Deklarative Modellierung und effiziente Optimierung (MOC 2012)
Ulrich Geske1, Armin Wolf2
1Universität Potsdam, 2Fraunhofer FIRST, Berlin Ulrich.Geske@uni-potsdam.de, Armin.Wolf@first.fraunhofer.de
Vorwort zum Workshop
"Deklarative Modellierung und effiziente Optimierung" (MOC 2012)
Ziel der Workshop-Reihe
Die Workshop-Reihe Deklarative Modellierung und effiziente Optimierung zielt insbesondere auf die Anwendung und Weiterentwicklung von Constraint-Technologien sowie auf die Verbindung bzw. Gegenüberstellung dieses Paradigmas mit anderen Optimierungsverfahren. Konkrete Themen, die für den Workshop eine Rolle spielen sind deshalb einerseits die constraint-basierte Modellierung und Lösung von Anwendungsproblemen, u.a. für die Werkstatt- und Materialbedarfsplanung, das Flottenmanagement und die Fahrplanoptimierung, Personal- und Stundenplanung, Variantenkonfiguration technischer Anlagen aber auch deren Software, die Optimierung verteilter Berechnungs- oder Geschäftsprozesse sowie die Lastverteilung, als auch andererseits die Implementierung von Constraint-Modellen, u.a. durch die Verwendung unterschiedlicher Constraint-Systeme, die Erweiterung von Konsistenztechniken, effiziente Suchverfahren, Nachweis der Optimalität, Visualisierung des Lösungsprozesse, inkrementelle Lösungsanpassung u.a. in Verbindung mit anderen Optimierungstechniken. Die für diesen Workshop akzeptierten Arbeiten demonstrieren einige der unterschiedlichen Aspekte des Workshop-Themas.
Beiträge zum diesjährigen Workshop
Der diesjährige Workshop zeigt anhand zweier Arbeiten die Spannbreite der Thematik.
Zum einen wird untersucht, wie ein als klassisch zu bezeichnendes Problem einer Scheduling-Aufgabe durch innovative Anwendung bekannter Konstrukte der Constraint-Programmierung gelöst werden kann. Zum anderen wird versucht, sich einer bisher nicht behandelbare verbale Anforderung an die zu findende Lösung mit Hilfe bekannter Techniken der Constraint-Programmierung approximativ zu nähern.
Die Arbeit von Anna Prenzel und Georg Ringwelski „Statistical Evaluation of the Usability of Decision-Oriented Graphical Interfaces in Scheduling Applications“. Darin wird untersucht, wie sowohl die Qualität der Lösung als auch die Erfolgsquote der Lösungsfindung durch Kombination eines automatischen Ablaufs mit interaktiver Einflussnahme verbessert werden kann. Der Vorteil der Interaktivität liegt auch darin, dass nicht explizit formulierte oder nicht formulierbare Randbedingungen noch berücksichtigt werden können. Die Art und Weise der Kombination von automatischer und interaktiver Lösungsfindung liegt im Fokus der vorliegenden Arbeit.
Christoph Beierle, Gabriele Kern-Isberner und Karl Södler beschreiben in der Arbeit
„Stepwise Optimization af a Constraint Logic Program for the Computation of Ranking Functions“ die Behandlung vager Bedingungen der Form „wenn A, dann normalerweise B“. Derartige Anforderungen werden z. B. bei der Realisierung intelligenter Agenten und deren Aktionen auf eventuell unvollständig erfasste Umgebungszustände erforderlich. Die Autoren beschreiben die Modellierung des Problems als Constraint- Programm, das zur Lösung mehrfache Optimierungsschritte erfordert. Im Zusammenhang mit dem Optimierungsprozess stellt sich die Frage nach der Effizienz, die u.a. durch verschiedene Benchmarks untersucht wird. Die Darstellung und Kommentierung des erforderlichen Programmcodes macht die Problembehandlung transparent.
