Default Logic Proposed by Ray Reiter (1980) go_Work → use_car Does not admit exceptions! Default rules go_Work : use_car use_car

More examples anniversary(X)  friend(X) : give_gift(X) give_gift(X) friend(X,Y)  friend(Y,Z) : friend (X,Z) friend(X,Z) accused(X) : innocent(X) innocent(X)

Default Logic Syntaxe A theory is a pair (W,D), where: –W is a set of 1st order formulas –D is a set of default rules of the form:  :  1, …,  n  –  (pre-requisites),  i (justifications) and  (conclusion) are 1st order formulas

The issue of semantics If  is true (where?) and all  i are consistent (with what?) then  becomes true (becomes? Wasn’t it before?) Conclusions must: –be a closed set –contain W –apply the rules of D maximally, without becoming unsupported

Default extensions  (S) is the smallest set such that: –W   (S) –Th(  (S)) =  (S) –A:Bi/C  D, A   (S) and  Bi  S → C   (S) E is an extension of (W,D) iff E =  (E)

Quasi-inductive definition E is an extension iff E =  i E i where: –E 0 = W –E i+1 = Th(E i ) U {C: A:B j /C  D, A  E i,  B j  E}

Some properties (W,D) has an inconsistent extension iff W is inconsistent –If an inconsistent extension exists, it is unique If W  Just  Conc is inconsistent, then there is only a single extension If E is an extension of (W,D), then it is also an extension of (W  E’,D) for any E’  E

Operational semantics The computation of an extension can be reduced to finding a rule application order (without repetitions).  = (  1,  2,...) and  [k] is the initial segment of  with k elements In(  ) = Th(W  {conc(  ) |    }) –The conclusions after rules in  are applied Out(  ) = {  |   just(  ) and    } –The formulas which may not become true, after application of rules in 

Operational semantics (cont’d)  is applicable in  iff pre(  )  In(  ) and   In(  )  is a process iff   k  ,  k is applicable in  [k-1] A process  is: –successful iff In(  ) ∩ Out(  ) = {}. Otherwise it is failed. –closed iff    D applicable in  →    Theorem: E is an extension iff there exists , successful and closed, such that In(  ) = E

Computing extensions (Antoniou page 39) extension(W,D,E) :- process(D,[],W,[],_,E,_). process(D,Pcur,InCur,OutCur,P,In,Out) :- getNewDefault(default(A,B,C),D,Pcur), prove(InCur,[A]), not prove(InCur,[~B]), process(D,[default(A,B,C)|Pcur],[C|InCur],[~B|OutCur],P,In,Out). process(D,P,In,Out,P,In,Out) :- closed(D,P,In), successful(In,Out). closed(D,P,In) :- not (getNewDefault(default(A,B,C),D,P), prove(In,[A]), not prove(In,[~B]) ). successful(In,Out) :- not ( member(B,Out), member(B,In) ). getNewDefault(Def,D,P) :- member(Def,D), not member(Def,P).

Normal theories Every rule has its justification identical to its conclusion Normal theories always have extensions If D grows, then the extensions grow (semi- monotonicity) They are not good for everything: –John is a recent graduate –Normally recent graduates are adult –Normally adults, not recently graduated, have a job (this cannot be coded with a normal rule!)

Problems No guarantee of extension existence Deficiencies in reasoning by cases –D = {italian:wine/wine french:wine/wine} –W ={italian v french} No guarantee of consistency among justifications. –D = {:usable(X),  broken(X)/usable(X)} –W ={broken(right) v broken(left)} Non cummulativity –D = {:p/p, pvq:  p/  p} –derives p v q, but after adding p v q no longer does so