11 november 2005 foundations of logic and constraint programming 1 negation: declarative...

11 November 2005 Foundations of Logic and Constraint Programming 1

Negation: Declarative Interpretation

Anoverview

• FirstOrderFormulasandLogicalTruth

• CompletionofPrograms

• SLDNF-resolution:Soundnessandrestrictedcompleteness

• ExtendedConsequenceOperator

• StandardModels

[Lloy87] J.W. Lloyd, Foundations of Logic Programming, Second Extended Edition,Springer,1987.

[ApBo94]KrzysztofAptandRolandBol,LogicProgrammingandNegation:ASurvey,JournalofLogicProgramming,19/20:9-71,1994.


First-Order Formulas

Given ranked alphabets F and for functiona and predicate symbols,respectively,andasetVofvariables,thesetof

(first-order) formulas(over,FandV)isinductivelydefinedasfollows:

IfatomATB ,F,V,thenAisaformula

If G1andG2areformulas,thenG1 , G1 G2(written G1,G2),G1 G2,G1

G2andG1 G2areformulas

IfGisaformulaandx V, then x Gand x G areformulas


Extended Notion of Logical Truth (1)

GivenaformulaG,andinterpretationIwithdomainD,andastate : V → D:

G is true in I under , written I |= G :

I |= p(t1, ..., tn) :(t1), ... , (t1) pI

I |= G :I | G

I |= G1 G2 : I |= G1 and I |= G2

I |= G1 G2 : I |= G1 or I |= G2

I |= G1 G2 : if I |= G2 then I |= G1

I |= G1 G2 : I |= G2 iff I |= G1

I |= xG : forevery d D: I |=’ G

I |= xG : forsome d D: I |=’ G

where ’ : V → Dwith’(x) = d and ’(y) = (y) forevery y V – {x}


Extended Notion of Logical Truth (2)

GivenaformulaG,setsofFormulasSandT,andinterpretationI,

andwherex1, ..., xkarethevariablesoccurringin G

x1, ..., xkG istheuniversal closureofG(abbreviatedto G)

I |= G : I |= G foreverystate

G is true in I (or I isa model ofG),written I |= G : I |= G

I isa model ofS,written I |= S : I |=G forevery G S

• T isa semantic (or logical) consequence ofS written S |= T :

everymodelof S isamodelof T ( I: I |= S implies I |= T)


No Negative Consequences of (Extended) Programs (1)

ConsiderprogramPmem:

member(x,[x|y]) member(x,[y|z]) member(x, z)

Then,e.gPmem|=member(a,[a,b])andPmem|member(a,[]).

Butalso,Pmem|member(a,[])since

HB {member},{|, [], a}|= Pmem andHB {member},{|, [], a}|member(a,[]).

Nevertheless,theSLDNFtreeofPmem {member(a, [ ])}issuccessful

member(a,[])

member(a,[])

failure

□ success


No Negative Consequences of (Extended) Programs (2)

Problem:

ForeveryextendedprogramP,thecorrespondingHerbrandBaseisamodel.

Hence,thereisnonegativegroundliteralAasalogicalconsequenceofP.

ButSLDNFtreeofPmem {A}, may be successful ! (?)

Hence, is SLDNF-resolutionsound?Whatdoessoundnessmeans?

Solution:

Strengthen P by completion (“replace implication by “equivalences”) to

comp(P)andcompare SLDNF-resolutionandcomp(P),insteadofP!


Completed Definitions (Example 1)

WhereasP |= precipitation, cold, snow %theleastHerbrandModel

comp(P) |= precipitation, cold, snow, holidays, sun, happy

P: happy snow, holidays happy sun

snow cold, precipitationcold precipitation

comp(P): happy ( snow, holidays ) sun snow cold, precipitation

cold true

precipitation true

holidays false

sun false


Completed Definitions (Example 2)

Then,e.g.comp(p) |= member(a,[a,b]), member(a,[ ]), member(a,[b,c])