Danksagung
Für die Unterstützung bei der Organisation und Durchführung des nunmehr fünften Workshops dieser Serie danken wir dem Programmkomitee und den externen Gutachtern, die insbesondere durch die Begutachtung der eingereichten Beiträge und die Unterstützung der Autoren durch Hinweise zu ihren Arbeiten zum Gelingen beigetragen haben: Christoph Beierle, FernUniversität Hagen; Stephan Frank, TU Berlin; Hans- Joachim Goltz, Fraunhofer FIRST, Berlin; Petra Hofstedt, Brandenburgische Technische Universität Cottbus; Walter Hower, Hochschule Albstadt-Sigmaringen; Ulrich John, SIR Dr. John UG, Berlin; Georg Ringwelski, Hochschule Zittau/Görlitz; Dietmar Seipel, Universität Würzburg.
Berlin, im Juni 2012
Ulrich Geske und Armin Wolf
Stepwise Optimization of a Constraint Logic Program for the Computation of Ranking Functions
Christoph Beierle1, Gabriele Kern-Isberner2, Karl S¨odler1
1Dept. of Computer Science, FernUniversit¨at in Hagen, 58084 Hagen, Germany
2Dept. of Computer Science, TU Dortmund, 44221 Dortmund, Germany Abstract:Ordinal conditional functions (OCFs) can be used for assigning a seman- tics to qualitative conditionals of the formifAthen (normally)B. The set of OCFs accepting all conditionals in a knowledge baseRcan be specified as the solutions of a constraint satisfaction problemCR(R). In this paper, we present three optimizations of a high-level, declarative CLP program solvingCR(R)and illustrate the benefits of these optimizations by various examples.
1 Introduction
Common sense knowledge like “birds are animals” or “birds fly” can be expressed as conditionals of the form “if Athen (normally) B”, formally denoted by (B|A). Such conditionals are very different from material implicationsA⇒B, and their semantics is often given by what can be seen as an epistemic state of an intelligent agent. In this paper, we consider Spohn’s ordinal conditional functions (OCF) [Spo88] providing a semantics for qualitative conditionals. Given a knowledge baseR={R1, . . . , Rn}of conditionals, we are interested in determining all OCFs that on the one hand acceptR, and that on the other hand, are as unspecific as possible. While for system Z [GMP93, GP96] a unique most unspecific OCF exists, here we consider the framework of c-representations [KI02]
that has been developed as a qualitative realization of the ideas underlying the probabilistic maximum entropy principle [Par94, PV97, KI98] and provides a generalization of system Z* [GMP93].
In [BKI11], as a challenge for constraint programming, the set of all c-representations acceptingRis specified via the set of solution vectors of a finite domain constraint satis- faction problemCR(R). For computing the solutions ofCR(R), in [BKS11, BKS12] a high-level, declarative CLP programGenOCFis developed. In this paper, we extend the work in [BKS11, BKS12] by developing three optimizationsCR(R)andGenOCF:
• We optimizeCR(R)by replacing inequations inCR(R)by equations such that no minimal solutions are lost (Sec. 4). This optimizations is motivated by the fact that one is mainly interested in minimal solutions ofCR(R).
The research reported here was partially supported by the Deutsche Forschungsgemeinschaft – DFG (grants BE 1700/7-2 and KE 1413/2-2).
• We optimizeGenOCF by simplifying minimum constraints over a sum of terms containing a 0 (Sec. 5). This optimization is possible since all solution values are non-negative.
• Finally, we further reduce the number of minimum constraints over a sum of terms by considering multiple terms and subterms (Sec. 6).
Before presenting the optimizations in detail, we recall the OCF background (Sec. 2) and the the basics ofCR(R)andGenOCF(Sec. 3) as far as is needed here. In Section 7, a first implementation of the different optimizations are evluated with respect to various examples, and Section 8 concludes and points out further work.
2 Background
We start with a propositional languageL, generated by a finite setΣof atomsa, b, c, . . ..
The formulas ofLwill be denoted by uppercase Roman lettersA, B, C, . . .. For concise- ness of notation, we will omit the logicaland-connective, writingABinstead ofA∧B, and overlining formulas will indicate negation, i.e.Ameans¬A. LetΩdenote the set of possible worlds overL;Ωwill be taken here simply as the set of all propositional interpre- tations overLand can be identified with the set of all complete conjunctions overΣ. For ω∈Ω,ω|=Ameans that the propositional formulaA∈ Lholds in the possible worldω.
