12 October 2006 Foundations of Logic and Constraint Programming 1 Operational Semantics An overview...

12 October 2006 Foundations of Logic and Constraint Programming 1

Operational Semantics

Anoverview

• Thelanguageofprograms

• Thecomputationmechanism

• Choicesandtheirimpact


Atoms and Herbrand Bases

The basic constructs of programs and queries are atoms (predicates).Syntacticaly, atoms are similar to functions, but defined over predicatesymbolsratherthanfunctionsymbols.

Given

thetermuniverseTUF,V(overvariablesVandFunctionsymbolsF)

arankedalphabetofpredicatesymbols

the termbaseTB ,F,V is (over , FandV) is thesmallestsetAofatomssuchthat

1. p A,ifp(0)

2. p(t1,t2, ... ,tn) Aifp (n)withn1,and

t1,t2, ... , tn TUF,V

Given the Herbrand Universe, HUF, and the ranked alphabet of predicatesymbols:

Herbrand BaseHB(over andF) : TB ,F,V


Programs and Queries

Fromatoms,bothprogramsandqueriescanbeconstructed.

Query: finitesequenceofatomsB1, B2, ..., Bn.

Empty Query : emptysequenceofatoms.

- Anemptyqueryisdenotedby□

H B (definiteclause) :

- H(theclausehead)isanatom,

- B(theclausebody)isaquery.

H (unitclause) :

- Theclausebodyisempty

(Definite) program : finitesetofdefiniteandunitclauses

Horn Clause : Clauseoranegatedquery


Intuitive Meaning of Clauses and Queries

Clauses are obtained from the atoms by connecting them with Booleanconnectives.Hencetheirintuitivemeaning:

AclauseH B1, .. , Bncanbeunderstoodastheformula

x1, ..., xk (B1 ... Bn H);or

x1, ..., xk (B1 ... Bn H)

wherex1, ... ,xkarethevariablesoccurringinH B1, .. , Bn.

AUnitclause,H ,encodesx1, ..., xk H.

Aquery,A1, ..., An,canbeunderstoodastheformula

x1, ..., xk (A1 ... An) ;or

x1, ..., xk (A1 ... An)

where,x1, ... ,xkarethevariablesoccurringinA1, ..., An.

Theemptyquery,□,isequivalenttotrue.


Queries and Negated Queries

Informally, one is interested in assessing whether a query is a logicalconsequence of a program. This is indeed equivalent to assess that thenegationofthequeryiscontradictorywiththeprogram.

Hence

x1, ..., xk (A1 ... An)

x1, ..., xk (A1 ... An)

x1, ..., xk ( A1 ... An)

x1, ..., xk (false A1 ... An )

x1, ..., xk (false (A1 ... An)

x1, ..., xk (false (A1 ... An)

A negated empty query is equivalent to false (since the empty query isequivalenttotrue)

Thecomputationmechanismoflogicprogrammingproveseither thataqueryisalogicalconsequenceoftheprogram thatthenegationofthequery,togetherwiththeprogram,entailsfalse.


What is computed

AprogramPcanbeunderstoodasasetofaxioms.

AqueryQcanbeinterpretedastherequestforfindinganinstance,Qθ,whichisalogicalconsequenceof P.

Asuccessfulderivationprovidessuchaθ

(bycompositionofthesubstitutionsperformedateachderivationstep)

Hence,thederivationisaproofofQθ(fromP).,i.e.

P |- Qθ


How it is computed

Acomputationisasequenceofderivationsteps.

In each step an atom A is selected from the current query and a program

clauseH Bischosen.

IfA andHareunifiable,andθisanmguofAand H,thenAisreplacedbyB

inthequery,andθisappliedtotheresultingquery.

Thecomputationissuccessfulifitendswiththeemptyquery.

The resultinganswersubstitution isobtainedbycompositionof themgusof

eachstep.


SLD-Derivations (proposicional)

Theschemadescribedassumesthat

atomsinthequeryareselectedaccordingtosomeSelectionrule,

derivationsarea(Linear)sequenceofresolutionsteps,andthat

theclausesintheprogramareallDefiniteclauses.

