Keywords Abstract.
1 Introduction
Based Programs with Static Global Variables
ManfredSchmidt-Schau
:knowledgerepresentation,semantics,rule-basedprogram,pro duction
rules,exp ertsystems,defaultrules,parallelexecution,three-valuedlogic
Pro ductionrulebased programminghasb een inusesincetheearlyexp ert sys-
tems([BFKM85,BS84])andcanb eseenasanforerunnerofthedeclarativepro-
grammingstyle.Nowadays,knowledgebasedsystemsarebasedonanintegration
oftwoormoreprogrammingstyles,likeintegrationoflogicandfunctionalpro-
gramming,which p ermits to use features of dierent programmingstyles, but
maylo osethedeclarativenessofrules.Thegoalof thispap eristodemonstrate
that the pro ductionrule style of programmingremains usefuland p owerful, if Fachb ereichInformatik
JohannWolfgangGo ethe-Universi tat
Postfach111932
D-60054Frankfurt
Germany
e-mail:schaus s@in formatik .uni -frankfurt.de
A rule based program is a set of pro ductio n rules of the
form\whenconditionthenassignment".Weprop osetousearestricted
form of rules as a rule-based programming paradigm. We sacrice the
p ossibil i ty ofexplici tely assignin g newvalues toavariable.Thegain is
thatthenalresultofrunningaprogramisindep endentfromtheorder
of rule execution and that rule execution can b e paralleli zed without
restrictions. Afurthergainisthatrules arereallydeclarative,andthus
lo cal changes have amorepredictabl e eectthan in general rulebased
programs.
Weprop oseasanaturalsemanticsasemi-lattice orderedbyinformation
content. Using this semantics we are able to show that rules can b e
indep endently executed, even in parallel in adistributed environment,
if only minor restriction s are satised. This semantics also gives clear
hints on the implementatio n of default rules that use meta-predicates
like unknown?intheirpreconditi on.
Thereareseveralinterestingsp eciali sati ons.Oneisthattheinformation
contentofavariablemayonlyb e\unknown"or\completelyknown",i.e.
thetruthvaluesofathree-valuedlogic.Anothersp eciali zatio nisinterval
arithmetic onnumb ers. Thismo dels thesituation wherethevalue ofa
variableisknowntob ewithin someinterval.
2 Syntax of Mini-RBL
j
j j j j j
j
j j j
j j j j
j
j j
j0j3j
j
< > < >
< > < >
< > < > < >< >< >
< > < >
< > < >
< >
< > < >< >< >
< >
< > < > < >< >< >
< >
< > =
< > < > < > < > < >
to viewglobalvariables ascontaining knowledge ab outanentity intheworld,
andtheprogramexecutiongeneratesmoreknowledgeusingtheavailableknowl-
edge. This excludes for examplethe usageof counters or variables that gather
evidence bymanipulatingaglobalvariable.
Dropping general assignmentsapp ears to b e rather restrictive, since most
programming techniques used in rule based systems cannot b e used, as they
dep end onmutableglobalvariablesorontheexecutionorderofrules.However,
thereareseveral advantages.Therule-basedprogramisindep endentoftheow
ofcontrol.Ononehandthisp ermitslo calvericationofthesoundness ofrules,
on the other hand the compiler can rearrange the rule execution in order to
improveeciency ortoexecute therulesconcurrently.
Therearetwotechniquestocomputevalues,oneisevaluationofexpressions,
likeinapplicativeprogramming,andtheotheristogatherinformationinglobal
variables by overwriting unknown variables with their value, when this value
b ecomes known. Hence there is an implicitway of mo difyingglobalvariables.
A naturalgeneralizationis an intervalarithmetic.The idea is that aprogram
computesconstraintsonthevaluesofglobalvariables.Thisisapracticallyuseful
extension,sinceingeneraltheinputisnotanexactvalue,forexamplemeasured
valuesareingeneralexactwithinsomegivenerror range.
Asuccessful instance oftheprogrammingparadigmisthesystemPro.M.D.
([PT88], [TP90]) that is in everyday use in a clinical lab oratory. Pro.M.D.is
arule based exp ert systemshell that issuccess fullyused with severaldierent
knowledgebasesforroutinediagnosisandclinicalchemistryanalyses.Itisimple-
mentedinProlog,andcompilesrulesintoPrologpredicates.Thispap erjusties
andclariestheprogrammingparadigmthatwasusedinPro.M.D.Furthermore
asemanticallycorrectusageandinterpretation ofdefaultrulesisdevelop ed.