By introducing a new binary operator|, we obtain the set(L | L) ={(B|A)|A, B∈ L}
ofconditionalsoverL.(B|A)formalizes “ifAthen (normally)B” and establishes a plau- sible, probable, possible etc connection between theantecedentAand theconsequence B. Here, conditionals are supposed not to be nested, that is, antecedent and consequent of a conditional will be propositional formulas. A conditional(B|A)is an object of a three- valued nature, partitioning the set of worldsΩin three parts: those worlds satisfyingAB, thusverifyingthe conditional, those worlds satisfyingAB, thusfalsifyingthe conditional, and those worlds not fulfilling the premiseAand so which the conditional may not be applied to at all. This allows us to represent(B|A)as ageneralized indicator function going back to [DeF74] (whereustands forunknownorindeterminate):
(B|A)(ω) =
1 if ω|=AB 0 if ω|=AB u if ω|=A
(1)
To give appropriate semantics to conditionals, they are usually considered within richer structures such asepistemic states. Besides certain (logical) knowledge, epistemic states also allow the representation of preferences, beliefs, assumptions of an intelligent agent.
Basically, an epistemic state allows one to compare formulas or worlds with respect to plausibility, possibility, necessity, probability, etc.
Well-known qualitative, ordinal approaches to represent epistemic states are Spohn’sordi- nal conditional functions, OCFs, (also calledranking functions) [Spo88], andpossibility distributions[BDP92], assigning degrees of plausibility, or of possibility, respectively, to
ω κ1(ω) κ2(ω) κ3(ω)
f ba 0 0 0
f ba 1 1 1
f ba 0 0 0
f ba 0 0 0
ω κ1(ω) κ2(ω) κ3(ω)
f ba 1 1 2
f ba 1 2 1
f ba 0 0 0
f ba 0 0 0
Figure 1: Different ranking functionκiaccepting the rule setRbirdsgiven in Example 1
formulas and possible worlds. In such qualitative frameworks, a conditional(B|A)is valid (oraccepted), if its confirmation,AB, is more plausible, possible, etc. than its refutation, AB; a suitable degree of acceptance is calculated from the degrees associated withAB andAB.
In this paper, we consider Spohn’s OCFs [Spo88]. An OCF is a functionκ : Ω → N expressing degrees of plausibility of propositional formulas where a higher degree denotes
“less plausible” or “more suprising”. At least one world must be regarded as being normal;
therefore,κ(ω) = 0for at least oneω ∈Ω. Each such ranking function can be taken as the representation of a full epistemic state of an agent. Each suchκuniquely extends to a function (also denoted byκ) mapping sentences and rules toN∪ {∞}and being defined by
κ(A) =
(min{κ(ω)|ω|=A} ifAis satisfiable
∞ otherwise (2)
for sentencesA∈ Land by κ((B|A)) =
(κ(AB)−κ(A) ifκ(A)6=∞
∞ otherwise (3)
for conditionals(B|A) ∈ (L | L). Note thatκ((B|A))> 0since anyωsatisfyingAB also satisfiesAand thereforeκ(AB)>κ(A). The belief of an agent being in epistemic stateκwith respect to a default rule(B|A)is determined by the satisfaction relation |=O given by:
κ|=O(B|A) iff κ(AB) < κ(AB) (4) Thus, (B|A)is believed in κ iff the rank of AB (verifying the conditional) is strictly smaller than the rank ofAB(falsifying the conditional). We say thatκacceptsthe condi- tional(B|A)iffκ|=O(B|A).
Given a knowledge baseR = {R1, . . . , Rn}of conditionals, a ranking functionκthat accepts everyRirepesents an epistemic state of an agent acceptingR. If there is noκthat accepts everyRi thenRisinconsistent. For the rest of this paper, we assume thatRis consistent.
For any consistentRthere may be many differentκaccepting R, each representing a complete set of beliefs with respect to every possible formulaAand every conditional (B|A).
