Luís Moniz Pereira Centro de Inteligência Artificial - CENTRIA Universidade Nova de Lisboa, Portugal Pierangelo Dell’Acqua Aida Vitória Dept. of Science and Technology - ITN Linköping University, Sweden

Motivation  Query answering systems are often difficult to use because they do not attempt to cooperate with their users.  We discuss the use of additional information about the user to enhance cooperative behaviour from query answering systems.

Idea  Consider a system whose knowledge is defined as: (P, R) P is a set of rules and R expresses preference information over the rules in P. When the rules in P conflict, then some rules are preferred over others according to R. P is used to derive conclusions and the preferences in R to derive the preferred conclusions.

Idea  Extra level of flexibility - if the user can provide preference information at query time: ?- (G,Pref ) Given (P,R), the system has to derive G from P by taking into account the preferences in R which are updated by the preferences in Pref.

Idea  Finally, it is desirable to make the background knowledge (P,R) of the system updatable in a way that it can be modified to reflect changes in the world (including preferences).

Update reasoning  Updates model dynamically evolving worlds.  Updates differ from revisions which are about an incomplete static world model.  Knowledge, whether complete or incomplete, can be updated to reflect world change.  New knowledge may contradict and override older one.

Preference reasoning  Preferences are employed with incomplete knowledge when several models are possible.  Preferences act by choosing some of the possible models.  They do this via a partial order among rules. Rules will only fire if they are not defeated by more preferred rules.

Preference and updates combined  Despite their differences preferences and updates display similarities.  Both can be seen as wiping out rules:  in preferences the less preferred rules, so as to remove models which are undesired.  in updates the older rules, inclusively for obtaining models in otherwise inconsistent theories.  This view helps put them together into a single uniform framework.  In this framework, preferences can be updated.

LP framework Atomic formulae: A atom not A default atom Formulae: every L i is an atom or a default atom generalized rule L 0  L 1  L n

LP framework Let N={ n 1,…, n k } be a set of constants containing a unique name for each generalized rule. Def. Prioritized logic program Let P be a set of generalized rules and R a set of priority rules. Then  =(P,R) is a prioritized logic program. Z is a literal n r <n u or not n r <n u priority rule Z  L 1  L n n r <n u means that rule r is preferred to rule u

Dynamic prioritized programs Let S={1,…,s,…} be a set of states (natural numbers). Def. Dynamic prioritized program Let (Pi,Ri) be a prioritized logic program for every i  S, then  =  {(Pi,Ri) : i  S} is a dynamic prioritized program. Intuitively, the meaning of such a sequence results from updating (P 1, R 1 ) with the rules from (P 2, R 2 ), and then updating the result with … the rules from (P n, R n )

Example: dynamic prioritized program This example illustrates the use of contextual preferences to select preferred models. (1) Suppose a scenario where John wants to buy a magazine. He can buy either a sport magazine (sm), a travel magazine (tm) or a financial magazine (fm). When John is at the office his preferred magazine is a financial magazine. sm  not fm, not tm (r 1 ) tm  not fm, not sm (r 2 ) fm  not sm, not tm (r 3 ) office (r 4 ) n1<n3  holiday n2<n3  holiday n3<n1  office n3<n2  office P1 R1

(2) Next, suppose that John goes on vacation. Now, John has two alternative magazines equally preferable: sport and travel magazine. not office (r 5 ) holiday (r 6 ) P2R2 Example: dynamic prioritized program

Queries with preferences  The ability to take into account the user information makes the system able to target its answers to the user’s goal and interests. Def. Queries with preferences Let G be a goal,  a prioritized logic program and  =  {(Pi,Ri) : i  S} a dynamic prioritized program. Then ?- (G,  ) is a query wrt. 