ThesequenceofresolutionstepsisthusknownasanSLD-derivation.

Inthepropositionalcase(novariables),given• Aprogram P• AqueryA, B, C(whereAmeansitcontainszeroormoreatoms).• AclauseB B

Bistheselectedatominthequery

Theresultingquery(A, B, C)iscalledtheSLD-resolvent.

ThenotationA, B, CA, B, Cmaybeusedinthiscase.


SLD-Derivations (example)

Givenprogram

1. happy :- sun, holidays.2. happy :- snow, holidays.3. snow :- cold, winter.4. cold :- winter5. precipitation :- holidays.6. winter.7. holidays.

ThefollowingisanSLD-derivationofquery?- happy

?- happy (clause 2)

?- snow, holidays (clause 3)

?- cold, winter, holidays(clause 4)

?- winter, winter, holidays (clause 6)

?- winter, holidays (clause 6)

?- holidays (clause 7)

?- □


SLD-Derivations (non proposicional)

SLDderivationscanbeextendedtofirst-orderpredicates(withvariables).

Given• Aprogram P• AqueryA, B, C• Aclausec P• AvariantH Bofc,variabledisjointwiththequery• AnmguθofBandH

SLD-resolventofA, B, Candcwrt.Bwithmguθ :

(A, B, C) θ

SLD-derivationstep:

(A, B, C) θc (A, B, C) θ

inputclause: variant H Bofc,

“clausecisapplicabletoatomB”



Example:Givenprogram

1. add(X,0,X).2. add(X,s(Y),s(Z)) :- add(X,Y,Z).

thefollowingisanSLD-derivationofquery?-add(s(0),W, s(s(s(0)))).

?-add(s(0),W, s(s(s(0)))).

(Variant a of Clause 2) {Xa/s(0),W/s(Ya),Za/s(s(0))}

?-add(s(0),Ya, s(s(0))).

(Variant b of Clause 2) {Xb/s(0),Ya/s(Yb),Zb/s(0))}

?-add(s(0),Yb, s(0)).

(Variant c of Clause 1) {Xc/s(0) ,Yb/0}

?- □



?-add(s(0),W, s(s(s(0)))). (2a) {Xa/s(0),W/s(Ya),Za/s(s(0))}

?-add(s(0),Ya, s(s(0))). (2b) {Xb/s(0),Ya/s(Yb),Zb/s(0))}

?-add(s(0),Yb, s(0)). (1c) {Xc/s(0) ,Yb/0}

?- □

Composingthesubstitutions,

(θ2a)(θ2b)=

{ Xa/s(0), W/s(s(Yb)), Za/s(s(0)),

Xb/s(0), Ya/s(Yb), Zb/s(0)) }

(θ2a)(θ2b)(θ1c)=

{Xa/s(0),W/s(s(0)),Za/s(s(0)),

Xb/s(0),Ya/s(0),Zb/s(0)),

Xc/s(0) ,Yb/0 }

theanswerW = s(s(0))isobtained.


SLD-Derivations Steps

1. Selection:

Selectanatominthequery

2. Renaming:

Rename(ifnecessary)theclause

3. Instantiation

Instantiatequeryandclausebyanmguoftheselectedatomandthehead

oftheclause

4. Replacement

Replacetheinstanceoftheselectedatombytheinstanceofthebodyof

theclause


SLD-Derivations

AmaximalsequenceofSLD-derivationsteps

Q0 θ1c1

Q1 θ2c2

Q2 ... Qn θn+1cn+1

Qn+1 ...

isan

SLD-derivationofP{Q0}:

Q0, Q1, ... , Qn+1, ...arequeries,eachemptyorwithoneatomselectedinit;

θ1, θ2, ... , θn+1, ...aresubstitutions;

c1, c2, ... , cn+1, ...areclausesofP;

ForeverySLD-derivationstep,standardisation apartholds.


Standardisation Apart

Variablesinaclauseareuniversallyquantified.Itisthusimportantthatwhena

clause isusedavariantof theclause isusedwith“fresh”variablesthathave

notbeenpreviously“used”.