Example 1 LetRbirds={R1, R2, R3}be the following set of conditionals:
R1: (f|b) birds fly
R2: (a|b) birds are animals R3: (a|f b) flying birds are animals
In Figure 1, three different OCFsκ1,κ2,κ3 acceptingRbirds are given. Thus, for any i∈ {1,2,3}andj ∈ {1,2,3}it holds thatκi|=ORj. In order to ilustrate the evaluation of beliefs, consider the conditional(a|bf)(“Are non-flying birds animals?”) that is not contained inR. Forκ3, we getκ3(abf) = 2andκ3(abf) = 1and thereforeκ3|=/O(a|bf) so that the conditional(a|bf)is not accepted byκ3. On the other hand, forκ2 we get κ2(abf) = 1andκ2(abf) = 2and thereforeκ2|=O(a|bf).
The full beliefs about non-flying birds being animals or not, represented by the condition- als(a|bf)and(a|bf), are given by the following table:
κ1|=/O(a|bf) κ2|=O(a|bf) κ3|=/O(a|bf) κ1|=/O(a|bf) κ2|=/O(a|bf) κ3|=O(a|bf)
An agent being in epistemic stateκ2believes that non-flying birds are animals and does not believe that non-flying birds are not animals. An agent being in epistemic stateκ3
does not believe that non-flying birds are animals and believes that non-flying birds are not animals. An agent being in epistemic stateκ1is completely indifferent with respect to non-flying birds being animals or not, since she considers a world where non-flying birds are animals as equally plausible (or equally surprising) as a world where non-flying birds are not animals.
Thus, Example 1 illustrates that every OCFκaccepting a knowledge baseRinductively completes the knowledge given byR, and that in general there are different ways of com- pleting the knowledge. In principle, one is interested in characterizing and determining the full set of accepting OCFs, and it is a crucial question whether someκis to be preferred to some otherκ′, or whether among the preferred ones there is a unique “best”κ.
3 Computing OCFs as Solutions of a Constraint Satisfaction Problem
3.1 The Constraint Satisfaction ProblemCR(R)
Different ways of determining a ranking function for a knowledge baseRare given by system Z [GMP93, GP96] or its more sophisticated extension system Z∗ [GMP93], see also [BP99]; for an approach using rational world rankings see [Wey98]. For quantitative knowledge bases of the formRx = {(B1|A1)[x1], . . . ,(Bn|An)[xn]} with probability valuesxiand with models being probability distributionsPsatisfying a probabilistic con- ditional(Bi|Ai)[xi] iffP(Bi|Ai) = xi, a unique model can be choosen by employing the principle of maximum entropy [Par94, PV97, KI98]; the maximum entropy model is a best model in the sense that it is the most unbiased one among all models satisfyingRx.
Using the maximum entropy idea, in [KI02] a generalization of system Z∗is suggested.
Based on an algebraic treatment of conditionals, the notion ofconditional indifferenceof κwith respect toRis defined and the following criterion for conditional indifference is given: An OCFκis indifferent with respect toR={(B1|A1), . . . ,(Bn|An)}iffκ(Ai)<
∞for alli∈ {1, . . . , n}and there are rational numbersκ0, κ+i , κ−i ∈Q, 16i6n,such that for allω∈Ω,
κ(ω) =κ0+ X
16i6n ω|=AiBi
κ+i + X
16i6n ω|=AiBi
κ−i . (5)
When starting with an epistemic state of complete ignorance (i.e., each worldωhas rank 0), for each rule(Bi|Ai)the values κ+i , κ−i determine how the rank of each satisfying world and of each falsifying world, respectively, should be changed:
• If the worldωverifies the conditional(Bi|Ai), – i.e.,ω |=AiBi –, thenκ+i is used in the summation to obtain the valueκ(ω).
• Likewise, ifωfalsifies the conditional(Bi|Ai), – i.e.,ω |=AiBi –, thenκ−i is used in the summation instead.
• If the conditional(Bi|Ai)is not applicable inω, – i.e., ω |= Ai –, then this conditional does not influence the valueκ(ω).
