CLAUSE-PROCESSOR

make or apply a :clause-processor rule (goal-level simplifier)
Major Section:  RULE-CLASSES

Also see define-trusted-clause-processor for documentation of an analogous utility that does not require the clause-processor to be proved correct. But please read the present documentation before reading about that utility. Both utilities designate functions ``clause-processors''. Such functions must be executable -- hence not constrained by virtue of being introduce in the signature of an encapsulate -- and must respect stobj and output arity restrictions. For example, something like (car (mv ...)) is illegal; also see signature.

We begin this documentation with an introduction, focusing on an example, and then conclude with details. You might find it most useful simply to look at the examples in distributed directory books/clause-processors/; see file Readme.lsp in that directory.

A :clause-processor rule installs a simplifier at the level of goals, where a goal is represented as a clause: a list of terms that is implicitly viewed as a disjunction (the application of OR). For example, if ACL2 prints a goal in the form (implies (and p q) r), then the clause might be the one-element list containing the internal representation of this term -- (implies (if p q 'nil) r) -- but more likely, the corresponding clause is ((not p) (not q) r). Note that the members of a clause are translated terms; see term. For example, they do not contains calls of the macro AND, and constants are quoted.

Note that clause-processor simplifiers are similar to metafunctions, and similar efficiency considerations apply. See meta, in particular the discussion on how to ``make a metafunction maximally efficient.''

Unlike rules of class :meta, rules of class :clause-processor must be applied by explicit :clause-processor hints; they are not applied automatically (unless by way of computed hints; see computed-hints). But :clause-processor rules can be useful in situations for which it is more convenient to code a simplifier that manipulates the entire goal clause rather than individual subterms of terms in the clause.

We begin with a simple illustrative example: a clause-processor that assumes an alleged fact and creates a separate goal to prove that fact. We can extend the hypotheses of the current goal with a term by adding the negation of that term to the clause (disjunctive) representation of that goal. So the following returns two clauses: the result of adding term as a hypothesis to the input clause, as just described, and a second clause consisting only of that term.

(defun note-fact-clause-processor (cl term) (declare (xargs :guard t)) ; optional, for better efficiency (list (cons (list 'not term) cl) (list term)))

As with :meta rules, we need to introduce a suitable evaluator; see defevaluator. Since we expect to reason about the function NOT, because of its role in note-fact-clause-processor as defined above, we include NOT in the set of functions known to this evaluator. We also include IF, as is often a good idea.

(defevaluator evl0 evl0-list ((not x) (if x y z)))

ACL2 can now prove the following theorem automatically. (Of course, :clause-processor rules about clause-processor functions less trivial than note-fact-clause-processor may require lemmas to be proved first!) The function disjoin takes a clause and returns its disjunction (the result of applying OR to its members), and conjoin-clauses applies disjoin to every element of a given list of clauses and then conjoins (applies AND) to the corresponding list of resulting terms.

(defthm correctness-of-note-fact-clause-processor (implies (and (pseudo-term-listp cl) (alistp a) (evl0 (conjoin-clauses (note-fact-clause-processor cl term)) a)) (evl0 (disjoin cl) a)) :rule-classes :clause-processor)

Now let us submit a silly but illustrative example theorem to ACL2, to show how a corresponding :clause-processor hint is applied. The hint says to apply the clause-processor function, note-fact-clause-processor, to the current goal clause and a ``user hint'' as the second argument of that function, in this case (equal a a). Thus, a specific variable, clause, is always bound to the current goal clause for the evaluation of the :clause-processor hint, to produce a list of clauses. Since two subgoals are created below, we know that this list contained two clauses. Indeed, these are the clauses returned when note-fact-clause-processor is applied to two arguments: the current clause, which is the one-element list ((equal (car (cons x y)) x)), and the user hint, (equal a a).

ACL2 !>(thm (equal (car (cons x y)) x) :hints (("Goal" :clause-processor (note-fact-clause-processor clause '(equal a a)))))

[Note: A hint was supplied for our processing of the goal above. Thanks!]

We now apply the verified :CLAUSE-PROCESSOR function NOTE-FACT-CLAUSE- PROCESSOR to produce two new subgoals.

Subgoal 2 (IMPLIES (EQUAL A A) (EQUAL (CAR (CONS X Y)) X)).

But we reduce the conjecture to T, by the :executable-counterpart of IF and the simple :rewrite rule CAR-CONS.

Subgoal 1 (EQUAL A A).

But we reduce the conjecture to T, by primitive type reasoning.

Q.E.D.

Summary Form: ( THM ...) Rules: ((:EXECUTABLE-COUNTERPART IF) (:EXECUTABLE-COUNTERPART NOT) (:FAKE-RUNE-FOR-TYPE-SET NIL) (:REWRITE CAR-CONS)) Warnings: None Time: 0.00 seconds (prove: 0.00, print: 0.00, other: 0.00)

Proof succeeded. ACL2 !>

That concludes our introduction to clause-processor rules and hints. We turn now to detailed documentation.