Thisguarantees that the inputclause isvariabledisjoint fromthe initialquery

andfromthesubstitutionsandinputclausesusedatearliersteps.

Formally:

Var(c’i ) (Var(Q0) j=1..i-1 (Var (θj Var(c´j )) =

fori 1,where c’i(variantofclauseCi P)istheinputclauseusedinthei-th

SLD-derivationstepQi θi+1ci+1 Qi+1


Result of a Derivation

Letξ =Q0 θ1c1 Q1 θ2

c2 Q2 ... Qn-1 θncn Qn beafiniteSLD-derivation.

ξ is successful:Qn = □

ξ failed:Qn □ andnoclauseisapplicabletoanyatomofQn.

Letξbesuccessful.

ComputedAnswerSubstitution (CAS) ofQ0(w.r.t.ξ)

: ( θ1 θ2 ... θn ) | Var(Q0)

ComputedInstanceofQ0(w.r.t.ξ)

: Q0 θ1 θ2 ... θn


Choices

AnumberofchoicescanbemadeinanySLD-derivation,namely

1. Choiceoftherenaming

2. Choiceofthemgu

3. Choiceoftheselectedatom

4. Choiceoftheprogramclause

Howdotheyinfluencetheresult?


Resultants: What is Proved after a Step

To assess the impact of the choices, it is convenient to assess what is

“logically”computedateachderivationstep.

Informally, toansweraqueryQ1, requiresansweringasubsequentqueryQ2.

HenceifQ2istrueforsomesubstitutionθsoisQ1.Moreformally,

ResultantAssociatedwithQ1 θ1 Q2 :Implication Q1θ1 Q2

Consider Aprogram P Aresultant R = Q A, B, C Aclause c AvariantH Bofc,variabledisjointwithR AnmguθofB andH

SLD – Resolvent ofresultantRandcwrtBwithmguθ:

Q (A, B, C) θ

SLD – Resultant step :

Q A, B, C θc Q θ (A, B, C) θ


Propagation

ConsideranSLD–Derivation

ξ =Q0 θ1c1 Q1 ... Qn θn+1

cn+1Qn+1 ...

Resultantantofleveliofξ, Ri :

Q0 θ1 θ2 ... θi Qi fori0

The resultant Ri describes what is proved wrt to the initial query Q0, after iderivationsteps.Inparticular

Nothinghasbeenproveninthebegining

R0:Q0 Q0

Thequeryhasbeenanswered,ifthederivationissuccessful

Rn :Q0 θ1 θ2 ... θn ifQn=□(since□=true)


Resultants: SLD - derivations

TheselectedatomofaresultantQ1 Qi isdefinedastheatomselectedinQi.

Lemma 3.12

AssumethatR θc R1andR’ θ’

c R’1aretwoSLD–resultantstepssuchthat

RisaninstanceofR’

InRandR’atomsinthesamepositionsareselected

thenR1isaninstanceofR’1.

Proof:See[Apt97],page55.

Sketch:IfRisaninstanceofR’,thenθ = θ’σ (forsomeσ)andR = R’ σ.


Resultants: SLD - derivations

Example:

Letusconsider

clausec:q(W,b):- s(W).

resultantR:p(a,Y) :- q(a,Z), r(Z,Y).

resultantR’:p(X,Y) :- q(X,Z), r(Z,Y).

atomq/2isselectedinbothresultants.

Now,ifatomq/2isselectedinbothresultants,itmustbe

θ = mgu(q(a,Z), q(W,b)) = {W/a, Z/b},andfromR,candθ

resultantR1:p(a,Y) :- s(a), r(b,Y).

θ’ = mgu(q(X,Z), q(W,b)) = {W/X, Y/b},andfromR’,candθ’

resultantR’1:p(X,Y) :- s(X), r(b,Y).

thusconfirmingthatR1isaninstanceofR’1.Moreover,itis

θ = θ’ {X/a} andso R1=R’1 {X/a}


Resultants: SLD-derivations

AsimilarresultcanbeobtainedforSLD-resolvents

Corollary 3.13

WhenQ θc Q1andQ’ θ’

c Q’1aretwoSLD–resultantstepssuchthat

QisaninstanceofQ’

InRandR’,atomsinthesamepositionsareselected

thenQ1isaninstanceofQ’1.