κ0is a normalization constant ensuring that there is a smallest world rank 0. Employing the postulate that the ranks of a satisfying world should not be changed and requiring that changing the rank of a falsifying world may not result in an increase of the world’s plausibility leads to the concept of ac-representation[KI02, KI01]: Any ranking function κsatisfying the conditional indifference condition (5) andκ+i = 0,κ−i >0(and thus also κ0= 0sinceRis assumed to be consistent) as well as
κ(AiBi)< κ(AiBi) (6) for all i ∈ {1, . . . , n} is called a (special) c-representation of R. Note that for i ∈ {1, . . . , n}, condition (6) expresses that κaccepts the conditional Ri = (Bi|Ai) ∈ R (cf. the definition of the satisfaction relation in (4)) and that this also impliesκ(Ai)<∞.
Thus, finding a c-representation forRamounts to choosing appropriate valuesκ−1, . . . , κ−n. In [BKS12] this situation is formulated as a constraint satisfaction problemCR(R) whose solutions are vectors of the form(κ−1, . . . , κ−n)determining c-representations ofR.
The formulation ofCR(R)requires that theκ−i are natural numbers (and not just rational numbers) and thatmin(∅) =∞.
Definition 2 [CR(R)[BKS12]] LetR = {(B1|A1), . . . ,(Bn|An)}. The constraint satisfaction problem for c-representations ofR, denoted byCR(R), is given by the con- junction of the constraints, for alli∈ {1, . . . , n}:
κ−i >0 (7)
κ−i > min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j − min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j (8)
A solution ofCR(R)is ann-tupel(κ−1, . . . , κ−n)of natural numbers, and withSolCR(R) we denote the set of all solutions ofCR(R).
Proposition 3 [BKS12] ForR = {(B1|A1), . . . ,(Bn|An)} let #»κ = (κ−1, . . . , κ−n) ∈ SolCR(R). Then the functionκdefined by (9) and denoted byκ#»κ, is an OCF that accepts R:
κ(ω) = X
16i6n ω|=AiBi
κ−i (9)
Example 4 LetRbirds={R1, R2, R3}be as in Example 1. From (8) we get κ−1 >0,
κ−2 >0−min{κ−1, κ−3}, κ−3 >0−κ−2
and sinceκ−i >0according to (7), the two vectors
sol1= (κ−1, κ−2, κ−3) = (1,0,1) sol2= (κ−1, κ−2, κ−3) = (1,1,0)
are two different solutions ofCR(Rbirds). The OCFκsol1 (resp. κsol2) induced bysol1 (resp.sol2) according to (9) isκ1(resp.κ2) as given in Example 1.
3.2 The CLP programGenOCF
In GenOCF [BKS12] a knowledge base R = {(B1|A1), . . . ,(Bn|An)} is given as a Prolog code file providing the predicates variables/1, indices/1 and conditional/3. If Σ = {a1, . . . , am} is the set of atoms, we assume a fixed or- deringa1< a2< . . . < amonΣgiven by the predicatevariables([a1,...,am]).
The fixed index ordering given byindices([1,...,n])ensures that the condition- als are ordered consecutively from 1 to n. Thus, the i-th conditional can be accessed byconditional(i,A,B), and in a solution vector[K1,...,Kn], thei-th compo- nentKi is theκ−-value for thei-th conditional. In the representation of a conditional, a propositional formulaA, constituting the antecedent or the consequence of the condi- tional, is represented byhAiwherehAiis a Prolog list[hD1i,...,hDli]. EachhDii represents a conjunction of literals such thatD1∨. . .∨Dlis a disjunctive normal form ofA. EachhDi, representing a conjunction of literals, is a Prolog list[b1,...,bm]of fixed lengthmwheremis the number of atoms inΣand withbk∈ {0, 1, _}. Such a list[b1,...,bm]represents the conjunctions of atoms obtained froma˙1∧a˙2∧. . .∧a˙m
by eliminating all occurrences of⊤, wherea˙k =ak ifbk =1,a˙k =ak ifbk = 0, and
˙
ak=⊤ifbk=_.
