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|
#|
The Universal Example
J Strother Moore, Jeff Golden, Hanbing Liu, and Sandip Ray
In this file, we establish that for any int-valued function f, and
any x, there exists an M5 schedule which induces computation of (f x) by a
"universal" M5 state. The upshot is that if one is willing to accept
a bytecode specification employing non-constructive existential
quantification of the schedule component, then we have one very simple
program which computes an entire set of interesting functions. Too
bad such a specification is not meaningful!
Here is the code array of the "universal" method:
(defconst *universal-program*
'((iconst_0)
(iconst_1)
(iadd)
(goto -2)))
Here is our specification of the "interesting set of functions":
(encapsulate (((f *) => *))
(local
(defun f (x)
(declare (ignore x))
0))
(defthm f-is-jvm-int-valued
(and (integerp (f x))
(>= (f x) (* -1 (expt 2 31)))
(<= (f x) (1- (expt 2 31))))))
It constrains f to be a function of one argument that returns a Java
int.
In the context of the encapsulation above, and definitions given below, we
proved:
(defthm universal-computes-f
(equal (top
(stack
(top-frame 0
(run (universal-schedule x)
*universal-state*))))
(f x)))
If you want to know the value of a particular such f at an n, you may
thus run the "universal program" according to the schedule generated
by the schedule-generator and then look at the top of thread zero's
active operand stack.
To recertify:
(include-book "utilities")
(certify-book "universal" 1)
Sun Jun 30 23:21:06 2002
|#
(in-package "M5")
(defconst *universal-state*
'(((0 ((0 NIL NIL
((INVOKESTATIC "Universal" "universal" 0)
(POP)
(RETURN))
JVM::UNLOCKED "Universal"))
SCHEDULED NIL))
((0 ("java.lang.Class" ("<name>" . "java.lang.Object"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(1 ("java.lang.Class" ("<name>" . "ARRAY"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(2 ("java.lang.Class" ("<name>" . "java.lang.Thread"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(3 ("java.lang.Class" ("<name>" . "java.lang.String"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(4 ("java.lang.Class" ("<name>" . "java.lang.Class"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0)))
(5 ("java.lang.Class" ("<name>" . "Universal"))
("java.lang.Object" ("monitor" . 0)
("mcount" . 0)
("wait-set" . 0))))
(("java.lang.Object" NIL ("monitor" "mcount" "wait-set")
NIL NIL (("<init>" NIL NIL (RETURN)))
(REF 0))
("ARRAY" ("java.lang.Object")
(("<array>" . *ARRAY*))
NIL NIL NIL (REF 1))
("java.lang.Thread"
("java.lang.Object")
NIL NIL NIL
(("run" NIL NIL (RETURN))
("start" NIL NIL NIL)
("stop" NIL NIL NIL)
("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 2))
("java.lang.String"
("java.lang.Object")
("strcontents")
NIL NIL
(("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 3))
("java.lang.Class" ("java.lang.Object")
NIL NIL NIL
(("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN)))
(REF 4))
("Universal" ("java.lang.Object")
NIL NIL NIL
(("<init>" NIL NIL (ALOAD_0)
(INVOKESPECIAL "java.lang.Object" "<init>" 0)
(RETURN))
("universal" NIL NIL
(ICONST_0)
(ICONST_1)
(IADD)
(GOTO -2))
("main" (|JAVA.LANG.STRING[]|)
NIL
(INVOKESTATIC "Universal" "universal" 0)
(POP)
(RETURN)))
(REF 5)))))
; Here is the code array of the method "universal":
(defconst *universal-program*
'((ICONST_0) ;;; 0
(ICONST_1) ;;; 1
(IADD) ;;; 2
(GOTO -2))) ;;; 3
; Here is our specification of the "interesting set of functions":
(encapsulate (((f *) => *))
(local
(defun f (x)
(declare (ignore x))
0))
(defthm f-is-jvm-int-valued
(and (integerp (f x))
(>= (f x) (* -1 (expt 2 31)))
(<= (f x) (1- (expt 2 31))))
:rule-classes nil))
; We provided the constraint above in terms of (expt 2 31) to make it
; understandable to outsiders. But here is the more useful version of
; the constraint.
(defthm intp-f
(intp (f x))
:hints (("Goal" :in-theory (enable intp)
:use f-is-jvm-int-valued)))
; Here is the notion of being poised at the top of the loop in the
; universal program.
