We saw in the Imp chapter how a naive approach to defining a function representing evaluation for Imp runs into difficulties. There, we adopted the solution of changing from a functional to a relational definition of evaluation. In this optional chapter, we consider strategies for getting the functional approach to work.

A Broken Evaluator

Here was our first try at an evaluation function for commands, omitting WHILE.

As we remarked in chapter Imp, in a traditional functional programming language like ML or Haskell we could write the WHILE case as follows:

    | WHILE b1 DO c1 END =>
        if (beval st b1) then
          ceval_step1 st (c1;; WHILE b1 DO c1 END)
        else st

Coq doesn't accept such a definition (Error: Cannot guess decreasing argument of fix) because the function we want to define is not guaranteed to terminate. Indeed, the changed ceval_step1 function applied to the loop program from Imp.v would never terminate. Since Coq is not just a functional programming language, but also a consistent logic, any potentially non-terminating function needs to be rejected. Here is an invalid(!) Coq program showing what would go wrong if Coq allowed non-terminating recursive functions:

     Fixpoint loop_false (n : nat) : False := loop_false n.

That is, propositions like False would become provable (e.g., loop_false 0 would be a proof of False), which would be a disaster for Coq's logical consistency. Thus, because it doesn't terminate on all inputs, the full version of ceval_step1 cannot be written in Coq — at least not without one additional trick...

A Step-Indexed Evaluator

The trick we need is to pass an additional parameter to the evaluation function that tells it how long to run. Informally, we start the evaluator with a certain amount of "gas" in its tank, and we allow it to run until either it terminates in the usual way or it runs out of gas, at which point we simply stop evaluating and say that the final result is the empty memory. (We could also say that the result is the current state at the point where the evaluator runs out of gas — it doesn't really matter because the result is going to be wrong in either case!)

We can improve the readability of this version by introducing a bit of auxiliary notation to hide the plumbing involved in repeatedly matching against optional states.

Relational vs. Step-Indexed Evaluation

Determinism of Evaluation Again

Using the fact that the relational and step-indexed definition of evaluation are the same, we can give a slicker proof that the evaluation relation is deterministic.