Example 5 The internal representation of the knowledge baseRbirdsfrom Ex. 1 is:
variables([f,b,a]). % propositional variables indices([1,2,3]). % list of indices for
% conditionals
% f b a f b a
conditional(1,[[_,1,_]],[[1,_,_]]). % (f|b) birds fly
conditional(2,[[_,1,_]],[[_,_,1]]). % (b|p) birds are animals conditional(3,[[1,1,_]],[[_,_,1]]). % (-f|p) flying birds are
% animals
Furthermore,GenOCFprovides the predicates
worlds(Ws) %Wslist of possible worlds
verifying worlds(i,Ws) %Wslist of worlds verifyingith conditional falsifying worlds(i,Ws) %Wslist of worlds falsifyingith conditional falsify(i,W) %worldW falsifiesith conditional
realising the evaluation of the indicator function (1) given in Sec. 2. The remaining part of GenOCFneeded forGenOCFO is the determination of vectors#»κas solutions ofCR(R).
The correspondingGenOCFpredicates closely follow the abstract specification ofCR(R) developed in Section 3.1 and are given in Figure 2.
4 Optimizing CR (R) by Transforming Inequations to Equations
The constraints inCR(R) given by (8) ensure that each conditional(Bi|Ai) ∈ Ris accepted. Since allκ−i are assumed to be natural numbers, we could replace the strict inequation (8) by
κ−i >1 + min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j − min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j (10)
without changing the set of solutionsSolCR(R). Generally, one is interested in minimal solutions, so one could be tempted so for minimizing the values of allκ−i , one could be tempted to replace the non-strict inequality in (10) by an equality as in:
κ−i = 1 + min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j − min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j (11)
However, using just (11), one might loose a solution in the case where the right hand side of the inequation (8) is negative since then (11) would require thatκ−i is negative. If (11) does not hold, the values of the vector(κ−i , . . . , κ−n)ensure that the conditional(Bi|Ai) is accepted, so that in that case no additional requirement onκ−i is needed.
Thus, we can use (11) only if 1 + min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j > min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j (12)
kappa(KB, K) :- % K is kappa vector of c-representation consult(KB), % for KB
constraints(K), % generate constraints for K labeling([], K). % generate solution
constraints(K) :-
indices(Is), % get list of indices [1,2,...,N]
length(Is, N), % N number of conditionals in KB
length(K, N), % gen. K = [Kappa_1,...,Kappa_N] of free var.
domain(K, 0, N), % 0 <= kappa_I <= N for all I accord. to (7) constrain_K(Is, K). % generate constraints according to (8) constrain_K([],_). % generate constraints for constrain_K([I|Is],K) :- % all kappa_I as in (8)
constrain_Ki(I,K), constrain_K(Is,K).
constrain_Ki(I,K) :- % gen. constr. for kappa_I as in (8) verifying_worlds(I, VWs), % all worlds verif. I-th cond.
falsifying_worlds(I, FWs), % all worlds falsif. I-th cond.
list_of_sums(I, K, VWs, VS), % VS list of sums for verif. worlds list_of_sums(I, K, FWs, FS), % FS list of sums for falsif. worlds minimum(Vmin, VS), % Vmin minium for verif. worlds minimum(Fmin, FS), % Fmin minium for falsif. worlds element(I, K, Ki), % Ki constraint variable for kappa_I Ki #> Vmin - Fmin. % constraint for kappa_I as in (8)
% list_of_sums(I, K, Ws, Ss) generates list of sums as in (8):
% I index from 1,...,N K kappa vector
% Ws list of worlds Ss list of sums
% for each world W in Ws there is S in Ss s.t.
% S is sum of all kappa_J with
% J \= I and W falsifies J-th conditional list_of_sums(_, _, [], []).
list_of_sums(I, K, [W|Ws], [S|Ss]) :- indices(Js),
sum_kappa_j(Js, I, K, W, S), list_of_sums(I, K, Ws, Ss).
% sum_kappa_j(Js, I, K, W, S) generates a sum as in (8):
% Js list of indices [1,...,N] I index from 1,...,N
% K kappa vector W world
% S sum of all kappa_J s.t.
