Análisis de argum entos 2.1 Parafraseo y diagramas
C. Argumentos entrelazados
In the previous section the operatorDT provided an ae-theoryT with a seman-
tics corresponding to expansions and the newly defined partial expansions. In the present section we will derive fromDT an additional operatorDstT on the set
Wof possible world structures and an operatorDst
T on the setBof belief pairs.
The fixpoint of these operators are called theextensions andpartial extensions
ofT, respectively. This terminology is of course not arbitrarily chosen; the ex- tensions of a default theory ∆ correspond precisely to the fixpoints ofDst
kon(∆).
Thus, the operator DT establishes a uniform semantics for both default and
autoepistemic logic. Thepr-least fixed point of DTst provides for the so-called
well-founded semantics of autoepistemic logic and is related to the extensions ofT in a similar way as the Kripke-Kleene fixpoint ofT is related to the expan- sions ofT.
Consider the functionD1
T(P, S) ={I| H
2
(S,P),I(T) =true}which we defined
in the previous section and constitutes the first entry ofDT(P, S). We can view
this function as an operator on possible world structures in the first variable as follows:
D1T(·, S) :P 7→ D
1
T(P, S).
Of course we will simply writeD1
T(P, S) forDT1(·, S)(P). Similarly defined, we
will consider the function D2
T(P, S) = {I | H2(P,S),I(T) = true} (the second
entry of DT(P, S)) as an operator on possible world structures in the second
variable:
D2
T(P,·) :S7→ D
2
T(P, S).
It is easy to see from the definitions that
D1 T(P, S) =D 2 T(S, P) and so D 1 T(·, S) =D 2 T(S,·). (5)
Lemma 3.16. Let T be an ae-theory and S a possible world structure. The operatorsD1
T(·, S)andD
2
T(S,·)are v-monotone.
Proof: Let P v P0. Then since also S v S it is obvious that (P, S) pr
(P0, S). By thepr-monotonicity ofDT it follows thatDT(P, S)prDT(P0, S)
and henceD1
T(P, S)v D
1
T(P0, S), which provesv-monotonicity ofD
1
T(·, S). By
(5),D2
T(S,·) is alsov-monotone.
Now that we have established monotonicity, we will look at the least fixed points provided by the Knaster-Tarski theorem. Let
DTst(S) =lfp(D1
T(·, S)) =lfp(D
2
T(S,·)).
Similar to DT this is a revision operator for possible world structures. The fixpoints ofDst
T are called extensions ofT.
Example 3.17. Recall example 3.5 and its continuation 3.14, in which we con- sidered the ae-languageLp with the only atompand let T ={¬Lp}, i.e., the
only thing the agent knows is that he doesn’t knowp. The relevant interpreta- tions are of courseIp:p7→trueandI¬p:p7→f alse. The four possible world
structures are thenAp ={Ip, I¬p},Ip = {Ip},I¬p ={I¬p} and ∅. We estab-
lished that to come to a stable set with the only knowledge that you don’t know
p, you can either decide to believe in nothing (Ap) or decide to believe in every-
thing (∅). In default logic, however, we cannot just decide to believe everything. This is illustrated in the extensions ofT. By definitionDst
T(Q) =lfp(D1T(·, Q)).
Now, from the table in example 3.14, we can read off entries ofD1
T(·, Q) very
easily. The results are that for anyQ⊆ Ap the fixpoints ofD1T(·, Q) are∅and
Ap, of which thev-least isAp. Thus the table ofDstT is
Q Ap I¬p Ip ∅
Dst
T(Q) Ap Ap Ap Ap
We see thatT has only one extension, which is believing in nothing, since that is the only justified belief we can deduce from not knowingp.
Consider the following operator onB:
DstT(P, S) = (D st
T(S), D
st
T(P)). (6)
Similar toDT, DstT is a revision operator for belief pairs. The fixpoints ofD st T
are calledpartial extensions. From (6) it follows immediately that
Dst T(P, P) = (D st T(P), D st T(P)). (7) Consequently, (P, P) is a fixpoint ofDst
Fig. 1. Operators associated with autoepistemic logic ([8]).
Theorem 3.18. LetT be an ae-theory. The operatorDst
T is anti-monotone and
the operator Dst
T is pr-monotone and symmetric. Also, for every consistent
belief pair(P, S),Dst
T(P, S)is also consistent.
