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|
; Copyright (C) 2014, ForrestHunt, Inc.
; License: A 3-clause BSD license. See the LICENSE file distributed with ACL2.
; Symbolic State Management -- Version 22
; J Strother Moore
; Fall/Winter, 2014/2015
; Georgetown, TX and Edinburgh, Scotland
;
; Stateman: Using Metafunctions to Manage Large Terms Representing Machine States
; J Strother Moore
; July, 2015
; The paper above gives several examples of the Stateman state management book.
; This file contains those examples in a form allowing the reader to
; interactively reproduce them. This is NOT a book. The reader interested in
; learning about the metafunctions in Stateman is urged simply to execute each
; of the commands below while or after reading the above paper and/or the code.
; The main motivation behind this file is to help users extend Stateman. It is
; helpful to see how to call and interpret the results of some of the functions
; in Stateman. By experimenting with the inputs provided below you can better
; understand some of the limitations of Stateman.
; Note: Before executing the next command you will need to connect to the
; directory containing stateman22.lisp or edit the command below to include the
; path.
(include-book "stateman22")
; Section 4. Examples
; This section of the paper shows 12 example executions, numbered (1) through
; (12). The first 7 are shown twice, first in untranslated form and then in a
; shorter notation. Here are all 12 of the examples suitable for execution at
; the top-level of ACL2 after including the Stateman book.
(meta-!I '(!I '123 st)) ;(1)
; (HIDE (!I '123 ST))
(meta-!R '(!R '0 '4 (R '16 '4 st) (HIDE (!I '123 ST))) ;(2)
nil state)
; (HIDE (!R '0 '4 (R '16 '4 ST) (!I '123 ST))) ;(st')
(meta-I '(I (HIDE (!R '0 '4 (R '16 '4 st) (!I '123 ST))))) ;(3)
; '123
(meta-R '(R '0 '4 (HIDE (!R '0 '4 (R '16 '4 st) (!I '123 ST)))) ;(4)
nil state)
; (R '16 '4 ST)
(meta-R '(R '2 '2 (HIDE (!R '0 '4 (R '16 '4 st) (!I '123 ST)))) ;(5)
nil state)
; (HIDE (ASH (R '16 '4 ST) '-16))
(meta-R '(R '8 '4 (HIDE (!R '0 '4 (R '16 '4 st) (!I '123 ST)))) ;(6)
nil state)
; (R '8 '4 ST)
(meta-R '(R '2 '4 (HIDE (!R '0 '4 (R '16 '4 st) (!I '123 ST)))) ;(7)
nil state)
; (HIDE (BINARY-+ (ASH (R '4 '2 ST) '16)
; (ASH (R '16 '4 ST) '-16)))
(meta-!R '(!R '8 '4 v st) nil state) ;(8)
; (HIDE (!R 8 4 v st))
(meta-R '(R '8 '4 (HIDE (!R '8 '4 v st))) nil state) ;(9)
; (HIDE (MOD (IFIX v) 4294967296))
; Before example (10) the paper says ``Now let the metafunction context encode
; the assumption that (R 16 4 st) is less than 16.'' To express this
; assumption, we create a fake metafunction context, mfc. The type-alist,
; which encodes our assumptions, is the second component of the mfc and all
; other components are irrelevant to Stateman. Our fake mfc is here
; stored in a state global variable:
(assign mfc (list nil `(((< (R '16 '4 st) '16) ,*ts-t*))))
; The type-alist shown above could have been written alternatively
; as ``(<= (R 16 4 st) 15)'', but that in fully translated form
; that is (NOT (< '15 (R '16 '4 ST))) and to be an effective type-alist
; entry the NOT is stripped off and encoded in the assigned type, i.e.,
; we say ``(< 15 (R 16 4 st)) has the false type-set'' or
; use the type-alsist `(((< '15 (R '16 '4 st)) ,*ts-nil*)).