% J \= I and W falsifies J-th conditional sum_kappa_j([], _, _, _, 0).
sum_kappa_j([J|Js], I, K, W, S) :- sum_kappa_j(Js, I, K, W, S1), element(J, K, Kj),
((J \= I, falsify(J, W)) -> S #= S1 + Kj; S #= S1).
Figure 2: CLP programGenOCFdetermining vectors as solutions ofCR(R)
1234567839ABCDE9F36751234567835426 1234567839ABC4DEAF77797483BF 1234567839A8BEAFE1234567839AB4EAF
1234567839A8BEAF3123456267797483BF 68!8394"#8454BEAE$%FE384524"#42 68&26'4 748!8394"#8454BEAE(%FE384524"#42748&26'4
#838#"#B$#83E$%FE$#83#838"#26 68!83&26'4
#838#"#B(#83E(%FE(#83#838"#26748!83&26'4
#35BEAEA8FEA81234567835 7687)26779 A8*+$#83(#831234567835267797483BF 123333333333333333333333333333333333333333333333332
1245678 9ABCDEF
123333333333333333333333333333333333333333333333332
22E2E2E2222222222212E2C2FD2929CD92DC2E2 22EEBE2E !"212E2EEB2#F2#C$E#E2%F$&2 229$99D72'2'E"22222222212'E2(FDCED2)CEC$92#F2*C++C,7 22'E2-3222E2.2E/2222212(FDCED2#F2*C++C,72C2E20"2
2221222C29ABCDEF
123333333333333333333333333333333333333333333333332 12581 9ABCDEF2
123333333333333333333333333333333333333333333333332
Figure 3: Optimization ofGenOCFby replacing inequations by equations
holds. Putting these constraints together yields κ−i = 1 + min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j (13)
−min{1 + min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j, min
ω|=AiBi
X
j6=i ω|=Aj Bj
κ−j}
Note that just as required, (13) reduces toκ−i = 0if (12) holds. Thus, our first optimization of CR(R)is to replace (8) by (13). Note that this optimization transforms a strictly- less-than relationship into an equation; thus it should be clearly distinguished from the modelling of a constraintx < ybyx+ 1 6 ywhich might be done by the underlying constraint solver.
For different notions of minimality, resulting from different orderings of the solution vec- tors #»κinSolCR(R), this optimization does not loose any minimal solution. For instance, this observation holds when#»κis compared to#»κ′by summing up the elements of the vec- tors, by a componentwise ordering of the vectors, or by a componentwise ordering of the full ranking functions induced by the vectors according to (9) (cf. [BKS12]).
A declarative implementation of this optimization inGenOCFis obtained by replacing the last three subgoals of in the predicateconstrain Ki/2(cf. Figure 2) is shown in Figure 3. Note that the new program code in Figure 3 closely follows the formulas given in (13).
It turned out that this optimization had a significant effect on the runtime ofGenOCFas
will be illustrated in Section 7.
5 Optimize Minimum Constraint Over List of Sums Containing 0
A typical set of constraints obtained from (8) may contain
κ−1 > min{κ−2 +κ−3,0} −min{κ−2,0} (14) Since allκ−i are non-negative, both minimum constraints of this inequation will always return 0 regardless of any other constraints. Thus, we can optimise both the first minimum constraintmin{κ−2 +κ−3,0}and the second minimum constraintmin{κ−2,0}in (14) by 0, and further evaluating0∗0to0transforms (14) into just
κ−1 >0 (15)
The optimisation is to avoid superfluous sum constraints where the minimum over a list of sums can be replaced by 0. Indead, this optimization can always be applied for any consis- tent set of conditionalsR. If a knowledge base is splitted intoordered partitions[GP96], all conditionals assigned to the first ordered partition give raise to this optimization. The tolerance condition [GP96] ensures that a verifying world for each conditional of the first orderd partition exists such that no other conditional of the whole knowledge baseRis falsified by this world. This leads to a sum of 0 as shown above.
