxt := Alpha_context.Gas.consume ctxt Typecheck_costs.parse_type_cycle  in  match fuel with  | Datatypes.O =>    Error_monad.error_value      (Build_extensible "Typechecking_too_many_recursive_calls" unit tt)  | Datatypes.S fuel => ...

We see that in parse_ty_aux we have an argument stack_depth, we can think of it as being a fuel that allows us to continue the processing. We do something similar for dep_parse_ty. First of all we define it without the #[bypass_check(guard)]. We add new argument fuel : nat, we chose nat because it is easy to work with natural numbers in Coq. We need to prefix the constructors O and S with Datatypes to avoid undesirable collisions and intersections. On each recursive step we consume one drop of fuel and continue execution, and when we've got no more fuel we abandon execution.

match fuel with | Datatypes.O =>

In the next step, we prove equality of our functions. Notice how we define stack_depth parameter for parse_ty_aux using fuel.

Fixpoint parse_ty_aux_dep_parse_ty_aux_eq  (ctxt : Alpha_context.context) (fuel : nat) (legacy : bool)  (allow_lazy_storage : bool) (allow_operation : bool) (allow_contract : bool)  (allow_ticket : bool) {ret_value : parse_ty_ret}  (node_value : Alpha_context.Script.node) : (fuel <= nat_number_10001)%nat ->  match (@dep_parse_ty_aux           ctxt fuel legacy           allow_lazy_storage allow_operation allow_contract           allow_ticket ret_value node_value,          parse_ty_aux            ctxt (number_10001 - Z.of_nat fuel) legacy            allow_lazy_storage            allow_operation allow_contract allow_ticket            ret_value node_value ) with    | (Pervasives.Ok (res, c), Pervasives.Ok (res',c')) =>        ret_fam_to_ex_ty_entrypoints res = res' /\ c = c'    | (Pervasives.Error e, Pervasives.Error e') => e = e'    | _ => False              end. Proof. ...

So summarizing the above: fuel or sometimes in articles or books, we can meet the term gas is a useful life hack for proof engineers, which helps to convince Coq that function terminates, and makes proofs lighter and shorter.