; Under the assuption above the paper goes on to show the input to example (10)
; using our ``==>+'' notation (with a ``dagger''), where the left-hand side of
; the ``==>+'' is:
; (R (+ 3200 (* 8 (R 16 4 st))) 8 ;(10)
; (HIDE (!R 3600 4 v (!R 8 4 w st))))
; ==>+
; ...
; Given the meaning of the adopted notation, you should meta-R to the fully
; translated version of the term shown above, our mfc, and state. That is,
; evaluate:
(meta-R '(R (BINARY-+ '3200 (BINARY-* '8 (R '16 '4 st))) '8 ;(10)
(HIDE (!R '3600 '4 v (!R '8 '4 w st))))
(@ mfc)
state)
; The paper then shows:
; ==>+
; (R (+ 3200 (* 8 (R 16 4 st))) 8 st)
; and goes on to say ``The ``+'' [``dagger''] on the transformation in (10)
; indicates that a side condition was generated. That side condition is (<= (R
; 16 4 st) 15), and it must be established before the replacement is made.''
; If you evaluate the meta-R expression above the result is:
; (IF (FORCE (NOT (< '15 (R '16 '4 ST)))) ; (IF (FORCE test)
; (R (BINARY-+ '3200 ; x'
; (BINARY-* '8 (R '16 '4 ST)))
; '8
; ST)
; (R (BINARY-+ '3200 ; x
; (BINARY-* '8 (R '16 '4 ST)))
; '8
; (HIDE (!R '3600 '4 V (!R '8 '4 W ST))))) ; )
; Recall that when a metafunction applied to x produces (IF test x' x) it is
; interpreted to mean ``if test can be established, replace x by x', else do
; not use the output of the metafunction.'' That is the situation here. The
; term playing the role of x' above is the term shown on the right-hand side of
; the ``==>+'' in the paper. The test is the translated form of (<= (R 16 4
; st) 15), however Stateman additionally FORCEs it because we know the test to
; be true.
; In a similar fashion, here is example (11)
(meta-!R
'(!R (BINARY-+ '3200 (BINARY-* '8 (R '16 '4 st))) '8 u ;(11)
(HIDE (!R '3600 '4 v
(!R '8 '4 w
(!R (BINARY-+ '3200 (BINARY-* '8 (R '16 '4 st))) '8 x
st)))))
(@ mfc)
state)
; The output will be:
; (HIDE (!R (BINARY-+ '3200
; (BINARY-* '8 (R '16 '4 ST)))
; '8
; U
; (!R '3600 '4 V (!R '8 '4 W ST))))
; As noted in the paper and for the reasons given, no side condition is
; necessary.
; Finally, example (12) of the paper is reproduced with:
(meta-R
'(R '3 '8 ;(12)
(HIDE
(!R '14 '5 x
(!R '0 '4 u
(!R '19 '8 y
(!R '9 '2 w
(!R '2 '4 v st)))))))
nil
state)
; and the output is:
; (HIDE (BINARY-+ (ASH (R '6 '3 ST) '24)
; (BINARY-+ (MOD (ASH (IFIX U) '-24) '256)
; (BINARY-+ (ASH (MOD (IFIX W) '65536) '48)
; (ASH (MOD (ASH (IFIX V) '-16) '65536)
; '8)))))
; Section 5: Ainni: Abstract Interpreter for Natural Number Intervals
; There are several examples given in this section.
