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

2. Prototypical Reasoning Most instances of a concept have a property bird(X) : flies(X) flies(X) anniversary(X)  friend(X) : give_gift(X) give_gift(X)

3. No-risk Reasoning A conclusion is obtained because the other alternative is a disaster accused(X) : innocent(X) innocent(X)

4. Buest-guess reasoning A convenient conjecture is made enrolled(X) /\ lecture(X,D): takesplace(X,D) takesplace(X,D)

5. More examples criminal(X)  foreigner(X) : expell(X) expell(X) friend(X,Y)  friend(Y,Z) : friend (X,Z) friend(X,Z)

6. Domain Knowledge Certain knowledge should also be expressed politicalRefugee(X) →  expell(X)

7. Hierarchies with exceptions Typically, mammals do not lay eggs Platypus are mammals Platpus lay eggs

8. Default Logic Syntax 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

9. Open defaults An open default is a default schema with free variables. A default schema defines a set of defaults:  :  1 , …,  n   For every grounding substitution 

10. What is the default theory? Given the default schema bird(X) : flies(X) flies(X) And the facts bird(tweety) and bird(sam). What is the default theory?

11. WRONG INTERPRETATIONS  bird(X) :  flies(X)  flies(X)  bird(X) :  flies(X)  flies(X)

12. 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

13. Motivating Example W={green,aaaMember} D = { green :  likesCars /  likesCars, aaaMember : likesCars/ likesCars } What can be concluded?

14. 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)

15. 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}

16. 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 

17. Operational semantics (cont’d)  is applicable in  iff pre(  )  In(  ) and   In(  )  is a process iff   k  ,  k is applicable in  [k] 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

18. Some properties A default theory may have none, one, or several extensions (W,D) has an inconsistent extension iff W is inconsistent –If an inconsistent extension exists, it is unique If W  Just  Conc is consistent, 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 Default logic is non-monotonic: expansion of W or D change extensions in unpredictable way

19. 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).

20. Normal theories Every rule has its justification identical to its conclusion:  :   Each process of a normal default theory is successful, thus normal theories always have extensions If D grows, then the extensions grow (semi-monotonicity) The union of two extensions of a normal default theory is inconsistent. Expanding the set of defaults of normal default theory does not decrease the number of extensions.

21. Limitations of Normal Default Theories 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!)

22. Priorities might be needed All birds are animals Normally, animals do not fly. Normally, birds fly. Alternatively, can be solved by semi-normal defaults!

23. Semi-normal default theories All defaults have the form:  :  /\   Unfortunately, we cannot guarantee anymore the existence of extensions, except if we impose additional constraints.

24. Problems of Default Logic 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

25. Justified Default Logic Tries to address the problem of non-existence of extensions, by resorting to the notion of maximal process. Let  and  be processes of a default theory. We say that  iff the set of defaults occurring in  is a proper subset of the ones in .  is a maximal process of T iff is successful and there is no successful process such that  <  A set of formula E is called a modified extension of T iff there is a maximal process of T such that = E = In(  )

26. Important results Every closed and successful process of a default theory is a maximal process. Therefore, every extension of T is a modified extension of T. Every default theory has at least one modified extension Let T be a normal default theory. Then E is an extension of T iff E is a modified extension of T.

27. Operational interpretation of JDL The difference between maximal and closed processes is subtle: –Closed processes are closed with respect to all applicable defaults. –Maximal processes are closed only under the application of defaults which preserve success. Maximal processes can be identified in process trees as succesful nodes that are –Closed and successful, or –That all immediate children are failed.

28. Example Find the extensions and modified extensions of default theory T=(W,D) with W={holidays,sunday} and D containing defaults sunday : goFishing /\  wakeUpLate goFishing holidays: wakeUpLate wakeUpLate

29. Constrained Default Logic What are the extensions of default theory T=({},{:p/q,:  p/r}) ? Constrained Default Logic ensures joint consistency of all justifications Requires extra definitions to avoid these cases.

30. CDL basic definitions Given a default theory T=(D,W) and  a sequence of defaults in D then let Con(  ) = Th(W  {  | is the consequent or a justification of a default occurring in  }) A default  :  1 /\ … /\  n /  is applicable to a pair of deductively closed sets of formulae (E,C) iff  E and  /\  1 /\ … /\  n is consistent with C. A pair of deductively closed sets of formulae (E,C) is closed under D iff for every default  :  1 /\ … /\  n /  in D that is applicable to (E,C) it is the case that   E and {  1, …,  n }  C.

31. Constrained Extensions  is a constrained process iff   k  ,  k is applicable in (In(  [k]),Con(  [k])) A closed constrained process  is a constrained process such that every default which is applicable to (In(  ),Con(  )) already occurs in . Definition: A pair of sets of formulae (E,C) is a constrained extension of T iff there is a closed constrained processs  of T such that (E,C)= (In(  ),Con(  ))

32. Advantages of CDL Default theories always possess constrained extensions Constrained Default Logic is semi- monotonic Constrained Default Logic satisfies a kind of orthogonality property.

33. Example Let W = {} and D = { true :p/q, true:  p/r, true:  q,  r /s } T has a single extension Th({q,r}) T has two modified extensions: –Th({q,r}) and –Th({s}) T has three constrained extensions –(Th({q}),Th({q,p})) –(Th({r}),Th({r,  p}) –(Th({s}),Th({s,  q,  r})

34. Relationships Let T be a default theory and E=In(  ) an extension of T. If E   Out(  ) is consistent then (E,Th(E   Out(  ) )) is a constrained extension of T. Let T be a default theory and E=In(  ) a modified extension of T. If E   Out(  ) is consistent then (E,Th(E   Out(  ) )) is a constrained extension of T. Let T be a default theory and (E,C) a constrained extension of T. Then there is a modified extension F of T such that E  F.

35. Computability and Complexity Default logic is undecidable (even for normal theories) Propositional default logic is decidable, however harder than propositional logic. The following problems are Σ 2 2 -complete (NP NP -complete): –Deciding whether T has an extension. –Deciding whether φ is contained in an extension of T. The following problem is Π 2 2 -complete (coNP NP -complete): –Deciding whether φ is contained in all extensions of T.