(defun poised-at-universal-loop (th s i)
(and (equal (status th s) 'SCHEDULED)
(equal (pc (top-frame th s)) 1)
(equal (program (top-frame th s)) *universal-program*)
(intp (top (stack (top-frame th s))))
(integerp i)
(<= 0 i)))
; Here is a schedule that will drive us from the top of the loop
; sufficiently to increment the top of the stack by i.
(defun universal-loop-sched (th i)
(if (zp i)
nil
(append (repeat th 3)
(universal-loop-sched th (- i 1)))))
(defun universal-loop-hint (th s i)
(if (zp i)
(list th s i)
(universal-loop-hint
th
(modify th s
:stack
(push
(int-fix (+ 1 (top (stack (top-frame th s)))))
(pop (stack (top-frame th s)))))
(- i 1))))
; These ought to be in utilities.lisp
(defthm universal-loop-behavior
(implies (poised-at-universal-loop th s i)
(equal (run (universal-loop-sched th i) s)
(if (zp i)
s
(modify
th s
:stack
(push (int-fix (+ i (top (stack (top-frame th s)))))
(pop (stack (top-frame th s))))))))
:hints (("Goal"
:induct (universal-loop-hint th s i)
:in-theory (enable zp))))
(defun poised-to-invoke-universal (th s i)
(and (equal (status th s) 'SCHEDULED)
(equal (next-inst th s) '(INVOKESTATIC "Universal" "universal" 0))
(equal (lookup-method "universal" "Universal" (class-table s))
'("universal" NIL NIL
(ICONST_0)
(ICONST_1)
(IADD)
(GOTO -2)))
(integerp i)
(<= 0 i)))
(defun universal-sched (th i)
(append (repeat th 2)
(universal-loop-sched th i)))
(defthm universal-is-correct
(implies (poised-to-invoke-universal th s i)
(equal (top
(stack
(top-frame th
(run (universal-sched th i) s))))
(int-fix i))))
(in-theory (disable universal-sched))
; So now I want to prove that if k is an int, then k = (int-fix i),
; for some natural number i. First I define the corresponding
; natural number:
(defun int-fix-nat (k)
(cond ((< k 0)
(+ (expt 2 32) k))
(t k)))
; Here I prove it is a natural.
(defthm natp-int-fix-nat
(implies (intp k)
(and (integerp (int-fix-nat k))
(<= 0 (int-fix-nat k))))
:hints (("Goal" :in-theory (enable intp))))
; Here I prove it does the job.
(defthm every-int-is-int-fix-nat
(implies (intp k)
(equal (int-fix (int-fix-nat k)) k))
:hints (("Goal" :in-theory (enable int-fix intp))))
(in-theory (disable int-fix-nat))
; Consequently, we can now prove that with a suitable schedule,
; universal computes our arbitrary int-valued function f.
(defun universal-schedule (x)
(universal-sched 0 (int-fix-nat (f x))))
(defthm universal-computes-f
(equal (top
(stack
(top-frame 0
(run (universal-schedule x)
*universal-state*))))
(f x))
:rule-classes nil
:hints (("Goal" :in-theory (disable top-frame))))
; Now I demonstrate that we can use it. Here is a concrete example of
; how universal can compute the int-fix of factorial. That value is
; what we proved of our true Java fact program.
(defun factorial (n)
(if (zp n)
1
(* n (factorial (- n 1)))))
(defthm integerp-factorial
(integerp (factorial n))
:rule-classes :type-prescription)
(defun universal-factorial-schedule (n)
(universal-sched 0 (int-fix-nat (int-fix (factorial n)))))
; Here I prove that (int-fix (factorial n)) satisfies the constraints
; on (f n).
(defthm relieve-the-contraints
(and (integerp (int-fix (factorial n)))
(>= (int-fix (factorial n)) (* -1 (expt 2 31)))
(<= (int-fix (factorial n)) (1- (expt 2 31))))
:rule-classes
((:linear :corollary
(>= (int-fix (factorial n)) (* -1 (expt 2 31))))
(:linear :corollary
(<= (int-fix (factorial n)) (1- (expt 2 31)))))
:hints (("Goal" :in-theory (enable int-fix))))
; And so now I can functionally instantiate our universal correctness
; theorem.
(defthm universal-computes-factorial
(equal (top
(stack
(top-frame 0
(run (universal-factorial-schedule n)
*universal-state*))))
(int-fix (factorial n)))
:hints (("Goal"
:use (:instance (:functional-instance
universal-computes-f
(f (lambda (n) (int-fix (factorial n))))
(universal-schedule universal-factorial-schedule))
(x n)))))
|