; The first example considers the translation of the term:
; (+ 288 (* 8 (LOGAND 31 (ASH (R 4520 8 st) -3))))
; which is
; (BINARY-+ '288
; (BINARY-* '8
; (BINARY-LOGAND '31
; (ASH (R '4520 '8 ST) '-3))))
; The paper says, ``In the absence of any contextual information, Ainni returns
; the natural number interval [288,536].'' Here is how you confirm that:
(ainni '(BINARY-+ '288
(BINARY-* '8
(BINARY-LOGAND '31
(ASH (R '4520 '8 ST) '-3))))
nil ; accumulated hypotheses so far
nil) ; type-alist containing assumptions
; the result is:
; (T NIL (INTEGERP (NIL . 288) NIL . 536))
; which (mv flg hyps interval), where flg is T meaning Ainni succeeded, the
; returned hyps is nil meaning there are no side conditions to relieve, and
; (INTEGERP (NIL . 288) NIL . 536) is a tau interval over the integers
; representing [288, 536]. A tau interval has five components: the domain
; (here INTEGERP), the lower relation (here nil, meaning ``<='', as opposed to
; t meaning ``<''), the lower bound (here 288), the upper relation (here nil)
; and the upper bound (here 536).
; The paper then says that the type-alist ``might assert that (R 4520 8 st) <
; 24, in which case Ainni determines that the term above lies in the interval
; [288,304].'' Here is that example:
(ainni '(BINARY-+ '288
(BINARY-* '8
(BINARY-LOGAND '31
(ASH (R '4520 '8 ST) '-3))))
nil ; accumulated hypotheses so far
`(((< (R '4520 '8 st) '24) ,*ts-t*)))
; and the output is:
; (T ((FORCE (NOT (< '23 (R '4520 '8 ST))))) ; flag and list of side conditions
; (INTEGERP (NIL . 288) NIL . 304)) ; interval
; where now we see one forced side condition and the interval [288,304].
; Three theorems are used to establish that Ainni is correct. Their names
; are:
; name informal interpretation
; ainni-correct-part-1 returned hyps are all pseudo-terms
; ainni-correct-part-2a returned interval is really an interval
; ainni-correct-part-2b value of input term lies in the returned interval
; and they correspond to the three bullets in the informal characterization of
; Ainni's correctness. By the way, the name of the actual evaluator used in
; Stateman is STATEMAN-EVAL, not the ``script E'' shown in the paper.
; Finally the paper, just for fun, shows a relatively large term. That term
; looks like (LOGIOR big-arg1 big-arg2 big-arg3). Below we show the translated
; forms of the three big-argi and use them to assemble big-term, where
; big-arg1, big-arg2, big-arg3, and big-term are state globals here:
(assign big-arg1
'(BINARY-LOGAND
'32
(ASH
(MOD
(ASH
(BINARY-LOGXOR
(BINARY-LOGIOR
(ASH
(ASH
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '0) '2)
'31)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '4026531840
(R '4520 '8 ST))
'-1)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-27) '2)
'26)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-28) '2)
'25)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '251658240
(R '4520 '8 ST))
'-3)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-23) '2)
'20)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-24) '2)
'19)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '15728640
(R '4520 '8 ST))
'-5)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-19) '2)
'14)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-20) '2)
'13)
(BINARY-LOGIOR (ASH (BINARY-LOGAND '983040 (R '4520 '8 ST))
'-7)
(ASH (MOD (ASH (R '4520 '8 ST) '-15) '2)
'8))))))))))))
'-8)
'24)
(ASH
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-16) '2)
'31)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '61440 (R '4520 '8 ST))
'15)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-11) '2)
'26)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-12) '2)
'25)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '3840 (R '4520 '8 ST))
'13)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-7) '2)
'20)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-8) '2)
'19)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '240 (R '4520 '8 ST))
'11)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-3) '2)
'14)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-4) '2)
'13)
(BINARY-LOGIOR (ASH (MOD (R '4520 '8 ST) '16) '9)
(ASH (ASH (R '4520 '8 ST) '-31)
'8))))))))))))
'-8))
(R (BINARY-+ '4376