Joinability function Def. Joinability at state s Let s  S + be a state,  =  {(P i,R i ) : i  S} a dynamic prioritized program and  =(P X,R X ) a prioritized logic program. The joinability function  s at state s is:   s  =  {(Pi,Ri) : i  S + } (Pi, Ri)if 1  i < s (Pi,Ri) =(P X, R X )if i = s (P i-1, R i-1 )if s < i  max(S + ) S + = S  { max(S) + 1 }

Example: car dealer Consider the following program that exemplifies the process of quoting prices for second-hand cars. price(Car,200)  stock(Car,Col,T), not price(Car,250), not offer (r1) price(Car,250)  stock(Car,Col,T), not price(Car,200), not offer (r2) prefer(orange)  not prefer(black) (r3) prefer(black)  not prefer(orange) (r4) stock(Car,Col,T)  bought(Car,Col,Date), T=today-Date (r5)

Example: car dealer When the company sells a car, the company must remove the car from the stock: not bought(volvo,black,d2) When the company buys a car, the information about the car must be added to the stock via an update: bought(fiat,orange,d1)

Example: car dealer The selling strategy of the company can be formalized as: price(Car,200)  stock(Car,Col,T), not price(Car,250), not offer (r1) price(Car,250)  stock(Car,Col,T), not price(Car,200), not offer (r2) prefer(orange)  not prefer(black) (r3) prefer(black)  not prefer(orange) (r4) stock(Car,Col,T)  bought(Car,Col,Date), T=today-Date (r5) n2 < n1  stock(Car,Col,T), T < 10 n1 < n2  stock(Car,Col,T), T  10, not prefer(Col) n2 < n1  stock(Car,Col,T), T  10, prefer(Col) n4 < n3

Example: car dealer Suppose that the company adopts the policy to offer a special price for cars at a certain times of the year. price(Car,100)  stock(Car,Col,T), offer (r6) not offer Suppose an orange fiat bought in date d1 is in stock and offer does not hold. Independently of the joinability function used: ?- ( price(fiat,P), ({},{}) ) P = 250 if today-d1 < 10 P = 200 if today-d1  10

Example: car dealer ?- ( price(fiat,P), ({},{not (n4 < n3), n3 < n4}) ) P = 250 For this query it is relevant which joinability function is used:  if we use  1, then we do not get the intended answer since the user preferences are overwritten by the default preferences of the company;  on the other hand, it is not so appropriate to use  max(S + ) since a customer could ask: ?- ( price(fiat,P), ({offer},{}) )

Joinability function  In some applications the user preferences in  must have priority over the preferences in . In this case, the joinability function  max(S+) must be used. Example: a web-site application of a travel agency whose database  maintains information about holiday resorts and preferences among touristy locations. When a user asks a query ?- (G,  ), the system must give priority to .  Some other applications need the joinability function  1 to give priority to the preferences in .

Conclusions  Novel logical framework:  update and preference information can be specified and used in query answering systems.  declarative semantics is stable model based.  procedural semantics based on a syntactical transformation (correct and complete).

Future work  A preference metalanguage that compiles the pairwise preference specification.  Detect inconsistent preference specifications.  How to incorporate abduction in our framework: abductive preferences leading to conditional answers depending on accepting a preference.  How to tackle the problem arising when several users query the system together.

Preferred stable models Let  =  {(Pi,Ri) : i  S} be a dynamic prioritized program, Q = { Pi  Ri : i  S }, PR =  i (Pi  Ri) and M an interpretation of P. Def. Default and Rejected rules Default(PR,M) = {not A :  (A  Body) in PR and M |  body } Reject(s,M,Q) = { r  Pi  Ri :  r’  Pj  Rj, head(r)=not head(r’), i<j  s and M |= body(r’) }

Preferred stable models Def. Unsupported and Unprefered rules Unsup(PR,M) = {r  PR : M |= head(r) and M |  body - (r)} Unpref(PR,M) is the least set including Unsup(PR, M) and every rule r such that:  r’  (PR – Unpref(PR, M)) : M |= r’ < r,  M |= body + (r’) and [not head(r’)  body - (r) or  (not head(r)  body - (r’) and M |= body(r))]

Preferred stable models Def. Preferred stable models Let s be a state,  =  {(Pi,Ri) : i  S} a dynamic prioritized program, and M a stable model of . M is a preferred stable model of  at state s iff M = least( [X - Unpref(X, M)]  Default(PR, M) ) where: PR =  i  s (Pi  Ri) Q =  { Pi  Ri : i  S } X = PR - Reject(s,M,Q)

Preferred conclusions Def. Preferred conclusions Let s  S + be a state and  =  {(P i,R i ) : i  S} a dynamic prioritized program. The preferred conclusions of  with joinability function  s are: (G,  ) : G is included in every preferred stable model of   s  at state max(S + )