Default Logic

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
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)

3
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
• j Y1, ,Yn
• g
• j (pre-requisites), Yi (justifications) and g
  (conclusion) are 1st order formulas

4
The issue of semantics

• If j is true (where?) and all Yi are consistent
  (with what?) then g becomes true (becomes? Wasnt
  it before?)
• Conclusions must
• be a closed set
• contain W
• apply the rules of D maximally, without becoming
  unsupported

5
Default extensions

• G(S) is the smallest set such that
• W ? G(S)
• Th(G(S)) G(S)
• ABi/C ? D, A ? G(S) and ?Bi ? S ? C ? G(S)
• E is an extension of (W,D) iff E G(E)

6
Quasi-inductive definition

• E is an extension iff E ?i Ei where
• E0 W
• Ei1 Th(Ei) U C ABj/C ? D, A ? Ei, ?Bj ? E

7
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

8
Operational semantics

• The computation of an extension can be reduced to
  finding a rule application order (without
  repetitions).
• P (d1,d2,...) and Pk is the initial segment
  of P with k elements
• In(P) Th(W ? conc(d) d ? P)
• The conclusions after rules in P are applied
• Out(P) ?Y Y ? just(d) and d ? P
• The formulas which may not become true, after
  application of rules in P

9
Operational semantics (contd)

• d is applicable in P iff pre(d) ? In(P) and ?Y ?
  In(P)
• P is a process iff ? dk ? P, dk is applicable in
  Pk-1
• A process P is
• successful iff In(P) n Out(P) .
• Otherwise it is failed.
• closed iff ? d ? D applicable in P ? d ? P
• Theorem E is an extension iff there exists P,
  successful and closed, such that In(P) E

10
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,CInCur,BOut
  Cur,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).

11
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!)

12
Problems

• No guarantee of extension existence
• Deficiencies in reasoning by cases
• D italianwine/wine frenchwine/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