(BINARY-+ (BINARY-* '8 (R '4536 '8 ST))
(BINARY-* '8
(UNARY-- (R '4528 '8 ST)))))
'8
ST))
'-40)
'256)
'-2)))
(assign big-arg2
'(ASH
(MOD
(ASH
(MOD
(ASH
(BINARY-LOGXOR
(BINARY-LOGIOR
(ASH
(ASH
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '0) '2)
'31)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '4026531840
(R '4520 '8 ST))
'-1)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-27) '2)
'26)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-28) '2)
'25)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '251658240
(R '4520 '8 ST))
'-3)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-23) '2)
'20)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-24) '2)
'19)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '15728640
(R '4520 '8 ST))
'-5)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-19) '2)
'14)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-20) '2)
'13)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '983040 (R '4520 '8 ST))
'-7)
(ASH (MOD (ASH (R '4520 '8 ST) '-15) '2)
'8))))))))))))
'-8)
'24)
(ASH
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-16) '2)
'31)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '61440 (R '4520 '8 ST))
'15)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-11) '2)
'26)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-12) '2)
'25)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '3840 (R '4520 '8 ST))
'13)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-7) '2)
'20)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-8) '2)
'19)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '240 (R '4520 '8 ST))
'11)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-3) '2)
'14)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-4) '2)
'13)
(BINARY-LOGIOR (ASH (MOD (R '4520 '8 ST) '16) '9)
(ASH (ASH (R '4520 '8 ST) '-31)
'8))))))))))))
'-8))
(R (BINARY-+ '4376
(BINARY-+ (BINARY-* '8 (R '4536 '8 ST))
(BINARY-* '8
(UNARY-- (R '4528 '8 ST)))))
'8
ST))
'-40)
'256)
'-2)
'32)
'-1))
(assign big-arg3
'(ASH
(MOD
(ASH
(MOD
(ASH
(BINARY-LOGXOR
(BINARY-LOGIOR
(ASH
(ASH
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '0) '2)
'31)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '4026531840
(R '4520 '8 ST))
'-1)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-27) '2)
'26)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-28) '2)
'25)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '251658240
(R '4520 '8 ST))
'-3)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-23) '2)
'20)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-24) '2)
'19)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '15728640
(R '4520 '8 ST))
'-5)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-19) '2)
'14)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-20) '2)
'13)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '983040 (R '4520 '8 ST))
'-7)
(ASH (MOD (ASH (R '4520 '8 ST) '-15) '2)
'8))))))))))))
'-8)
'24)
(ASH
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-16) '2)
'31)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '61440 (R '4520 '8 ST))
'15)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-11) '2)
'26)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-12) '2)
'25)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '3840 (R '4520 '8 ST))
'13)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-7) '2)
'20)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-8) '2)
'19)
(BINARY-LOGIOR
(ASH (BINARY-LOGAND '240 (R '4520 '8 ST))
'11)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-3) '2)
'14)
(BINARY-LOGIOR
(ASH (MOD (ASH (R '4520 '8 ST) '-4) '2)
'13)
(BINARY-LOGIOR (ASH (MOD (R '4520 '8 ST) '16) '9)
(ASH (ASH (R '4520 '8 ST) '-31)
'8))))))))))))
'-8))
(R (BINARY-+ '4376
(BINARY-+ (BINARY-* '8 (R '4536 '8 ST))
(BINARY-* '8
(UNARY-- (R '4528 '8 ST)))))
'8
ST))
'-40)
'256)
'-2)
'2)
'4))
(assign big-term
`(BINARY-LOGIOR
,(@ big-arg1)
(BINARY-LOGIOR
,(@ big-arg2)
,(@ big-arg3))))
; To see Ainni bound big-term, evaluate
(time$ (ainni (@ big-term) nil nil))
; the result is:
; (EV-REC *RETURN-LAST-ARG3* ...) took
; 0.00 seconds realtime, 0.00 seconds runtime
; (125,536 bytes allocated).
; (T NIL (INTEGERP (NIL . 0) NIL . 63))
; Note that the returned interval is [0,63].
