Logical representation of preference & nonmonotonic reasoning Jérôme Lang

Logical representation of preference & nonmonotonic reasoning Jérôme Lang Institut de Recherche en Informatique de Toulouse CNRS - Université Paul Sabatier - Toulouse (France) • About the meaning of preference • The need for compact representations and the role of logic • Some logical languages for compact preference • representation (a brief survey with examples) • Preference representation and NMR • Other issues

About the meaning of preference • The need for compact representations and the role of logic • Some logical languages for preference representation • Preference representation and NMR • Other issues

preference • has different meanings in different communities • in economics / decision theory: • preference = relative or absolute satisfaction • of an individual when facing • various situations

preference • has different meanings in different communities • in economics / decision theory: • preference = relative or absolute satisfaction • of an individual when facing • various situations • in KR / NMR • preference = [weak] [strict] order • with various meanings • A is more plausible / believed than B • preferential models, preferential entailment etc. • rule A has priority over rule B

« preference » • has different meanings in different communities • in economics / decision theory: • preference = relative or absolute satisfaction • of an individual when facing • various situations • in KR / NMR • preference = [weak] [strict] order • with various meanings • A is more plausible / believed than B • preferential models, preferential entailment etc. • rule A has priority over rule B relative (ordinal) uncertainty control

binarypreferences S = G  G cardinal preferences u : S   utility function ordinal preferences  preorder on S fuzzy preferences R fuzzy relation on S R : S  S  [0,1] Preference structure: represents the preferences of an agent over a set S of possible alternatives

About the meaning of preference • The need for compact representations + the role of logic • Some logical languages for preference representation • Preference representation and NMR • Other issues

Complex domains: a state is defined by a tuple of values for a given set of variables Example : preferences on airplane tikcets option = (destination, price, dates, number-changes) preferentially interdependent variables Combinatorial explosion: prohibitive number of alternative 50 destinations, 10 price ranges, 10 departure dates and 10 return dates, 0/1/2 changes 150 000 alternatives Need for concise representations for preferences

Representation and elicitation of preferences agent (user, client…) INTERACTIVE ELICITATION concise form REPRESENTATION preference relation on S preferred alternatives

Why (propositional) logic? • prototypical compact & structured language • good starting point • expressive power • + closeness to human intuition • elicitation issues • efficient and well-studied algorithms • (+ tractable fragments etc.) • optimization issues (find optimal alternatives)

About the meaning of preference • The need for compact representations + the role of logic • A brief survey on propositional logical languages • for preference representation • Preference representation and NMR • Other issues

S = { |  K }    Some logical languages for preference representation 1a. “Basic” propositional representation K propositional formula set of possible alternatives 2 positions maximum to be filled 4 candidates A,B,C,D K = ( A   B)  ( A  C)  ( A  D)  ( B   C)  (B  D)  ( C   D) K = [  2 : A, B, C, D]

         Some logical languages for preference representation 1a. “Basic” propositional representation K B = {1, …, n} set of goals K  1  …  n  such that  « good » states  K   impossible states K   (1  …  n )  such that  « bad » states

Some logical languages for preference representation 1a. “Basic” propositional representation K = [  2 : A, B, C, D] G = { (A  B), (B  C),  D } I would like to hire A or to hire B; if B is hired then I would prefer not to hire C; I would like not to hire D (A,B,  C,  D) hire A and B (A,  B, C,  D) hire A and C (A,  B,  C,  D) hire A only ( A,B,  C,  D) hire B only « good » states

   Some logical languages for preference representation 1b. “Basic” propositional representation + cardinality K B = {1, …, n} set of goals For all   Mod(K), uB() = | {i,  i } |

(A,B) (A, C) (A) (B) (A, D) (B, C) (B, D) (C) ( ) (C, D) (D) Some logical languages for preference representation 1b. “Basic” propositional representation + cardinality K = [  2 : A, B, C, D] ; G = { (A  B), (B  C),  D } u()=3  u()=2  u()=1

      Some logical languages for preference representation 1c. “Basic” propositional representation + inclusion K B = {1, …, n} set of goals For all  ,  ’  Mod(K)    ’ if and only if { i ,  j } i }  { j ,  ’

(A,B) (A, C) (A) (B) (C) ( ) (B, C) 12 23 13 (C, D) (D) Some logical languages for preference representation 1c. “Basic” propositional representation + inclusion K = [  2 : A, B, C, D] ; G = { (A  B), (B  C),  D } 2 3 1 123   (A, D) (B, D)   2

   Some logical languages for preference representation 2. Propositional logic + weights K B = { (1, x1 ), …, (n , xn ) } I propositional formula xi > 0 reward xi * xi < 0 penalty Example: additive weights For all   Mod(K),  xi uB() = i  1 .. N i 

   Some logical languages for preference representation 2. Propositional logic + weights K B = { (1, x1 ), …, (n , xn ) } I propositional formula xi > 0 reward xi * xi < 0 penalty Example: additive weights For all   Mod(K), other aggregation functions  xi uB() = i  1 .. N i 