P: member(x,[x|y]) member(x,[y|z]) member(x,z)

disjoint([],x) disjoint([x|y],z) member(x,z), disjoint(y,z)

comp(P): x1 x2 member(x1,x2)

x,y (x1 = x, x2 = [x|y]) x,y,z (x1 = x, x2 = [y|z]), member(x,z)x1 x2 disjoint(x1,x2)

x (x1 = [], x = x) x,y,z (x1 = [x|y], x2 = z,

member(x,z), disjoint(y,z)) (plusthestandardaxiomsforequalityandinequality)


Completion (1)

Thecompletionofextended programP (denotedby comp(P)), isthesetof

formulasconstructedfromPbythefollowing6steps:

1. Associatewitheveryn-arypredicatesymbolpasequenceofpairwisedistinct

variablesx1, ... , xnwhichdonotoccurinP.

2. Transformeachclausec = p(t1, ... , tn) B into

p(x1, ... , xn) x1, = t1, ... , xn, = tn, B

3. Transformeachresultingformulap(x1, ... , xn) G into

p(x1, ... , xn) z G

wherezisasequenceoftheelementsofVars(c).


Completion (2)

4. Foreveryn-arypredicatesymbolp,let

p(x1, ... , xn) z1 G1 , . . . , p(x1, ... , xn) zm Gm

beallimplicationsobtainedinstep3(m0).

4.a) Ifm>0,thenreplacethesebytheformula

x1, ... , xn p(x1, ... , xn) z1 G1 . . . zm Gm

(ifsomeGiisempty,replaceitbytrue.)

4.b) Ifm = 0, i.e predicate p had no defining clause, then add theformula

x1, ... , xn p(x1, ... , xn) false.


Completion (3)

5. Addthestandard axioms of equality

5.1 [ x = x] %reflexivity

5.2 [ x = y → y = x ] %simetry

5.3 [ x = y , y = z → x = z ] %transitivity

5.4 [ xi = y → f(x1, ..., xi, ... , xn) = f(x1, ..., y, ... , xn) ] %substitutivity

5.5 [ xi = y → p(x1, ..., xi, ... , xn) p(x1, ..., y, ... , xn) ] %substitutivity

6. Addthestandard axioms of inequality

6.1 [ x1 y1 ... xm ym →f(x1, ..., xi, ... , xn) f(y1, ..., yi, ... , yn) ]

6.2 [ f(x1, ..., xi, ... , xn) g(y1, ..., yi, ... , yn) ] (whenf g )

6.3 [ x t ] (whenx isapropersubtermof t)

– Noticethataxioms6.1to6.3arearestrictionofFOequalitytotheUNA(Unique Names Assumptiom)–“differentnamesdenotedifferententities”


Soundness of SLDNF-Resolution

GivenextendedprogramP,extendedqueryQandsubstitution:

| var(Q)isacorrect answer substitutionofQ: comp(P) |= Q

Qisacorrect instance ofQ: comp(P) |= Q

Theorem (Lloyd, 1987)

IfthereexistsasuccessfulSLDNF-derivationofP {Q}withCAS then

comp(P) |= Q

Corollary (Lloyd, 1987)

IfthereexistsasuccessfulSLDNF-derivationofP {Q},then

comp(P) |= Q


Incompleteness of SLDNF (Inconsistency)

Thus,comp(P) { p p } “=“ { false }

Hence, comp(p) |= p and comp(p) |= p

In this case comp(P) is inconsistent, as it has no model, since forevery I, I | comp(P)

In this case of inconsistency, SLDNF-resolution is incomplete, since there is

neitherasuccessfulSLDNF-derivationforP {p},norforP { p},

P: p p


Incompleteness of SLDNF (Non Strictness)

Thus,comp(P) { p q q, q q } “=“ { p true }

Hence, comp(p) |= p.

Inthiscase,ofnon-strictness(seelater),SLDNF-resolutionisincomplete,since

thereisnosuccessfulSLDNF-derivationforP {p}.

Notice that in the absence of one of the first two clauses, there would be no

incompleteness,sincep true wouldnotbein comp(P).

P: p q

p q

q q


Incompleteness of SLDNF (Floundering)

Thus,comp(P) { x1 p(x1) x (x1 = x, q(x)) , x1 q(x1) false } “=“

{ x1 p(x1) true , x1 q(x1) false }

Hence, comp(p) |= x1 p(x1) .