Weintro duceasmallrule-based languageMini-RBL,whichcanalsob eseen as
a subset of the Pro.M.D-language.Every program consists of a set of rules of
theform"IF cond THEN action ".
cond ::= b-expr
b-expr ::=NOT b-expr b-expr b-op b-expr
F T UNKNOWN c-expr b-var
(known? b-expr ) (known? n-expr )
b-op ::=AND OR EQUIV IMPL
c-expr ::= n-expr n-cmp n-expr
n-cmp ::=GT GE EQ LE LT
n-expr ::= numb ers n-expr n-op n-expr
UNKNOWN n-var
n-op ::=+
action ::= b-var := b-expr n-var := n-expr
i
1 2
2
Denition3.1
x y y x
D
unk now n D
a;b D c a c b c
d a;b d c
a;b l ub a;b
a a
a a a i
3 Semantics of production rules with static global
variables
rule-based Mini-RBL program
An admissable semantic domain is a partially ordered set,
such that
i.) Thereisaleastelement,called in .
ii.) For ,ifthereissome with and ,then thereisaunique
leastelement with .Thiselement iscalledleastupper boundof
(denoted ).
iii.) Furthermore for every increasing chain of elements, there
existsaleastupper bound with forall .
numb ers.For ease of notation, we write \x := t" for rules of the form \IF T
THENx:=t".
Syntactically,UNKNOWNis aconstant. Its meaning is that thevalue ex-
ists,but is notknown.The values ofthe Bo oleanand arithmeticfunctions for
unknown argumentscan b edetermined inastraightforwardway.For Bo olean
op erators,thisgivestruthtables usingthreetruthvalues.Theusualb ehaviour
ofarithmeticop eratorsis thattheyevaluatetoUNKNOWN,ifat leastonear-
gumentisUNKNOWN(cf.section 6).Thesyntaximpliesthat thereis anite
setofglobalvariables.Dep endingonthepurp ose someofthemmayb eviewed
asinput,outputorauxiliary.
A is a set of rules that corresp onds to the
syntax ab ove.Theexecution ofarulebased programconsistsof ringrules, if
theirconditionistrue,untilruleringdo esnotmo difyglobalvariablesanymore.
Iftheconditionevaluatestounknownorfalse,thentheruledo es notre.This
executionof Mini-RBLprograms is guided by theprinciplethat the execution
sequence ofrulesshouldnotinuencetheresult.Aconsequence isthat explicit
assignmentto variablesis forbidden.Variableshaving thevalueunknown may
b e overwritten with a value, an assignment of the samevalue to avariable is
alsop ermitted.Ifavariableisalreadyassigned avalue(except unknown), and
a further assignment tries to assign a dierent value,then the program stops
witherror\conictingfacts". Thiserroris seenasaprogrammingerror that is
dynamicallydetected.
n order to provide a denotationalsemantics [Sto81] for pro duction rules with
staticglobalvariables(andthusalsoforMini-RBLprograms),wemo delthein-
formationcontentofvariables.Therefore,weuseanupp er,completesemilattice
asasemanticdomain. meansthat containsmoreinformatinthan .
Thereissomerelatedworkintheareaofasynchronousandparallelpro cesses
(see [Gla90] and [GK91]), where anenvironmentusing theprinciple of single-
assignmentvariablesisinvestigated.However,thisworkdo esnotconsiderthree-
valuedlogicnoralatticeofinformationcontent.
( )
...
>
>
>
> >
>
> >
> > >
3
>
> 3
>
0
3
1 1
1 2
0
0
0
0
0 0 1 2 0 0
1 2
1
0
+1 Denition3.2
Denition3.3
Lemma 3.4
n
n n
i i
n
n
n
n
n n
n
n n
i
n n n
n
n
n
i
i i i
i
i i
i
k
i i m m
f 2 j g f 2 j g f g
>
>
! > > )
!
2
!
2
! 2
2 2 2
2
!
2
2
2
x Dx<fal se x Dx<tr ue unk now n
D
D
D v ; ;v w ; ;w
v w i
D
D
D ' D
D ' v w 'v 'w
' D D v 'v
v D
D 8
' D D 8
' ;' ; ' D 8
8 D D w D ;
W w W w W 'w W
' 8 8 w
D 8 D
D 8 w
w D c ;c ;c ; c w
' ;' ; ' 8
c 'c i ; ;
c W
l ub8 w
c
W c
w W c w 'w
c c 'c
imalelements, and .
Let beanadmissablesemanticdomain.Afunction
is , i and . A function
is ,i itismonotonic and for
all .
Let be an admissable semantic domain. Let be a set of
functions .A sequenceofapplicationsof isasequence
with such that every function in occurs innitely often in
this sequence.