   Some logical languages for preference representation 2. Propositional logic + weights K B = { (1, x1 ), …, (n , xn ) } I propositional formula xi > 0 reward xi * xi < 0 penalty For all   Mod(K), uB() = F ({xi | , i  1 .. N} ) i 

      i  Some logical languages for preference representation 2. Propositional logic + weights K B = { (1, x1 ), …, (n , xn ) } I propositional formula xi > 0 reward xi * xi < 0 penalty For all   Mod(K), uB() = F ( G ({xi |, i  1 .. N, xi > 0} ) , H ({xj |, i  1 .. N , xi < 0} ) )  j bipolarity

Some logical languages for preference representation 2. Propositional logic + (additive) weights K = [  3 : A, B, C, D, E] ; G = { (B  C, +5) ; (A  C, +6) ; (A  B, -3) ; (D  E, -3) ; (D, +10) ; (E, +8) ; (A, +6) ; (B, +4) ; (C, +2) } only B and C can teach logic only A and C can teach databases A and B would be in the same group (to be avoided) idem for D and E D is the best candidate E is the second best etc.

Some logical languages for preference representation 2. Propositional logic + weights  = (A, D, E,B, C) K = [  3 : A, B, C, D, E] ; G = { (B  C, +5) ; (A  C, +6) ; (A  B, -3) ; (D  E, -3) ; (D, +10) ; (E, +8) ; (A, +6) ; (B, +4) ; (C, +2) } +6 -3 u() = +27 +10 +8 +6

Some logical languages for preference representation 2. Propositional logic + weights ’ = (C, D, E,A, D) K = [  3 : A, B, C, D, E] ; G = { (B  C, +5) ; (A  C, +6) ; (A  B, -3) ; (D  E, -3) ; (D, +10) ; (E, +8) ; (A, +6) ; (B, +4) ; (C, +2) } +5 +6 u(’) = +31 +10 +8 +2

Some logical languages for preference representation 2. Propositional logic + weights u()  K = [  3 : A, B, C, D, E] ; G = { (B  C, +5) ; (A  C, +6) ; (A  B, -3) ; (D  E, -3) ; (D, +10) ; (E, +8) ; (A, +6) ; (B, +4) ; (C, +2) } 31 (C,D,E) (A,C,D) (A,B,D) (A,D,E) (B,C,D) (A,C,E) (B,D,E) (C,D) 29 28 27 24 23

increasing priority Some logical languages for preference representation 3a. Propositional logic + priorities K B1 … Bp B =  B1, …, Bp  stratification of B K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C,  (D  E),  (D  E)} 1 2 3 4 B2 = {D,A} B3 = {E} B4 = {B, C} 5 6 7 8 9

3a. Propositional logic + priorities K = [  3 : A, B, C, D, E] ; B1 = {B  C, A  C,  (A  B),  (D  E)} 1 2 3 4 B2 = {D,A} B3 = {E} B4 = {B, C} 5 6 7 8 9 « Best-out » ordering u () = min i,  violates at least a formula of Bi ( = +  if there is no such i)

3a. Propositional logic + priorities  = (A, B, C,D, E) K = [  3 : A, B, C, D, E] ; B1 = {B  C, A  C,  (A  B),  (D  E)} 1 2 3 4 u () = 1 B2 = {D, A} B3 = {E} B4 = {B, C} 5 6 7 8 9 « Best-out » ordering u () = min i,  violates at least a formula of Bi

3a. Propositional logic + priorities  = (A, C, D,B, E) K = [  3 : A, B, C, D, E] ; B1 = {B  C, A  C,  (A  B),  (D  E)} 1 2 3 4 u () = 3 B2 = {D,A} B3 = {E} B4 = {B, C} 5 6 7 8 9 « Best-out » ordering u () = min i,  violates at least a formula of Bi

3a. Propositional logic + priorities K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 « leximin » ordering [Benferhat et al. 93] w > w’ iff (w satisfies more formulas of B1 than w’) or (w and w’ satisfy the same number of formulas of B1, and w satisfies more formulas of B2 than w’) or (w et w’ satisfy the same number of formulas of B1 and of B2, and w satisfies more formulas of B3 than w’) etc.