In this case, of floundering, SLDNF-resolution is incomplete, since there is no

successfulSLDNF-derivationforP {p(x1)}

Note:SLDNFblocksquery q(x),contrarytowhatPrologdoes!

P: p(x) q(x)


Incompleteness of SLDNF (Unfairness)

Thus,comp(P) { r p, q , p p, q false } “=“ {r false , q false }

Hence, comp(P) |= r .

Inthiscase,SLDNF-resolutionmightbeincomplete,sincethereisnosuccessful

SLDNF-derivationforP {r}whentheleftmostselectionrule(asinProlog)is

adopted (unfairness of the selection rule).

P: r p, q

p p


Dependency Graphs

Adependency graphDpofanextendedprogramP

:

directedgraphwithlabellededges,where

ThenodesarethepredicatesymbolsofP

Theedgesareeitherpositive(labelled+)ornegative(labelled-);

p →+ q edgein Dp :

P containsaclause p(s1, ..., sm) L, q( t1, .., tn), R

p →- q edgein Dp :

P containsaclause p(s1, ..., sm) L, q( t1, .., tn), R


Strict, Hierarchical and Stratified Programs

Given an extended program P, with dependency graph Dp, with predicatesymbolsp, q,andanextendedqueryQ:

pdepends evenly / oddlyonq:

There is a path in Dp from p to q with an even (including 0) / oddnumberofnegativeedges.

Pisstrict wrt. Q :

NopredicatesymboloccurringinQdependsbothevenlyandoddlyonapredicatesymbolintheheadofaclauseinP.

Pishierarchical :

NocycleexistsinDp

P isstratified :

Nocyclewith a negative edgeexistsinDp


Strict, Hierarchical and Stratified Examples (1)

P1: p p

• pdependsonitself (oddly)

• P1isstrict(noevenandodddependencyofponanypredicate)

• P1 isnot hierarchical(thereisacycleinP1)

• P1isnot stratified(thereisacyclewithanegativeedgeinP1)

• pdependsevenly(0)andoddly(1)onq,whichdependsonitself(evenly)

• P2isnot strict(thereisanevenandanodddependencyofponq

• P2 isnot hierarchical(thereisacycleinP2,, asqdependsonitself)

• P2isstratified(thereisnocyclewithanegativeedgeinP2)

P2: p qp qq q

p

+

-

q +

p -


Strict, Hierarchical and Stratified Examples (1)

• pdependsoddlyonq

• P3isstrict(noevenandodddependencyofpredicatep/1onanypredicate)

• P1 ishierarchical(thereisnocycleinP3)


• rdependsevenlyonqandp,thatdependsevenlyonitself

• P4isstrict(nopredicatedependsevenlyandoddlyonanotherpredicate)

• P2 isnot hierarchical(thereisacycleinP4,theselfdependencyofp)


p/1

-

P3: p(x) q(x) q/1

P4: r p, q

p pr

+

+p +

q


Restricted Completeness of SLDNF-resolution (1)

Theorem(Lloyd,1987)

LetPbeahierarchicalandallowedprogramandQanallowedquery.

If comp(P) |= Q for some such that Q is ground, then there is a successfulSLDNF-derivationofP {Q}withCAS

Note: Theoremdoesnotholdifanarbitraryselectionruleisfixed.

Theselectionrulemustbesafe (doesnotselectnegativeliteralsthatarenotground).


Restricted Completeness of SLDNF-resolution (2)

Theorem(Lloyd,1987)

LetPbeastratifiedandallowedprogramandQanallowedquery,suchthatPisstrictwrt.Q

If comp(P) |= Q for some such that Q is ground, then there is a successfulSLDNF-derivationofP {Q}.WithCAS

Note: Theoremdoesnotholdifanarbitraryselectionruleisfixed.

Theselectionrulemustbesafeandfair.


