[TUHS] Formal Specification and Verification (was Re: TUHS Digest, Vol 33, Issue 5)

[Perry E. Metzger]

Sorry for the thread necromancy, but this is a _very_ important
topic. Perhaps it doesn't belong on tuhs but rather on coff.

This is a pretty long posting. If you don't care to read it, the TL;DR
is that formal specification and verification is now a real
discipline, which it wasn't in the old days, and there are systems to
do it in, and it's well understood.

On 2018-08-06 at 08:52 -0700, Bakul Shah wrote:
>
> What counts as a "formal spec"? Is it like Justice Potter
> Stewart's "I know it when I see it" definition or something
> better?

At this point, we have a good definition. A formal specification is a
description of the behavior of a program or piece of hardware in a
precise machine-readable form that may be used as the basis for fully
formal verification of the behavior of an implementation. Often these
days, the specification is given in a formal logic, such as the
predicative calculus of inductive constructions, which is the logic
underlying the Coq system.

Isabelle/HOL is another popular system for this sort of work. ACL2 is
(from what I can tell) of more historical interest but it has
apparently been used for things like the formal verification of
floating point and cache control units for real hardware. (It is my
understanding that it has been many years since Intel would dare
release a system where the cache unit wasn't verified, and the only
time in decades it tried to release a non-verified FPU, it got the
FDIV bug and has never tried that again.) There are some others out
there, like F*, Lean, etc.

Formal specifications good enough for full formal verification have
been made for a variety of artifacts along with proofs that the
artifacts follow the specification. There's a fully formally verified
C compiler called CompCert for example, based on an operational
semantics written in Coq. There's another formal semantics for C
written in K, which is a rather different formal system. There's a
verified microkernel, seL4, whose specification is written in
Isabelle/HOL. There's a fully formal specification of the RISC V, and
an associated verified RTL implementation.

Generally speaking, a formal specification:

1) Must be machine readable
2) The semantics of the underlying specification language must
   themselves be extremely precisely described. You can't prove the
   consistency and completeness of the underlying system (see Gödel)
   but you _can_ still strongly specify the language.
3) Must be usable for formal (that is, machine checkable) proofs that
   implementations comply with the spec, so it has to be sufficiently
   powerful. Typical systems are now equivalent to higher order logic.

From: "Hellwig Geisse" <hellwig.geisse at mni.thm.de>
Sent:Mon, 06 Aug 2018 18:30:30 +0200
>
>  For me, a "formal spec" should serve two goals:
>  1) You can reason about the thing that is specified.

Yes.

2) The spec can be "executed" (i.e., there is an
>  interpreting mechanism, which lets the spec behave
>  like the real thing).

Not always reasonable.

First, it is often the case that a spec does not describe execution at
all. See, for example, the specification of a sorting function I give
at the end of this message: it simply says "a sorting function is a
function such that, for all inputs, the return is a non-decreasing
permutation of the input". This is not executable. It is a purely
descriptive property, and you cannot extract an executable algorithm
from the spec.

Second, even when a spec amounts to a description of execution, a
proof assistant often cannot actually execute it. For example,
although you can reason about non-terminating execution in pCiC (and
thus Coq), programs written in a strongly normalizing lambda calculus
that can be used as a logic must terminate, or functions that did not
terminate would be inhabitants of all types and the logic would be
inconsistent. Thus, you cannot execute a program with infinite loops
inside Coq, although you can reason about them (and indeed, you can
reason about coinductively defined objects like infinite execution
traces.)

On Mon, 06 Aug 2018 14:19:31 -0700 "Steve Johnson" <scj at yaccman.com>
wrote:
> I take a somewhat more relaxed view of what a spec should be:
> It should describe a program with enough completeness that a
> competent programmer could write it from the spec alone.

I think this is a bit more relaxed than is currently accepted.

> The formal systems I have seen would roll over and die when
> presented with even a simple compiler.__

I don't know what this means. If it is that there aren't
implementations of languages like pCiC, that's not true, see Coq. If
it means no one can formally specify a compiler, given that formally
verified compilers exist, that's also not true.

The "final theorem" proving the correctness of CompCert depends on
having an operational semantics of both C and the target architecture,
and says (more or less) that the observed behavior of the input
program in C is the same as the observed behavior of the output
program (say in ARM machine language). This is a serious piece of
work, but it is also something that has actually been done -- the
tools are capable of the task.

> Additionally, being able to specify that a particular function be
> carried out by a heapsort, for example, would require that the
> formalism could describe the heapsort and prove it correct.__ These
> don't grow on trees...

Formally verifying a couple of sorting algorithms described in Coq is
a exercise for an intro level class on formal verification. I've done
it. Once you have the proper primitives described, the specification
for a sorting algorithm in Coq looks like this:

Definition is_a_sorting_algorithm (f: list nat -> list nat) :=
  forall al, Permutation al (f al) /\ sorted (f al).

That says "the property "is_a_sorting_algorithm" over a function from
lists of natural numbers to lists of natural numbers is that the
output is a permutation of the input in which all the elements are in
non-decreasing order." The definitions in question are very
precise. For example, one definition of sorted (the property of being
a non-decreasing list) is:

Definition sorted (al: list nat) :=
 forall i j, i < j < length al -> nth i al 0 <= nth j al 0.

and the property of being a permutation, which is relatively
complicated inductively defined property, is:

Inductive Permutation {A : Type} : list A -> list A -> Prop :=
    perm_nil : Permutation
  | perm_skip : forall (x : A) (l l' : list A),
                Permutation l l' ->
                Permutation (x :: l) (x :: l')
  | perm_swap : forall (x y : A) (l : list A),
                Permutation (y :: x :: l) (x :: y :: l)
  | perm_trans : forall l l' l'' : list A,
                 Permutation l l' ->
                 Permutation l' l'' ->
                 Permutation l l''.

Coq starts out with basically nothing defined, by the way. Notions
such as "natural number" and "list" are not built in. Peano naturals
are defined in the system thusly:

Inductive nat : Type :=
  | O : nat
  | S : nat -> nat.

The underlying basis of Coq (i.e. the Predicative Calculus of
Inductive Constructions) is a dependently typed lambda calculus that's
astonishingly simple, and the checker for proofs in the system is only
a few hundred lines long -- the checker is the only portion of the
system which needs to be trusted.

In recent years, I've noted that "old timers" (such as many of us,
myself included) seem to be unaware of the fact that systems like Coq
exist, or that it is now relatively (I emphasize _relatively_) routine
for substantial systems to be fully formally specified and then fully
formally verified.


Perry
-- 
Perry E. Metzger		perry at piermont.com