IA159 Formal Verification Methods
Theorem Prover ACL2
Jan Strejˇcek
Department of Computer Science
Faculty of Informatics
Masaryk University
Theorem provers
Theorem provers are software tools, which
assist human experts in construction of formal proofs
can be used in software and hardware verification
work only with statements (and corresponding axioms and
inference rules) in a suitable formal notation
Do they work automatically?
only simple theorems can be proven fully automatically
nearly all proofs result from interaction of a tool and a user
success depends primarily on user’s skill
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ACL2
“A Computational Logic for Applicative Common Lisp”
ACL2 is
1 a functional programming language based on
Common Lisp
2 a first-order, quantifier-free mathematical logic
3 a mechanical theorem prover
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History of ACL2
1971: Robert S. Boyer and J Strother Moore created
Nqthm - the first theorem prover for Lisp
1989: Boyer and Moore started to work on ACL2
since 1993, ACL2 is systematically developed by Matt
Kaufmann and J Strother Moore
now in version 7.0 (March 2015)
winner of VSTTE 2012 Software Verification Competition
ACL2 is available under a license based on BSD-3-Clause
http://www.cs.utexas.edu/users/moore/acl2/
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ACL2 achievements
ACL2 has been used to verify that
the functionality of FPU in AMD K5, Athlon, and Opteron
(described on register-transfer level, RTL) follows the
corresponding IEEE standard
the microarchitectural model of a Motorola DSP processor
implements a given microcode engine and that certain
microcode in ROM implements certain DSP algorithms
the microcode for the Rockwell Collins AAMP7 implements
a given security policy concerning process separation
the bytecode produced by the Sun compiler javac on
certain simple Java classes has the claimed functionality
a BDD package written in Lisp is sound and complete
a Lisp program that checks proofs produced by the Ivy
theorem prover is sound
. . .
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Back to the ground
the proof of correctness of the floating point division
microcode in AMD K5 required approx. 1200 lemmas
the verification has not been done directly on RTL of the
FPU, but on its automatically translated Lisp model;
correctness of the translation has been “verified” by
80 000 000 test computations
the proof of correspondence between the Motorola DSP
microarchitecture and its microcode engine involved
formulas of 25 MB per formula; finding one subtle
generalization took Moore many days
. . .
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The mechanics of using ACL2
Two buffers in Emacs
1 buffer with definitions and lemmas, typically concluding
with the main theorem we want to prove
2 shell with ACL2
Typical working cycle
1 send the subsequent definition or theorem to ACL2
2 if it succeeds, go to 1
3 inspect the output of ACL2 and analyze the failure
4 if the command is faulty (e.g. with a syntax error), fix it;
if the command is a theorem ACL2 is unable to prove, then
suggest, formulate, and prove additional lemmas and then
try to prove the theorem again
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Syntax of ACL2
Supported data objects
numbers (integer, rational and complex)
characters
strings ("Hello world!")
symbols (t, nil, ’ok, ’quick-sort,. . . )
ordered pairs
Lists
in fact nested pairs: 1, 2, 3, nil or 1, 2, 3
written in list notation: ’(1 2 3) or ’(1 2 . 3)
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Syntax of ACL2
Some primitive (built-in) functions
(cons x y) constructs the ordered pair x, y
(car x) left component of x, if x is a pair; nil otherwise
(cdr x) right component of x, if x is a pair; nil otherwise
(consp x) t if x is a pair; nil otherwise
(if x y z) z if x is nil; y otherwise
(equal x y) t if x is y; nil otherwise
Meaning of the single quote mark
’(car x) evaluates to the list car, x, nil
(car x) application of the function car to x
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Syntax of ACL2
Function definition
(defun f (a1 a2 . . . an) β)
creates the function f with arguments a1, a2, . . . , an
and body β
(Built-in) Lisp definitions of standard logic connectives
(defun not (p) (if p nil t))
(defun and (p q) (if p q nil))
(defun or (p q) (if p p q))
(defun implies (p q) (if p (if q t nil) t))
(defun iff (p q) (and (implies p q)
(implies q p)))
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Examples of recursive function definitions
dup - duplicates each element in a list
(defun dup (x)
(if (consp x)
(cons (car x)
(cons (car x)
(dup (cdr x))))
nil))
app - concatenates two lists
(defun app (x y)
(if (consp x)
(cons (car x) (app (cdr x) y))
y))
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Axioms in ACL2
Some primitive (built-in) axioms
1 t = nil
2 x = nil → (if x y z) = y
3 x = nil → (if x y z) = z
4 (equal x y) = nil ∨ (equal x y) = t
5 x = y ↔ (equal x y) = t
6 (consp x) = nil ∨ (consp x) = t
7 (consp (cons x y)) = t
8 (consp nil) = (consp t) = (consp ’ok) =
(consp 0) = . . . = nil
9 (car (cons x y)) = x
10 (cdr (cons x y)) = y
11 (consp x) = t → (cons (car x) (cdr x)) = x
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Proofs in ACL2 (a sketch)
ACL2 contains
ordinals up to ωωω···
a well-founded relation o< on such ordinals
axioms definining the size of ACL2 objects (measured with
the function acl2-count)
and particularly
definition principle
induction principle
simplification based on
rewrite rules
linear arithmetic rules (inequality chaining)
. . . (approx. 12 kinds of rules in total)
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Definition principle
when a recursive function definition is submitted, ACL2
must prove that a there is a well-founded measure such
that arguments of recursive calls are decreasing with
respect to this measure
existence of such a measure ensures that the evaluation of
the function terminates after a finite number of steps.