Fair Selection Rule

AnextendedselectionruleRisfair:

foreverySLDNF-treeFviaRandforeverybranchinF:

Either isfailed;or

For every literal L occurring in a query of , (some further instantiatedversionof)Lisselectedwithinafinitenumberofderivationsteps

Examples:

Selectionrule“select leftmost literal”isunfair(depth-firstsearch)

Selectionrule“select leftmost literal to the right of the literals introduced in the previous derivation step, if it exists, otherwise select leftmost literal”isfair(breadth-firstsearch)


Extended Consequence Operator

LetPbeanextendedprogramandIaHerbrandinterpretation.Then

TP(I) : { H | H B ground(P), I |= B}

WhenPisadefiniteprogram,then

TP ismonotonic

TPIscontinuous

TPhastheleastfixpointM(P)

M(P)=TP

In case of extended programs all these properties are lost, since TP is not

monotonic.


Extended Consequence Operator is Not Continuous

LetI = {q }

ThenTP(I) = { },

Hence,itisnotthecasethat I TP(I).

Therefore,TPisnotmonotonic(norcontinuous)forPa(duetonegation).

Pa: p q

Pb: p p

LetI1 = { } and I2 = {p }

ThenTP(I1) = I1 = { },and TP(I2) = I2 = {p},

Therefore,TPismonotonic(andcontinuous)forPb.(andallwithnonegation)


Extended TP-Characterization (1)

Lemma 4.3([ApBo94]):

LetPbeanextendedprogramandI aHerbrandinterpretation.Then

I |= P iff TP (I) I

Proof.

I |= P

iffforeveryH B ground(P) : I |= B implies I |= H

iffforeveryH B ground(P) : I |= B implies H I

iffforeveryground atom H : H TP (I) implies H I

iff TP (I) I



Definition

Let F and be ranked alphabets of function and predicate symbols,

respectively,let = beabinarypredicatesymbol(for“equality”),andletI

beaHerbrandinterpretationforF and .

Then I= :I { t = t | t HUFis called a standardized Herbrand

interpretationforF and {=}



I |= comp(P) iff TP (I) I



Proof sketch of Lemma 4.4:

I= |= comp(P)

iffforeverygroundatomH:

I |= H (H B ground(P)) B )

(since I= isamodelforstandardaxiomsofequalityandinequality)

iffforeverygroundatomH:

H I I |= B forsomeH B ground(P)

iffforeveryground atom H :

H I implies H TP (I)

iff TP (I) I


Extended TP-Characterization: Example (1)

Let I0 = { }, I1 = {p }, I2 = {q }, I3 = {p, q },

Then TP(I0) = { p }, TP(I1) = { p }, TP(I2) = { }, TP(I3) = { }.

TP (I1 ) = I1 . Hence, I1 |= comp(Pa) and I1 |= Pa

TP (I2 ) I2. but TP (I2 ) I2. Hence, I2 |= Pa,butI2 | comp(Pa)

TP (I3 ) I3. but TP (I3 ) I3. Hence, I3 |= Pa,butI3 | comp(Pa)

TP (I0) I0 . Hence,I0 | Pa

Pa: p q comp(Pa) = {p, q}


Extended TP-Characterization: Example (2)

Pb: p q

q qcomp(Pb) = {p q}

Let I0 = { }, I1 = {p }, I2 = {q }, I3 = {p, q },

Then TP(I0) = { p }, TP(I1) = { p }, TP(I2) = { q }, TP(I3) = { q }.

TP (I1 ) = I1 . Hence, I1 |= comp(Pb) and I1 |= Pb

TP (I2 ) = I2 . Hence, I2 |= comp(Pb) and I1 |= Pb

TP (I3 ) I3. but TP (I3 ) I3. Hence, I3 |= Pb,butI3 | comp(Pb)

TP (I0) I0 . Hence,I0 | Pa


Completion may be Inadequate

Thus,comp(P) { ill ill, infection , infection true } “=“

{ill ill, infection true }

isinconsistent(ithasnomodels).

Hence, comp(P) |= healthy

But,I ={ ill, infection } istheonlyHerbrandmodelofP.

Hence, P | healthy

ill ill, infection

infection


Non-Intended Herbrand Models

P1hasthreeHerbrandmodels

M1 = {p}, M2 = {q},andM3 = {p, q}

P1hasnoleast,buttwominimalmodels:M1andM2.