Let be an admissable semantic domain and let
be a nite set of strongly monotonic functions. Let W := for
some . Consider a chain with that isderived
with a fair sequence of applications of functions , such that
for .
Then the limitof isa leastupper bound of and viceversa.
Thislimitelementisalso denoted as
Proof.
= =
canb edescrib edasacompleteupp ersemilatticethatcontainsthedomain
of athree valuedlogic(see [Hah93],[Urq86]).Furthermorethere isno greatest
element in . A motivation forthe omissionof agreatest element is that the
domain of b o olean values could only b e extended if some algebraic rules are
sacriced.
Theordering on is the pro ductordering ( ... ) ( ... ) i
for all . This pro duct ordering satisesconditions i)-iii)of Denition
3.1.Inorderto avoiddealingwithpartialfunctions,weextend thedomain
to b e a latticebya topelement (which could b e interpreted as inconsistent
knowledge)that isgreater thanallother elements.Thecorresp onding setwith
the extended ordering is denoted by( ) . Elementsthat arenot equalto
are calledprop er elements.
:( )
( ) monotonic =
: ( ) ( ) strongly monotonic
( )
Wedonotenforcethatfunctionsarecontinuousinthesenseoflatticetheory.
:( ) ( ) fair
...
Let : ( ) ( ) b e anite set of functions.For ( ) we
inductivelydene aset with , andif ,then ( ) for all
.Weabbreviate thissetas ( ).
: ( )
( ) ( )
( ) ... =
...
= =1 2 ...
( ( ))
Note that by the strong monotonicity criterion, ( ) is an increasing
chain.
1.) Firstwe showthatforevery elementin thereissomegreater element
inthechain.Wemakeinductiononthenumb erofapplications.Assumethat
forsome ,we have anelement . Nowconsider . Since the
chain isfair,there isan indexminthechain( ),such that = ,
2
0 1 0
1
1 1
0 0
0
0
0
0
>
>
> >
>
0
3
>
>
3 2
! f g !
2
6
2
>
2 2
2
Denition3.5
Corollary3.6
m k m m m
i i
i
n
i
i i
n
i
n n
n n
i
i n
i
n
n
new old
W
c
c W
D W
c W
c W c W
D
IFTHENcond;t i n
cond D unk now n;tr ue;fal se t D D
w v ; ; ;v D
IFTHENcond;t w
l ub w ; v ; ;v ;t w ; ;v if condw tr ue
w if condw tr ue
8 IFTHENcond;t
8 ' 8
w W 8 w
W w
W P l ubW
P
w D 8 ' 8
'w w
8 w D
w 8 8 w
v t
v t
v l ub v ;t
' w 'w
We dene on .
The operator for accepts two functions as
arguments: , and . For
,then
A isaset offunctions of the form .
a.) A D-program is ,iall arestronglymonotonic.
b.) Let bethe input tothe programand let .Then thereare two
cases
i.) hasonly asupperbound.Thentheprogramis on .
ii.) hasaproperupperbound.Thentheoutputof is .Wedenote
thisoutput as .
c.) An element isax-pointforthe D-program ,iforall ,
wehave .
d.) A D-program on the input ,i thereis some x-
point for inthe set .
The result of an admissable D-program is independent of the
sequence,inwhichrulesaretriedandred,aslongasthesequenceisafairone.
This holds for the three possible types of behaviour: Termination with success,
termination with failure, and non-termination. In the case of non-termination
the programapproximatesthe result.
have shownthat every elementin isdominated bysomeelementof the
chain ( ).
2.) Theconverseis trivial,since everyelementin isalsoin .
3.) Note that due to the denition of ( ) , the set alwayshas an upp er
b ound. The limitof the chain is also a least upp er b ound of , since
and( )dominateseveryelementof .
D-programs ( )
( ) 1
:( ) :( )
=( ... ... ) ( )
( )( )=
( ( ... ( ) ... )) ( )=
( )=
D-program ( )
admissable
=
contradictory
( )
[[ ]]
( )
( )=
terminates ( )
( )
Thisdenitioninterprets rulesas follows.Iftheconditionoftheruleissat-
ised, then the assignment := inthe assignmentpart do es notreplace the
valueofthevariable bythevalueof ,butcomputestheleastupp er b oundof
theoldandnewvalue,i.e., := ( ).Thisconformswiththeintuition
of information gathering.The interesting p ointis that on therighthand side,
everyrule lo callysatisestherestriction ( ).
ThesemanticsofaMini-RBLprogramisaD-program,wherethedomainD
isconstructedfromthedomainofathree-valuedlogic,andthenumb ers,where
unknownisthesmallestelement,andnumb erscannotb e compared.Everyrule
isinterpreted asafunction.