The signature of a clause-processor function, CL-PROC, must have one of the following forms. Here, each st_i is a stobj (possibly state) while the other parameters and results are not stobjs (see defstobj). Note that there need not be input stobjs in [3] -- i.e., k can be 0 -- and even if there are, there need not be output stobjs.

[1]  ((CL-PROC cl) => cl-list)

[2] ((CL-PROC cl hint) => cl-list)

[3] ((CL-PROC cl hint st_1 ... st_k) => (erp cl-list st_i1 ... st_in))

In [3], we think of the first component of the result as an error flag. Indeed, a proof will instantly abort if that error flag is not nil.

We next discuss the legal forms of :clause-processor rules, followed below by a discussion of :clause-processor hints. In the discussion below, we use lower-case names to represent specific symbols, for example implies, and we use upper-case names to represent more arbitrary pieces of syntax (which we will describe), for example, CL.

If a :rule-classes specification includes :clause-processor, then the corresponding term must have the form

(implies (and (pseudo-term-listp CL)
              (alistp A)
              (EVL (conjoin-clauses <CL-LIST>)
                    A))
         (EVL (disjoin CL) A))
where: EVL is a known evaluator (see defevaluator); CL and A are distinct non-stobj variables; and CL-LIST-EXPR is an expression representing the clauses returned by the clause-processor function CL-PROC, whose form depends on the signature of that function, as follows. For cases [1] and [2] above, CL-LIST-EXPR is of the form (CL-PROC CL) or (CL-PROC CL HINT), respectively, where in the latter case HINT is a non-stobj variable distinct from the variables CL and A. For case [3], CL-LIST-EXPR is of the form
(clauses-result (CL-PROC CL HINT st_1 ... st_k))
where the st_i are the specific stobj names mentioned in [3]. Logically, clauses-result returns the cadr if the car is NIL, and otherwise (for the error case) returns a list containing the empty (false) clause. So in the non-error case, clauses-result picks out the second result, denoted cl-list in [3] above, and in the error case the implication above trivially holds.

The clause-processor function, CL, must have a guard that ACL2 can trivially prove from the hypotheses that the first argument of CL is known to be a pseudo-term-listp and any stobj arguments are assumed to satisfy their stobj predicates.

Next we specify the legal forms for :clause-processor hints. These depend on the signature as described in [1] through [3] above. Below, as above, CL-PROC is the clause-processor function, and references to ``clause'' refer to that exact variable (not, for example, to cl). In each of the three cases, the forms shown for that case are equivalent; in particular, the :function syntax is simply a convenience for the final form in each case.

Signature [1], ((cl-proc cl) => cl-list):

:clause-processor CL-PROC
:clause-processor (:function CL-PROC)
:clause-processor (CL-PROC clause)
or any term macroexpanding to (CL-PROC clause).

Signature [2], ((cl-proc cl hint) => cl-list):

:clause-processor (:function CL-PROC :hint HINT)
:clause-processor (CL-PROC clause HINT)
or any term macroexpanding to (CL-PROC clause HINT), where HINT is any term with at most CLAUSE free.

Signature [3], ((CL-PROC cl hint ...) => (erp cl-list ...))

:clause-processor (:function CL-PROC :hint HINT)
:clause-processor (CL-PROC clause HINT st_1 ... st_k)
or any term macroexpanding to (CL-PROC clause HINT st_1 ... st_k), where HINT is any term with at most CLAUSE free.

A :clause-processor hint causes the proof to abort if the result returned by evaluating the suitable CL-PROC call, as above, is not a list of clauses, i.e., a list of (translated) term lists. The proof also aborts if in case [3] the first (erp) value returned is not nil, in which case erp is used for printing an error message as follows: if it is a string, then that string is printed; but if it is a non-empty true list whose first element is a string, then it is printed as though by (fmt @0 (list (cons #0 erp)) ...) (see fmt). Otherwise, a non-nil erp value causes a generic error message to be printed.

If there is no error as above, but the CL-PROC call returns clause list whose single element is equal to the input clause, then the hint is ignored since we are left with the goal with which we started. In that case, the other prover processes are then applied as usual.

You can see all current :clause-processor rules by issuing the following command: (print-clause-processor-rules).

The following paper discusses ACL2 clause-processors at a high level suitable for a non-ACL2 audience:

M. Kaufmann, J S. Moore, S. Ray, and E. Reeber, ``Integrating External Deduction Tools with ACL2.'' Proceedings of the 6th International Workshop on the Implementation of Logics (IWIL 2006) (C. Benzmueller, B. Fischer, and G. Sutcliffe, editors), CEUR Workshop Proceedings Vol. 212, Phnom Penh, Cambodia, pp. 7-26, November 2006, http://ceur-ws.org/Vol-212/. Expanded version to appear in the Journal of Applied Logic.