Example:

From Q: q(a,Z), r(Z,Y) and Q’: q(X,Z), r(Z,Y) if atom q/2 is

selectedinbothresolvents,togetherwithclausec:q(W,b):- s(W),then

Q1:s(a), r(b,Y)andQ’1:s(X), r(b,Y)


Similar SLD-derivations

Considertwo(initialfragments)ofSLD–derivations

ξ =Q0 θ1c1 Q1 ... Qn θn+1

cn+1Qn+1 ...

ξ’ =Q’0 θ’1c1 Q’1 ... Q’n θ’n+1

cn+1Q’n+1 ...

ξ and ξ’ are similar :

lenght(ξ ) = lenght(ξ’ );

Q0 andQ’0 arevariants,

inQi andQ’i atomsinthesamepositionsareselected(foriin0..n)


A Theorem on Variants

Theorem 3.18:

Consider two similar SLD – derivations ξ and ξ’ . Then for every i 0, the

resultants Ri and R’i of level i of ξ and ξ’, respectively, are variants of each

other.

Proof(byinduction):

Base case (i=0) :

R0:Q0 Q0 and R’0:Q’0 Q’0 .ButQ0 is a variant of Q’0 (bydefinitionof

similarderivations),henceQ0 Q0isavariantofQ’0 Q’0 .

Induction case (i > i+1) :

LetRi θi+1ci+1

Ri+1 and R’ i θ’i+1ci+1

R’i+1.Then

Ri isavariantofR’i(inductionhypothesis)

Ri isaninstanceofR’iandvice-versa(definitionofvariant)

Ri+1 isaninstanceofR’i+1andvice-versa(lemma3.12)

Ri+1 isavariantofR’i+1(definitionofvariant)


Answer Substitutions of Similar SLD-derivations

Corollary 3.19:

Consider two similar SLD – derivations of Q0 with computed answers

substitutionsθ and σ . ThenQ0θ andQ0 σarevariantsofeachother.

Proof:

Ifthederivationsaresuccessful,thentheirfinalresultantsare

ξ :Q0θ □ and ξ ‘:Q0 σ □ .

Then,bytheorem3.18,Q0θ and Q0 σ arevariants.

Hence,

Choices of type 1 (choice of a renaming)

Choices of type 2 (choice of an mgu)

do not influence - modulo renaming – on the statement proved by a

successful SLD–derivation.


Atom Selection in Queries

Let INIT be the set of all initial fragments of all possible SLD-derivations in

whichthelastqueryisnon-empty.

Aselectionrule,isafunctionwhichforeveryξ< INIT yieldsanoccurrenceof

anatominthelastqueryof ξ<

AnSLD-derivationξisviaaselectionruleRifforeveryinitialfragmentξ< ofξ

endingwithanon-emptyqueryQ,R (ξ< ) istheselectedatomofQ.

Example:

InProlog,theruleusedis“select the leftmost atom”


Switching Lemma

Lemma 3.32

ConsideranSLD-derivation

ξ =Q0 θ1c1 Q1 ... Qn θn+1

cn+1 Qn+1 θn+2cn+2 Qn+2 ...

where Qn includesatoms A1 and A2

A1 istheselectedatomofQn

A2θn+1 istheselectedatomofQn+1

Thenforsome Q’n+1, θ’n+1 and θ’n+2 thereisanSLD-derivation

ξ’ =Q0 θ1c1 Q1 ... Qn θ’n+1

cn+1Q’n+1 θ’n+2cn+2Qn+2 ...

where A2 istheselectedatomofatoms Qn

A1 θ’n+1 istheselectedatomofQ’n+1

θ’n+1 θ’n+2 =θn+1 θn+2

Proof:see[Apt97,page65].