>
>
0 0
0
2
111111
0 3
n
n
n
low low hig h low hig h hig h
4 Concurrent Execution of Rules D
w w D
D D
w D
:
ptt>
ptt
v
q
q ;v
q v q =q q q ; : v =q q
v Corollary 3.8
Example 3.9
Example 3.10
{
{
Ifeveryprop erlyincreasingchainin isnite,thenthisholdsfor( ) .
Obviously,ifprop erlyincreasingchainsstartingwith arenitein( ) ,then
every inniteapplication
07
40
40
:=[1 ]
:=[( ) ( + )+ 05 ( + )]
Theparallelexecutionmo delofrulesinwhichwehavesharedmemoryandrule
execution canb e inprinciplesequentialized, isalreadycovered bytheprevious ing chain in is nite.Anadmissable rule-basedprogramterminateson input
,if everyproperly increasing chainstartingwith in isnite.
Proof.
The resultof an admissable rule-based program is independent
of the sequence, in whichrules are triedand red, aslong as the sequence is a
fair one. This holdsfor the three possible types of behaviour: Terminationwith
success,terminationwithfailure,and non-termination.
The following rules are taken from a rule based knowledge base
for the diagnostics of blood coagulation. The result isa certain combination of
symptoms.
IF quick THENquick-patho:=true
IF THENptt-patho:= true
IF THENptt-normal :=true
IF ptt-pathoANDtzt-pathoAND
tzy-patho ANDbzt-patho THENresult-combination-I :=true
Hereknowledge isstructuredin adecision treeand thusismoreof apropo-
sitionaltype.Theprogramterminates,sinceatsomepointintime,allvariables
arecomputed.
An exampleof anon-terminating,but sensibleD-programcon-
sists of the following tworule to compute the square-rootof some number. We
assumethatvaluesarenumbersthatarerestrictedbyintervals.Thelub-operation
inthe domainistheintersectionofintervals.Thefunctions thatextractthe val-
uesofupperandlowerboundsoftheintervalsarepermittedonlyintheexpression
on therighthandside ofthe assignment.Wedenotethe lowerand upperbound
of an intervalvariableby the suxes lowand high. Let be the input variable
and be thedesiredoutput.The rulesof the programare:
Thesetwounconditionalrulesformavalidprogramtoapproximatethesquare-
rootof the input . Since these two rules are unconditional, our interpretation
guarantees thatthe programisadmissable.
>
>
0
> >
3
2 2
2
2
2 2
1 2 2
2 21 2 1 1 2 +1 2
+1
0
0
1 2
1 1
n
n
; ;i ;i ;i ;n
i
i i
i i i i i i
i i
i i i
j
n n
i i
i
i i i
Denition4.1
Denition4.2
Theorem4.3
{
l ub
M D
8 D
b b M j b
l ubb ; b ; ;b ;b ;b ; ;b :
' 8 b M b 'b
b M b M
M
M M
M c c l ub M
M
M i b M
' 8 j c M c ' b
'
D 8
D w D
M
l ub8 w
c ;c ; M
b M
b c
Let D be anadmissable domain. The model forthe distributed
memoryisamultiset ofelementsin .Theprogramisrepresentedbya
set ofstronglymonotonicfunctions on .Therearethreetransformation
rulesforprocessing:
i.) Let and be twoelements from . Select someindex and replace
by
ii.) Takesome ,some and replace by .
iii.) Select some and addacopy of to .
This produces asequence of computations, consistingof multisets . Such
asequence iscalled . For bettersyntactic manipu-
lations,wecouldalsoremembertheexactoperationsthatleadfrom to ,
butthis isnotnecessaryforourpurposes.
From the sequence , we can dene a sequence as .
Wecallthis the sequencecorrespondingto .
A parallelexecution sequence is ifor all ,
and thereis some and some such that .
Let be an admissable semantic domain, let be a set of
strongly monotonic functions on , let be some element from
(theinput), and let beafair parallelexecution sequence.Then the corre-
spondingcompressedsequenceisanascending chainwith thelimit .
Proof.
can mo del concurrent access inshared memoryon the samevariable,or with
distributed memory,where thesamevariable mayb e inthe systemmorethan
once on dierent pro cessors . If we consider such a situation, it may not b e
p ossibleto simulate theexecutionbya sequentializedruleexecution. Sincewe
concentrateonprovingcorrectnes sofexecution,notoneciency,wewillnotuse
amo del,wherepro cessorsandchannelsareexplicit,butasimpliedmo delthat
onlymakesthedierentvalues atdierentplacesofthesamevariableexplicit.