the definition is admitted by ACL2 (as a new axiom) only if
the existence of such a measure is proven; a user can
assist with the proof
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Induction principle (a sketch)
Structural induction on lists and binary trees
If we want to prove ϕ(x, y), it is sufficient to prove
1 base case
ϕ(x, y) holds in the case that x is an empty tree
2 induction step
if x = (l, r) and ϕ(l, y) and ϕ(r, y), then ϕ(x, y)
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Induction principle (a sketch)
Induction on binary trees in ACL2
If we want to prove (ϕ x y), it is sufficient to prove
1 base case
(implies (not (consp x)) (ϕ x y))
2 induction step
(implies (and (consp x)
(ϕ (car x) y) ;induction hypothesis 1
(ϕ (cdr x) y)) ;induction hypothesis 2
(ϕ x y)) ;induction conclusion
induction hypothesis can be any (ϕ δ α) such that we
can prove
(implies (consp x)
(o< (acl2-count δ) (acl2-count x))
axioms imply that (car x), (cdr x) are smaller than x
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A proof
(defun treecopy (x)
(if (consp x)
(cons (treecopy (car x))
(treecopy (cdr x)))
x))
Theorem: (equal (treecopy x) x).
Proof: Name the formula above *1.
Perhaps we can prove *1 by induction. One induction scheme
is suggested by this conjecture - namely the one that unwinds
the recursion in treecopy.
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A proof
If we let (ϕ x) denote *1 above then the induction scheme
we’ll use is
(and (implies (not (consp x)) (ϕ x))
(implies (and (consp x)
(ϕ (car x))
(ϕ (cdr x)))
(ϕ x))).
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A proof
This induction is justified by the same argument used to admit
treecopy, namely, the size of x is decreasing according to a
certain well-founded relation. When applied to the goal at hand
the above induction scheme produces the following two
nontautological subgoals.
Subgoal *1/2
(implies (not (consp x))
(equal (treecopy x) x)).
But simplification reduces this to t, using the definition of
treecopy and the primitive axioms.
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A proof
Subgoal *1/1
(implies (and (consp x)
(equal (treecopy (car x)) (car x))
(equal (treecopy (cdr x)) (cdr x)))
(equal (treecopy x) x)).
But simplification reduces this to t, using the definition of
treecopy, and the primitive axioms.
That completes the proof of *1.
Q.E.D.
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Simplification of Subgoal *1/1
(implies (and (consp x) ;hypothesis 1
(equal (treecopy (car x)) (car x)) ;hypothesis 2
(equal (treecopy (cdr x)) (cdr x))) ;hypothesis 3
(equal (treecopy x) x)).
(treecopy x) = (if (consp x) ;treecopy definition
(cons (treecopy (car x))
(treecopy (cdr x)))
x)
= (if t ;hypothesis 1
(cons (treecopy (car x))
(treecopy (cdr x)))
x)
= (cons (treecopy (car x)) ;axioms 1 and 2
(treecopy (cdr x)))
= (cons (car x) ;hypothesis 2
(treecopy (cdr x)))
= (cons (car x) ;hypothesis 3
(cdr x))
= x ;axiom 11 and
;hypothesis 1
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Proofs in ACL2: induction
ACL2 uses heuristics to choose a suitable induction
scheme
induction scheme is based on recursively defined function
occurring in the theorem
the resulting scheme can be a combination of two or more
recursive schemes used in the theorem
the user can specify an induction scheme with a hint
choosing the right induction is crucial to a successful proof
even more important is to choose the right theorem to
prove by induction - the theorem has to be general enough
in order to provide a sufficiently strong induction hypothesis
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Proofs in ACL2: simplification via rewriting
simplification means the reduction of the formula to some
preferred form by the use of rewrite rules
rules are derived from axioms, definitions, and previously
proved theorems
a definition generates the rule rewriting function calls by
the instantiated body of the function
a formula of the form
(implies (and hyp1 . . . hypn) (equal l r))
generates the rule replacing instances of l by the
corresponding instance of r, provided the corresponding
instances of hyp1, . . . , hypn rewrite to t
equivalent formulae (like (equal l r) and (equal r l));
may give rise to radically different rules
some rule combinations can lead to cyclic rewriting
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Proofs in ACL2: simplification via inequality chaining
there is a large set of rewrite rules allowing to put
arithmetic expressions into a preferred form
there are books (i.e. collections of such rules) for
elementary algebraic properties of numbers, modulo
arithmetic, floating point arithmetic, . . .
ACL2 also maintains a graph of terms involved in the
current formula, where edges correspond to inequalities
edges are added by a decision procedure for linear
arithmetics
when submitting a theorem, we can specify what kind of
rule should be generated when the theorem is proved (it is
a rewrite rule by default)
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Proofs in ACL2: simplification via inequality chaining
Example
consider a theorem concluding with (<= O (* x x))
if it is used to generate a rewrite rule, the rule replaces
certain instances of (<= O (* x x)) by t
this rule cannot be used to prove
(implies (and (< 0 a) (rationalp b))
(< 0 (+ a (* b b))))
if we say that an arithmetic rule should be gerenated from
the theorem, the the rule can be used to add some edges
to the graph of inequalities
There are many other kinds of simplification.
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ACL2 working cycle
when ACL2 receives a syntactically correct theorem, it
1 simplifies the theorem
2 uses an induction
3 go to 1
it exits the cycle if
the simplification results in t, or
there is no suggested induction scheme
if the theorem is proved, a corresponding rule is derived
. . . and everything is vividly commented
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ACL2
Demonstration
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Coming next week
Model checking: quick overview
Model checking once again?
Yes, but very quickly.
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