; To see Ainni bound the three arguments separately do
(ainni (@ big-arg1) nil nil) ; (t nil [0,32])
(ainni (@ big-arg2) nil nil) ; (t nil [0,15])
(ainni (@ big-arg3) nil nil) ; (t nil [0,16])
; Because of Ainni the meta-< metafunction can easily prove that
; the big-term is less than 64.
(meta-< `(< ,(@ big-term) '64) nil state)
; returns 'T.
; Section 6: Syntactic Simplification of MOD Expressions
; This section introduces in passing the notion of a ``syntactic integer''
; expression. The idea is that we need to quickly determine whether an
; expression always returns an integer. Here are some examples and
; counterexamples:
(syntactic-integerp '(BINARY-+ '3 (R A B C))) ; = T
(syntactic-integerp '(BINARY-* '3 (R A B C))) ; = T
(syntactic-integerp '(MOD (R A B C) '123)) ; = T
(syntactic-integerp '(BINARY-+ X (R A B C))) ; = NIL
(syntactic-integerp '(BINARY-* '3 X)) ; = NIL
(syntactic-integerp '(MOD (R A B C) X)) ; = NIL
(syntactic-integerp '(R X Y Z)) ; = T
(syntactic-integerp '(BINARY-LOGAND X Y)) ; = T
(syntactic-integerp '(BINARY-LOGXOR X Y)) ; = T
(syntactic-integerp '(BINARY-LOGIOR X Y)) ; = T
(syntactic-integerp '(ASH (R A B C) X)) ; = T
(syntactic-integerp '(ASH X (R A B C))) ; = NIL
(syntactic-integerp (@ big-term)) ; = T
; The definition is quite simple. See (pe 'syntactic-integerp)
; Here are examples of the 7 rewrite rules implemented by meta-MOD. The second
; argument to meta-MOD is the metafunction context and it can tested with our
; previously set global variable (@ mfc) which assumes (R '16 '4 ST) < 16. In
; these examples we use Z as a generic term not known to be a syntactic integer
; and we use (R ...) as generic terms known to be syntactic integers.
; In the first example below, meta-mod makes no change to the expression
; because it cannot determine that the first argument to MOD is always an
; integer -- that is, X is not a syntactic integer expression. But in the next
; example the first argument to MOD is a syntactic integer so simplification
; occurs.
(meta-MOD '(MOD Z '0) nil state) ; no change
(meta-MOD '(MOD (R A B C) '0) nil state) ; simplifies
; = (R A B C)
(meta-MOD '(MOD '25 '7) nil state)
; = '4
(meta-MOD '(MOD (MOD Z '7) '17) nil state)
; = (MOD Z '7)
(meta-MOD '(MOD (MOD (R A B C) '15) '3) nil state)
; = (MOD (R A B C) '3)
(meta-MOD '(MOD (R A '2 C) '65536) nil state) ; 256^2 = 65536
; = (R A '2 C)
; In the following test, we use (R 16 4 ST) which is in general larger
; than 65536, but we specify (@ mfc) as the context and that context
; assumes (R 16 4 ST) < 16. So (conditional) simplification occurs.
(meta-MOD '(MOD (R '16 '4 ST) '65536) (@ mfc) state)
; = (IF (FORCE (NOT (< '15 (R '16 '4 ST))))
; (R '16 '4 ST)
; (MOD (R '16 '4 ST) '65536))
(meta-MOD '(MOD (BINARY-+ (R U V W)
(BINARY-+ (MOD (R A B C) '15)
(R I J K)))
'3)
nil state)
; = (MOD (BINARY-+ (R U V W) (BINARY-+ (R A B C) (R I J K))) '3)
; This last example illustrates the value of Ainni. Recall that (@ big-term)
; is a quite large term and yet this simplification occurs very quickly. We
; test the returned value rather than display it to save space in this file.
(equal (meta-MOD `(MOD ,(@ big-term) '64) nil state)
(@ big-term))
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