Theideaofourmo delistoassumethateverypro cessorhasitsownmemory
foralltheglobalvariables.Thenallop erationscanb emo delledusingthreekinds
oftransformations.Therstis thattwopro cessorscommunicateand exchange
part oftheirknowledgeto construct the oftheirvalues.Thesecond isthat
somepro cessor res somerule. The third is that some pro cessor starts a new
pro cessorwithacopyof itsmemory.
( )
( )
( ( ... ... ))
( )
aparallelexecutionsequence
( ) ( ) := ( )
compressed ( )
( ) fair
( )
The fairness condition is rather natural and forces every function to b e
appliedasoftenasrequired.Itprevents,forexample,uselesscomputationswhich
aredominatedbycopy-op erations
( ) ( )
( )
( ( ))
Let ...b ethecompressedsequencecorresp ondingto .Nowcon-
structafairsequentialsequence( ) from asfollows:
:=
3
i i
{
nf g
2
+1 +1
+1 +1
1 1
+1
+1 +1 +1 +1 +1
+1 +1
+1
1 1 +1
1 1 +1 +1
0
5 Monotonic Operatorsand Truth-Preserving Conditions
i i i i
i i i i
i i
i i
i i
i i
i i i i i i i
i i i i i
i i
i i
j i
i i
i i
i i
i i i i i i
i j j i
M M ' b ' b
b
c b
b c i i b c
M M
c c b b b c M
M ' b 'b ' c c
' c l ub M
c l ub M d ;' d d
'
i j c b
d
d c d 'd ' 8
' b
d b i d b d 'd 'b b
d M c b
l ub 8 w
t
k now n k now n
cond
If isconstructed from byapplicationof ,then := ( ).
Thesequence ( ) is fairinthe sequential sense, since it is also fairin the
parallelsense.Nowwe showthatthetwosequence s ( )and( )havethesame
limit.
1.) for all : B y induction on . We have = , hence the relation
holds. If is constructed from by a lub or copy op eration, then
= and = , hence . If is constructed from
by an application of a function , then = ( ) ( ) ,
since is monotonic. The last inequation holds, since = ( ), and
= ( ( ))forsome .
Sinceevery musto ccurinnitelyoften,wecan constructafairsequential
sequence thatdominatestheparallelsequence ofcompressedenvironments.
2.) Forevery thereexists some such that :
By thefairnesscondition,we canconstruct asequence ( ) as follows.We
let = and theelementthat isgreater than ( )forsome .
We can cho osethe samesequence of 'sas forthesequence ( ).Thuswe
get byinductionon ,since = and ( ) ( )= .
Since iscontainedinsome ,wealsohave
Nowwe haveshownthatafairparallelsequence hasthesamelimitasafair
sequentialsequence ,hence thelimitis equalto ( ( )).
Theorem4.3showsthatifanadmissablerule-basedprogramisconcurrently
executed in a distributed environment, then the result is the same as in the
sequentialcase,ifsomefairnessconditionshold.Aterminationconditioninthe
parallelexecution case could b e to stop, ifthere is somepro cessor that fullls
aterminationconditionsimilarto thesequentialone,i.e.,thereisnomorepro-
gramrulethatmakesprogressinthecompressedsequence.Thisconditionisnot
eective asadistributedalgorithm.Theinvstigationofpragmaticaltermination
testsisb eyondthescop eofthispap er.Usingstrongerconditions,itisp ossibleto
showthattheknowledgeofthewholedistributedcomputationisalsoeventually
availableonsomesinglepro cessor
Inthissectionwe exhibitsomegeneralconditionswhicharesucientto ensure
the monotonicityofthe functions that corresp ond torules. Therst condition
isthattheexpression ontherighthandsideofanassignmentmustcorresp ond
to a monotonicfunction.This excludes, for example,the use of functions like
?intheassignmentpart,since ?isnotmonotonic.Afurthercon-
ditionwhichwemusthave isthatevery inaruleremainstrue,afteritwas
evaluated to true. In the following we investigate these requirements in more
detail.
>
>
n
i
n new
old
n
n m
n
!
)
2
!
2f
g
_ ^ :
0 3
>
!
_ ^
! f g
Denition5.1
Theorem5.2
Lemma 5.3 {
{
Lemma 5.4
Lemma 5.5
' D D v w
'v tr ue ' w tr ue
D D x y x y
x;y D fal se tr ue fal se tr ue
R v t cond
D D v
l ubv ;t
R
D
' v w cond v ;condw
unk now n;unk now n ; unk now n;fal se ; fal se;false; unk now n;tr ue ;
fal se tr ue ; tr ue tr ue '
v l ub w ;t v
l ubv ;t v l ub w ;t w v w t v t w
condv ;condw
tr ue;fal se tr ue;unk now n l ub v ;t v w
t v w
; ; ;= <; ; ; ;>
D D
fal se unk now n tr ue
pr ov ed tr ue tr ue fal se
uuf fal se tr ue unk now n
exact d;I
d
' D unk now n;fal se;tr ue
A function is called , i
and implies that .