However: M1andnotM2 , isthe“intended“modelofP1

P1: p q

BecauseTpisnotmonotonicnorcontinuous,itisnotpossible,ingeneraltogeta

fixpoint,noraleastmodelfromanextendedprogram.Nevertheless,thereare

minimalmodels,althoughnotnecessarilyunique.


Supported Herbrand Interpretations

AHerbrandinterpretation I is supported

:

Forevery H I thereexistssome H B ground(P) suchthat I |= B

(intuitively: B isan explanation forH )

Example:

M1 = {p, q },M2 = {p ,q},andM3 = {p, q}

M1isasupportedmodelofP1 (q istheexplanationfor p)

M2isnosupportedmodelofP1 (thereisnoexplanationfor q)

Notethat TP(M2) = M1 andthat TP(M1) = M1.

P1: p q


Non-Intended Supported Models

P2hasthreeHerbrandmodels

M1 = {p}, M2 = {q},andM3 = {p, q}

P2hastwosupportedHerbrandmodels:M1andM2.

Infact,both M1 andM2 areminimalmodelsfor comp(P2) = {p q}

However: M1andnotM2 , isthe“intended“modelofP2.

M1iscalledthestandardmodelofP2 (cf.laterslide)

Ingeneral,itisnotpossibletodefinestandardmodels,unless the programs are stratified!

P2: p q

q q


Stratifications

LetPbeanextendedprogramandDp itsdependencygraph,

PredicatesymbolpisdefinedinP :P containsaclausep(t1, ..., tm) B

P1 ... Pn = PisastratificationofP:

Pi foreveryi 1.. n

Pi Pj foreveryi,j 1.. n,withi j

foreverypdefinedinPiandedgep →+ q in Dp :

qisnotdefinedinnj=i+1Pj

foreverypdefinedinPiandedgep →- q in Dp :

qisnotdefinedinnj=iPj


An extended program is stratified iff it admits a (not necessarily unique)stratification.


Stratifications Example (1)

P1 P2 P3 isastratificationofP, where

P1 = { num(0) , num(s(x)) num(x) }

P2 = { zero(0) }

P3 = { positive(x) num(x) , zero(x) }

zero(0)

positive(x) num(x), zero(x)

num(0)

num(s(x)) num(x)


Stratifications Example (2)

P admitsnostratification

since

strat(even) > oddbutstrat(odd) strat(even)

even(0)

even(x) odd(x), num(x)

odd(s(x)) even(x)

num(0)

num(s(x)) num(x)


Standard Models (Stratified Programs)

Let

IbeanHerbrandinterpretationandasetofpredicatesymbols:

I | : I p(t1, ..., tm) | p P, t1, ..., tm groundterms)

P1 ... Pn beastratificationofastratificationofextendedprogramP,

Then

M1 : leastHerbrandmodelofP1suchthat

M1 | { p | p notdefinedin P } =

M2 : leastHerbrandmodelofP2suchthat

M2 | { p | p definednowhereorin P1 } = M1

...

Mn : leastHerbrandmodelofPnsuchthat

Mn | { p | p definednowhereorin P1 ... Pn-1 } = Mn-1

WecallMP = Mnthestandard modelof P


Standard Models: Example

Let P1 P2 P3 with

P1 = { num(0) , num(s(x)) num(x) }

P2 = { zero(0) }

P3 = { positive(x) num(x) , zero(x) }

beastratificationofP. Then

M1 = { num(t) | t HU{s,0} }

M2 = { num(t) | t HU{s,0} } {zero(0)}

M3 = { num(t) | t HU{s,0} } {zero(0)} {positive(t) | t HU{s,0} – {0} }

HenceMP = M3isthestandardmodelof P

zero(0)

positive(x) num(x), zero(x)

num(0)

num(s(x)) num(x)


Properties of Standard Models

Theorem 6.7([ApBo94]):

ConsiderastratifiedprogramP.Then

MP doesnotdependonthechosenstratificationof P,

MP isaminimalmodelof P,

MP isasupportedmodelof P,

Corollary:

ForastratifiedprogramP, comp(P)admitsaHerbrandModel.

11 november 2005 foundations of logic and constraint programming 1 negation: declarative...

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