To realise this optimisation, the predicateconstrain Ki/2 (cf. Figure 2) must be modified:
• It must be checked whether a verifying world exists such that no other conditional is falsified by that world
((once((verify(I,World), \+ falsify(_,World)))) and dually, if such a falsifying World exists:
((once((falsify(I,World), \+ (falsify(J,World),J \= I))))
• In case of such a world could be found, a list with only one constraint to sum 0 will be created. Otherwise the original predicatesverifying worlds/2(as well as falsifying worlds/2andlist of sums/4will be used to create the list of sum constraints.
The modification is implemented using two new predicatesverifying sumlists/3 andfalsifying sumlists/3. Calls to these new prediactes replace the calls to verifying worlds/2andfalsifying worlds/2inconstrain Ki/2. The optimized version ofconstrain Ki/2that also takes into account the first optimiza- tion presented in Section 4 is given in Figure 4 and 5.
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6 Reduce Number of Sum Constraints
Suppose that from the constraints inCR(R)we have obtained the constraint κ−2 > min{0, κ−6 +κ−7, κ−4 +κ−5, κ−4 +κ−5 +κ−6, κ−1, κ−1, κ−1, κ−1}
−min{κ−3, κ−3, κ−3 +κ−4, κ−3 +κ−4, κ−1, κ−1, κ−1, κ−1} (16)
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Figure 5: Optimization ofGenOCFw.r.t the minimum of list of sums (part 2/2)
(In fact, this constraint arises for one of the examples given in [M¨ul04]). With the opti- mization given in the previous section, (16) can be replaced by
κ−2 > min{0} −min{κ−3, κ−3, κ−3 +κ−4, κ−3 +κ−4, κ−1, κ−1, κ−1, κ−1} (17) A further simplification is to remove duplicate sum constraints, yielding
κ−2 > min{0} −min{κ−3, κ−3 +κ−4, κ−1} (18) Furthermore, it is evident that a minimum constraintmin{κ−3, κ−3 +κ−4}may be reduced tomin{κ−3}sinceκ−i >0. So if all constraint variables of a sum constraint are also part of another sum constrain, the larger sum constraint is superfluous and does not have to be issued to the constraint solver. Thus, our third optimization is to avoid
• duplicate sum constraints
• sum constraints greater or equal to other sum constraints
in order not to post a big bulk of redundant constraints, speeding upGenOCFbefore the constraint solver starts to calculate solutions and aiming at improving the overall perfor- mance ofGenOCF.
Based on the optimization in the previous sections, the new predicates verifying sumlists/3andfalsifying sumlists/3will be modified.
Generally, the way to to avoid spare sum constraints and to identify only relevant sum constraints will be changed. Two new predicates verify falsify stripped/2 and falsify falsify stripped/2 will replace the predicatesverifying worlds/2andfalsifying worlds/2. With these new predicates the search for verifying and falsifying worlds changes: The search for verifying resp. falsifying worlds is defined by constraints, so the iteration to search falsify- ing worlds for each verifying world is obsolet (see predicatefalsifying except/3).
With predicate stripedList all sum constraints greater or equal to other sum constraints are removed. The complete implementation of this optimization is given in Figures 6, 7, and 8.
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Figure 6: Optimization ofGenOCFby reducing sum constraints (part 1/3)
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7 Implementation and First Evaluation
We implemented the three optimizations developed above and applied them to the sythetic knowledge bases of the formkb synth<n> c<2n−1>.plwhich were introduced in [BKS12]. In Figures 9 and 10, results of applyingGenOCFwith and without the opti- mizations are shown. It is intesting to note that the first optimizations has the biggest effect on the runtimes, while the other two optimizations are still significant.
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Figure 10: Runtimes ofGenOCFwith optimizations for knowldege bases with 10 variables and increasing number of conditionals
8 Conclusions and Further Work
We presented three stepwise optimizations of a high-level declarative CLP program com- puting the solutions of a constraint satisfaction problem specifying ranking functions that accept the conditionals of a knowlegde base. A first implementation and its applications to various examples illustrates the effects of the optimizations. The biggest effect is obtained by a sharpening of the original constraint satisfaction problem; this optimizations trans- forms inequations to equations, but without loosing any minimal solutions. Among our current work is the development of further refinements and a more detailed investigation of the benefits of the different optimizations.
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