Let \IF cond THEN " be a rule. Consider
as a function: , and let the assignment be interpreted as
.
If cond istruth-preservingand tismonotonic,then isstronglymonotonic
on .
Proof.
The functions , and aremonotonic.
the functions and arithmeticcomparisons aremono-
tonic, wheredivisionby zeroresultsin aglobalerror,i.e., .
The followingfunctions are -monotonic:
i.) tupling,projection,and compositionof -monotonic functions: .
ii.) the known?-function that is for ,otherwise
iii.) The function ,that is for ,otherwise .
iv.) The functions and .
v.) The function ,that is for ,otherwise .
vi.) In thecontextof intervalarithmetic,the function ,thatistrueif
the intervalIislessor equal .
The following functions are
truth-preserving:
i.) -monotonic functions.
: truth-preserving
( )= ( )=
Let b eanextensionoftheorder on and ,suchthat
for and fortheb o oleanvalues and ,such that
isthe smallest such ordering. We willalso freely use this notionsfor tuple-
valuedfunctions,inthiscaseeverycomp onentmustsatisfythetruth-preserving
condition.
Usingtruth-prese rvingconditionswecan givearathergeneralconditionfor
aruletob e stronglymonotonic:
= =
( ) :=
( )
( )
Let b eatruth-preservingconditionandlet .Then( ( ) ( ))
( ) ( ) ( ) ( )
( , ) ( , ) . In therst threecases thefunction is theiden-
tity.Inthelastthree cases truth-preservation requires that ( ( ))or
( ( )) ( ( )).Thetwoequationshold,since and ( ) ( ).
It is instructive to consider the forbidden cases that ( ( ) ( )) is
( ) or( ).Inthesecases theinequation ( ( ))
isrequired,whichholdsonlyintheveryrestricted casethat ( ) .
+ =
?
( )
Thefollowinglemmadescrib essometruth-preser vingfunctionsthatarep er-
mittedasconditionsinadmissablerules.
:
1 2 3 k
h k
k
h k h
6 Application to fuzzy values Corollary 5.6
{
{
Example 5.7
{
{
!
!
!
!
:
:
_ ^ :
_ ^
^
_: ^ _ :
:
_
x:f g x f D D
g D D
f D D
g D D v w D
g v g w g v g w f
f g v f g w
k now n
x k now n x
; ;
; k now n
k now n x k now n y
x>y k now n x k now n x x x;y ; x>
y
f f unk now n fal se f fal se
unk now n f tr ue tr ue
k now n f x k now n f fal se fal se
k now n f unk now n tr ue
k now n x
tr ue unk now n tr ue
; ; ;
iii.) The composition of a -monotonic function and a
monotonic function .
Proof.
Thefollowingconditions inMini-RBL aretruth-preserving.
Anarbitrarycondition, whereonlythe booleanoperators areused
Acondition,whichcanbeseenasthe compositionofafunctionthatiscom-
posedsolelyof ,and ,andanothertuple-valuedBooleanfunction
that ismonotonic.
These examples show that composition is not compatible with
truth-preservationand with .
Thecompositionoftwotruth-preservingfunctionsmaybenottruth-preserving.
For example, let be dened such that
and . Then f istruth-preserving.Thecomposition
isnottruth-preserving,since and
.
the composition of a monotonic with a -monotonic function may be not
-monotonic. The standard exampleis .
( ( )) :
:
Weprove onlythelast part,theotherones aretrivial.Let :
b e -monotonicand : b emonotonic.Let b eelementsin .
Then ( ) ( ),hencealso ( ) ( ).Since isalso -monotonic,wehave
( ( ) ) ( ( )),hence thecomp ositionistruth-preser ving.
Thefunction ismonotonic,butnot -monotonicandnottruth-preser ving.
Thesameholdsforlogicalimplication.Thefunction ?is notmonotonic,
but -monotonicandtruth-preserving.Thefunction ( ). ( ?( ))isalso
nottruth-preser ving.
?
An example for a truth-prese rving condition is ?( ) ( ?( )
( )),whichiscomp osedas ?( ) ( ?( ) ) ( (
)).It isnothardto syntactically checkthese conditionsduring compilationof
aMini-RBLprogram.