3a. Propositional logic + priorities: leximin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 0 1 1 (A,C) 3 1 1 0 (A,D) 3 (B,C) 3 0 0 2 (C,D) 3 1 0 1

3a. Propositional logic + priorities: leximin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 0 1 1 (A,C) 3 1 1 0 (A,D) 3 (B,C) 3 0 0 2 (C,D) 3 1 0 1 (D,E) 1 1 1 0

Some logical languages for preference representation 3b. Propositional logic + ordered disjunction K [Brewka, Benferhat & Le Berre 02] Y= (j1  j2 …jp ) ideally j1; otherwise (sub-ideally) j2 otherwise j3 etc. B =  Y1, …, Yp 

         Some logical languages for preference representation 3b. Propositional logic + ordered disjunction K Y= (j1  j2 …jp ) ideally j1; otherwise (sub-ideally) j2 otherwise j3 etc. For all   Mod(K), j1 disu(, Y) = 0 if   j1  …  ji-1  ji = i if  = p+1 if   j1  …  jn

For all , ’  Mod(K),  >B ’ iff disu (, B) <leximin disu (’, B) Some logical languages for preference representation 3b. Propositional logic + ordered disjunction K B =  Y1, …, Yp  disu (, B) =  disu (, Y1), …, disu (, Yp) 

Some logical languages for preference representation 3b. Propositional logic + ordered disjunction K B =  F1, …, Fp  K = [  2 : A, B, C, D, E] ;  = (A,E) ’ = (A,B) F1 : (B C)  (B  C) 3 2 F2 : (A C)  (A  C) 2 2 F3 :  (D  E) 1 1 F4 :  (A  B) 1 2 F5 : D  A  E  B  C 2 2 F5 : (= 2 : A,B,C,D,E)  (= 2 : A,B,C,D,E) 1 1

Some logical languages for preference representation 3b. Propositional logic + ordered disjunction K B =  F1, …, Fp  >B K = [  2 : A, B, C, D, E] ;  = (A,E) ’ = (A,B) F1 : (B C)  (B  C) 3 2 F2 : (A C)  (A  C) 2 2 F3 :  (D  E) 11 F4 :  (A  B) 1 2 F5 : D  A  E  B  C 2 2 F5 : (= 2 : A,B,C,D,E)  (= 2 : A,B,C,D,E) 11

3. Propositional logic + priorities K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B, C} 5 6 7 8 9 « discrimin » ordering [Brewka 89] strict inclusion w > w’ iff { j B1, w satisfies j }  { j B1, w’ satisfies j } or ( { j B1, w satisfies j } = { j B1, w’ satisfies j } and { j B2, w satisfies j }  { j B2, w’ satisfies j }) etc.

3. Propositional logic + priorities : discrimin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 - 6- 9 (A,C) 123- (B,C) 123- - -- 89 (A,D) -234 5 -- 9 (C,D) 12-4 5 -- 9

3. Propositional logic + priorities: discrimin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 - 6- 9 (A,C) 123- (B,C) 123- - -- 89 (A,D) 2--4 5 -- 9 (C,D) 12-4 5 -- 9

3. Propositional logic + priorities: discrimin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 - 6- -9 (A,C) 123- (B,C) 123- - -- 89 (A,D) -234 5 6- -- (C,D) 12-4 5 -- -9

3. Propositional logic + priorities: discrimin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 incomparable - 6- -9 (A,C) 123- (B,C) 123- - -- 89 (A,D) -234 5 6- -- (C,D) 12-4 5 -- -9

3. Propositional logic + priorities: discrimin ordering K = [  2 : A, B, C, D, E] ; B1 = {B  C, A  C, A  B, D  E} 1 2 3 4 B2 = {D} B3 = {A,E} B4 = {B,C} 5 6 7 8 9 B1 B2 B3 B4 leximin discrimin best-out no - 6- -9 (A,C) 123- yes yes (B,C) no no yes 123- - -- 89 (A,D) -234 5 6- -- yes yes yes yes no yes (C,D) 12-4 5 -- -9

Some logical languages for preference representation 4. Propositional logic + distances • K • B = 1, …, p  • d: S  S   • d (, ’) = d (’, ) • d (, ’) = 0 iff  = ’ • (example: d = Hamming distance) • d (, i ) = min {(, ’) | ’ satisfies K  i } • d (, B) = F (d (, 1), d (, 2), …, d (, n) ) •   ’ iff d (, B)  d (’, B) distance-based merging

Some logical languages for preference representation 5. « ceteris paribus » preferences [von Wright 63; Hansson 66; Doyle & Wellman 91] g : j > y For any two states w, w’ such that - w satisfies g j y - w’satisfies g j  y - w and w’coincide on « irrelevant » variables then  >B ’ (+ transitive closure)

Some logical languages for preference representation 5. « ceteris paribus » preferences g : j > y For any two states w, w’ such that - w satisfies g j y - w’satisfies g j  y - w and w’coincide on « irrelevant » variables then  >B ’ (+ transitive closure) e.g. variables outside Var (g)  Var (j)  Var (y) (more sophisticated definitions are possible)

5. « ceteris paribus » preferences K = { (coffee  tea) } B = {coffee: sugar >  sugar ; tea:  sugar > sugar ; T : coffee > tea >  coffee   tea ; T : croissant >  croissant } (coffee, sugar, croissant) (coffee, sugar, croissant) (coffee, sugar, croissant) (coffee, sugar, croissant) (tea, sugar, croissant) (tea, sugar, croissant) (tea, sugar, croissant) (tea, sugar, croissant) (coffee, tea, , croissant) (coffee, tea, , croissant)

Logical representation of preference & nonmonotonic reasoning Jérôme Lang