Independence of Selection Rule

Theorem 3.33

Let ξ be a successful SLD-derivation of P {Q0}. Then for every selection

ruleR , thereexistsasuccessfulSLD-derivation ξ’ofP {Q0} viaR such

that

CAS ofQ0 (wrt ξ) = CAS ofQ0 (wrt ξ’)

ξ andξ’ areofthesamelength

Hence, choices of type 3 (choice of a selected atom) have no influence in

case of successful queries.


Independence of Selection Rule

Proof Sketch (of Theorem 3.33):(By induction on the number of derivation steps)

Base case (i=1) : If thederivationhasonlyonestep, thanQ0 hasasingleatomandany

rulemustselectit.

Induction case (i > i+1) : Assume that the first i stepsof thesuccessfulderivationξ aresimilar to the

firsti stepsobtainedbyusingruleR..

Assume further that in the i+1th stepξ selectsatomA inQi, whereas ruleR

selectsatomB. SinceBmustbeselectedinsomesubsequentstepjofξ(i +1 j n),itis

ξ =Q0 ... Qi[A] θi+1

ci+1 Qi+1 ... Qj[B] θj+1

cj+1 Qj+1 ... Qn-1 Qn = □

Applying the switching lemma j-i times to ξ the successful derivation ξ’ is

obtained

ξ’ =Q0 ... Qi[B] σi+1

c’i+1 Q’i+1[A] ... Q’n -1Qn = □.

forwhichthefirsti+1stepsaresimilartothoseobtainedbyusingruleR.


SLD-Trees and Search Space

Selectionrule Rvariant independent:

For all initial fragments of SLD-derivations that are similar, R chooses the

atominthesamepositionofthelastquery.

Examples:

Prolog’s rule “select leftmost atom” in the query is variant

independent.

Aselection rulesuchas “select the rightmost atom with predicate

symbol p with arity k, otherwise select the leftmost atom” isalso

variantindependent

Aselectionrulesuchas“select leftmost atom if variable X appears

in the query, otherwise select rightmost atom” is not variant

independent


SLD-Trees and Search Space

SLD-Tree forP {Q} viaselectionrule R:

ThebranchesareSLD-derivationsofP {Q} viaR

Every node Q with selected atom A has exactly one descendant for everyclausecofP,whichisapplicabletoA.

ThisdescendantisaresolventofQandcwrtA.

SLD-Tree successful:

AnSLD-treethatcontainsthesuccessleaf□.

SLD-Tree finitelyfailed:

AnSLD-treethatisfiniteandnotsuccessfully.

TheSLD-Tree via“leftmost selection rule”correspondstoProlog’sstrategyforsearchingforsolutions..


The Branch Theorem

Theorem 3.38

ConsideranSLD-treeT forP {Q0}viaavariantdependentselectionrule

R.TheneverySLD-derivationofP {Q0}viaRissimilartoabranchinT.

Hence, choices of type 4 (choice of a program clause) have no influence

on the search space as a whole.

Nevertheless,thesamesearchspacecanbesearcheddifferentlybydifferent

strategies.Forexample,

Ifaninfinitebranchistothe“left”ofanysuccessnode,the“left to right

depth first search”ofPrologdoesnotsucceed(notermination).

Iftheorderoftheclausesischanged,thanitisasifPrologwere“right to

left depth first search”, thus finding the solution (before entering the

endlessloop).


The Branch Theorem

Proof Sketch (of Theorem 3.38):

Letξ =Q0 Q1 Q2 ... beanSLDderivationof P {Q0}viaR

Byinductiononi0,abranch(withnodesQ’0, Q’1, Q’2, ...)inTsimilartoξ should be found:

Base case (i=0) :

Q’0 = Q0(inparticulartheyarevariants).

Induction case (i > i+1) :

Byinductionhypothesis,Q0 ... Qi issimilartoQ’0 ... Q’i

BydefinitionofT ,theexistenceofQ’iimpliestheexistenceofQ’i+1(applythesameclauseastoQi).

By variant independence, in Qi and Q’i atoms in the same position areselected,soQ0 ... Qi+1 isalsosimilartoQ’0 ... Q’i+1..

12 October 2006 Foundations of Logic and Constraint Programming 1 Operational Semantics An overview...

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