( ) = ( ) =
( )=
?( ( )) ?( ( ))=
?( ( ))=
( ?( ))
Ifweapplythemetho dtonumb erswhichareknowntob einsomeinterval,then
the semi-latticeissimplythelatticeof intervalsorderedbythesup erset order-
ing,andtheemptyintervalisomitted.Itisnothardtodeterminetheop eration
of functions onintervalsgiven functions on elements, thefunctions aresimply
liftedtosets.Thismetho densures thatfunctionsonintervalsaremonotonic.If
anintervalcontainsonlyone element,thenthevaluecanb e consideredunique,
andhenceknown.Forexample,thisgivestheequations =
and [1 2]+[4 5]=[5 7].It is also p ossible to have a b etter approximation,if
i i
i i
i i
_:
3
f 0 3 g
:
: 7 The Usage of Defaults
x x x
tr ue
A ;A B B A
A B unk now n
A ;B unk now n
k now n Exact e;x
tr ue x e
k now n
IL
l eng th ; l eng th ; ; ;
k now n exact
IL
k now n x
NO T k now n v v
NO T k now n v v
v v
k now n x in0,regardlessof thevalueof . Thesameholdsfor which can always
b e replacedby . This viewofthesemantics ofunknown p ermitsto useall
algebraic lawsto evaluateexpressions. However,dierent strength ofalgebraic
manipulationsmayresultindierentoutcomesofthesameprogram.Forexam-
ple,inthecaseofintervalarithmetic,thedistributivelawincombinationwitha
mo dulardenitionof and+maydecrease theexactness ofapproximations.If
acompilerusessuchalgebraicmanipulationsorsimplications,thenthisshould
onlyb edone,ifab etterapproximationresults.
Suchanimprovementcanalsoexhibithidden\conictingfacts",forexample,
lettherulesb e := 2 := ( ) Withoutalgebraicsimplications,
the Mini-RBL program started with = = terminates with
= 2 = . After algebraic simplication, the execution is halted
with\conictingfacts".
Intheintervalcaseanop eratorcorresp ondingto ?wouldb e ?( ),
which is ,ifthelengthoftheintervalfor isnotgreater than .Itcan b e
used ina similar way as the ? -op erator. Supp ose, we have a function
, that computes the length of an interval. This function is not monotonic,
since [0 3]=3, [1 2]=1,we have [0 3] [1 2],but not3 1in
theinformationordering . Thusthe functions ? and ? aretruth-
preservingandcanb eusedinconditions,iftheyarenotunderanegation.The
function ,andtheextractionofupp er andlowerb oundaryoftheintervalsis
notmonotonicandshould b eillegalinconditions.
Ifexpressions like ( ?( ))areusedtoimplementdefaults,thentheorder
ofruleexecutionplaysarole. Forexample,considerthetworules
IF ( ?( ))THEN :=1
IF ( ?( ))THEN :=2
Ifwe usethemwithoutprecaution,thentheresultmayb e =1,or =2,
dep endingonthesequenceofexecution,andtherealsoisnop ossibilitiytodetect
thiserrordynamically,since onceone rulehasred,b othrulesb ecomeinappli-
cable.Aswe haveseen inthelastparagraph,theconstruction ( ?( ))is
nottruth-prese rving,and thusdo es notsatisfytheconditionsoftheorem5.2.
Toremedythiswep ermitsuchdefault-rules,splittheprogramintoa\normal
part" and a \default part". The normal execution of the program consists of
ringtherulesinthenormalpart,andto retherulesinthedefaultpart only
inthe case, where the normal part of the programhas terminated. One must
ensure,however,thatthedefaultsarenotconicting,sinceotherwise,onewould
reintro duce adep endence oftheresult onthe orderofexecution. Theexample
ab oveis such asetofconicting defaults. Acorrect interpretation wouldb e to
keep the variable values xed, collect all results of the default rules, and add
the lub of all the outcomes, with a certain bad chance of getting as answer
\conictingfacts".
3
3
Constraints
8 Comparison with other Approaches fal se
unk now n
notk now n A A
B ;C unk now n
A unk now n
A x A
usual cycleofringnormalrulesuntiltheprogramterminates,or theprogram
stops afterringthedefaultrules.Therearesomereasons (seeb elow)toprefer
thesecond one.
Another, more subtle metho d would b e to also p ermit the application of
default rulesiftheprogram statusensuresthat thedefaultconditionsare true,
and cannot b echangedto byother rules.Inapracticalimplementation,
thiscanb edonebystoppingtheprograminterpretation,insp ectingthecurrent
program status, and then executing all the defaults. In order to have a clean
lo op breaking metho dsuch that theresults do notdep end on thetime of the
break,itmustb eensured thattheapplicationofdefaultsb ehavesasifthelimit
wasalreadyreached.Thismetho dofusingdefaultsensuresthatprogramshave
auniqueresult.