Proof: LetP vS and letP0 =DTst(P) =lfp(D1
T(·, P)) andS0 =D st
T(S) =
lfp(D1
T(·, P)). Clearly (becauseP0 is a fixpoint ofD
1
T(·, P)) P0=D1
T(P0, P). (8)
Also, bypr-monotonicity of DT and since (P0, S)pr (P0, P) it follows that
DT(P0, S)prDT(P0, P) and hence D1 T(P0, S)v D 1 T(P0, P). (9) By (8) and (9) D1 T(P0, S) v P0, i.e. P0 is a prefixpoint of D 1 T(·, S). By lemma 3.16, D1
T(·, S) is v-monotone. Then by the Knaster-Tarski theorem, lfp(mathcalDT1(·, S))≤xfor all v-prefixpoints ofmathcalD1T(·, S). It follows thatS0vP0, which concludes the anti-monotonicity ofDTst.
Let (P, S)pr(P0, S0). It follows directly from the anti-monotonicity ofDstT
thatDst T(S, P)prDstT(S0, P0), i.e. D st T(P, S)prDTst(P0, S0), hence concluding monotonicity ofDst T. The symmetry ofD st
T is immediate from the definition.
The last statement is immediate from proposition 3.11.
Since we now established monotonicity ofDst, we will again investigate the
least fixpoint, which is called thewell-founded fixpoint ofT and is denoted by
W F(T). Denecker et al. chose the name because the semantics specified by the well-founded fixpoint is closely related to the well-founded semantics for default logic and logic programming.
The following corollary states for the well-founded fixpoint similar state- ments to what corollary 3.13 stated for the Kripke-Kleene fixpoint; W F(T)
approximates all (if any) partial extensions of T, is consistent, and provides a sufficient argument for the uniqueness of a partial extension.
Corollary 3.19. Let T be an ae-theory. Then
(a) the fixpointW F(T)is consistent, i.e. W F(T)1vW F(T)2,
(b) for every partial extension (P, S)of T,W F(T)pr(P, S),
(c) ifW F(T)is complete and thusW F(T) = (P, P)for someP, then it is the unique partial extension ofT andP is the unique extension of T. Proof: SinceW F(T) can be reached by iteratingDst
T over (A,∅) (the least,
and consistent, element ofB) (a) follows directly from theorem 3.18. (b) is a direct result from the fact thatW F(T) is thepr-least fixpoint ofDTst and (c)
is a direct result from (a) and (b).
Example 3.17 (continued) Because of the work in examples 3.5, 3.14 and 3.17 it is now very easy to find the partial extensions ofT ={¬Lp}. Since the only extension ofT isAp, we know from the definition thatDTst(P, S) = (Ap,Ap)
for allP and S. Thus the only partial extension is (Ap,Ap). This in partic-
ular complies with the corollary above and the theorem below on the relation
between the four different operators.
The following theorem explains the relation between the partial extensions and the partial expansions of a theory T. The order v on belief pairs in the theorem is the component-wise extension of the knowledge ordering on possible world structures which we briefly mentioned earlier 3.2.2: (P, S) v(P0, S0) if and only ofPvP0 andSvS0.
Theorem 3.20. Let T be an ae-theory. Then
(a) partial extensions ofT are also partial expansions of T, (b) KK(T)prW F(T),
(c) every partial extension of T is av-minimal partial expansion of T. Proof: For (a), let (P, S) be a partial extension, i.e. a fixpoint ofDst
T. Then P =Dst T(S) = lfp(D 1 T(·, S)) and consequently, D 1 T(P, S) = P. Similarly, S = lfp(D1
T(·, P)) and consequently, S = DT1(S, P) = D2T(P, S), hence concluding
thatDT(P, S) = (P, S), proving (a). SinceKK(T) is the least fixpoint of DT,
(b) follows directly from (a).
For (c), assume that (P, S) is a partial extension of T. Then (P, S) is a fixpoint of Dst
T and hence S = lfp(D1(·, P)). Also assume that (P0, S0) is a
partial expansion ofT such that (P0, S0)v(P, S). Then we have to show that (P0, S0) = (P, S). Since P0 vP, it follows that (S0, P)pr (S0, P0). By pr-
monotonicity ofDT we get thatD1T(S0, P)v D
1
av-prefixpoint ofD1(·, P). Since by lemma 3.16D1
T(·, P) isv-monotone, it fol-
lows by the Knaster-Tarski theorem thatlfp(D1(·, P))vxfor allv-prefixpoints. It follows thatS =lfp(D1(·, P))vS0 and thus that S =S0. By similar argu-
mentation we get thatP vP0 and thus that P=P0, proving (c).
Direct results of theorem 3.20 are that extensions ofTare also expansions of
T and that every extension of T is av-minimal expansion ofT. This of course complies with the idea that extensions are a stronger concept than expansions.