Ifthecompilermakes useofalgebraic lawsto simplifyexpressions,and the
programusesdefaults,thentheoutcomeofthewholeprogrammaychange,since
forexample,therstexecutionofallnon-defaultrulesmaypro duce for
acertain variable,but after algebraicsimplication,thevalueb ecomes known.
In the rst case, a default may b e applicable, wheras in the second case the
default rulesarenolongerapplicable.Inordertogiveanexample,consider the
followingrules:
A:=B*C
IF ( ?( ))THEN :=1
A criticalinputis =0 = .Theusual implementationof isthat
remains .Theprogramterminates,andthenthedefaultisredand
gives = 1. If the algebraic law 0 = 0 is applied, then is assigned 0,
and the default rule is not applicable. For sensible programming, this means
that defaultsshould onlyb e used as a last resort, for examplefor preparing a
readable formoftheoutput, butnotasanormalprogrammingdevice.
Inthisparagraphwewanttocommentonthep ossibilitiestoimplementtheMini-
RBLmetho dinotherparadigms.Theissueisadequacy ofsuch anemb edding.
(1) (cf. [Win92]) can express dep endencies b etween variables and
p erform the same computations for a subset of the Mini-RBL programs.
Constraintsare notdirected, whereas rulesare directed inthe sense, that
a single rule has a xed input and as output a new information on the
variable in the assignment. Thus Mini-RBL programs cannot b e used in
a backward direction. The advantage of Mini-RBL programs is that also
cyclic dep endencies can b e enco ded and lead to a stable output or to an
approximation of the output, whereas constraint systems require a direct
metho dfor nding thesolution,or if constraint propagation is used, some
kindof oscillationmayo ccur,ifcyclicdep endencies aregiven.
k now n pr ov ed exact
notk now n x 9 Conclusion
Acknowledgements Functionalprogramming
Rulesasimplications
ofalogicprogramamountstocomputingnewfacts,thusapproximatingthe
minimalmo del in a xed-p oint computation. Globalvariables as in Mini-
RBLcan b eenco ded as predicates andcorresp ondingfacts.Themanipula-
tion must b edone using Assert, Retract (or setval, getval),and ap ossible
computationof least upp er b ounds would b e ametho dto mo difyasserted
facts. This manipulation metho ds are not a p ossible in a pure subset of
Prolog, hence itis notclear,howtheexistence ofaminimalmo delcan b e
guaranteed.
(3) .([BW88],[Dav92])Purefunctionalprogrammingas
abasisforMini-RBLhasseveralproblems.Non-mutablevariablesarene,
however,evenmo difyingglobalvariablesfromunknowntoavalueisnotp os-
sible inpurefunctionallanguages.Ifthe functionallanguagep ermitssome
non-puretechniquesinitsI-O-pro cessing(cf.[SA92]),thentheemulationb e-
comesmoreadequate.However,thep ossiblecyclicityofvariabledep enden-
ciesinMini-RBLcannotb edirectlytranslatedintothefunctionalparadigm,
and mustb e checked inorder to prevent innitelo ops. This translation is
studied indetailin[Mo o93].Thep olymorphictyp e-disciplineoffunctional
programmingiseasilyadaptable toextensions ofMini-RBL,aslongas the
information-orderiscompatiblewiththetyp estructure.
(4) .Ifpro ductionrulesareseen astwo-valuedlogicalim-
plications,thenthe corresp ondingrulebased programshave theprop erties
ofb eing indep endent ofthesequence ofexecution.Inordertosimulateour
programmingparadigmusingimplicationalrules,thelogicmustb eatleast
athree-valued one.In this case,there mustb e somenon-determinism(see
theexamplesinSection7),andtheresults ofthispap erareapplicable.
Wehaveshownthatsomerestrictionsontheusageofpro ductionsystemsp ermits
tohaveacleanandnicesemanticsofPSG-programs.Theconsequence sarethat
theresultofaprogramisindep endentofthesequenceoftheexecutionofrules.
A further result is that parallelization of rule execution, even ina distributed
environment,leadsto thesameresultsas sequentialexecution.
A clear distinction can b e made b etween legal and illegal usage of several
op erators, like the ?, ?, and ?- op erator. Furthermore, the
usageof ( ?( ))for defaultscan b ep ermitted ifthe interpretationof
defaultrulesdiersfromtheinterpretationof otherrules.
IwouldliketothankBernhardPohl,FrankPupp e,Sven-EricPanitzandMarko
Sch utzforreadingadraftofthispap er.
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