In many ways Rescher’s method of reasoning from maximal consistent subsets has anticipated many recent developments in AI. One in particular is the nonmonotonic formalism developed by Reiter in [151]. In this section, we’ll recap the connection be- tween Rescher’s method and Reiter’s original formalism for default reasoning. Since the publication of [151], Reiter’s formalism has been revised and extended (see Schaub [163] for a summary). Many of these new developments have been shown to be ex- pressively equivalent to various forms of belief revision formalism. We’ll not be able to summarise all these new developments here. But since many of these extensions are theoretically grounded in some form of reasoning from maximal consistent sub- sets, Belnap’s methodological criticism is still in force here. In any case, we’ll focus on the standard default formalism of Reiter. We’ll begin by recalling some standard definitions.
called facts andDis a set of default rules of the form: A:B1, . . . , Bn
C
Intuitively, the meaning of the rule is that if Ais provable and eachBiis consis- tent, then we may concludeC.Ais called aprerequisiteof the default rule, eachBiis a
justificationof the default rule andCis aconsequentof the default rule. A default rule without justification is equivalent to an inference rule in standard logics. Hence the requirement of justification is responsible for the nonmonotonicity of default reason- ing.
Example 3.2.1
F ={head light on} F0 ={head light on, brake light fails}
D=
head light on: component 1 ok, ..., component n ok electrical system ok
From the observed fact that the head light is working properly andin the absence of evidence to the contraryit is reasonable to conclude that the electrical system is working properly. But of course this conclusion can be defeated when we observe the failure of some other component of the system even though the head light may still be in operation. So in this sense a default rule is defeasible when new observation violates some of the justifications of the rule.
The key concept in default reasoning is the notion of an extension. An extension of a default theory is a deductive closure (under classical logic) of the facts together with the consequents of the applicable default rules. Intuitively we can think of an extension as a possible scenario according to the information provided by the facts and the applicable default rules. More formally an extension for a default theory is defined as follows:
Definition 3.2.1
(Reiter [151]) Let W = hD,F i be a default theory. For any deductively closed set S ⊆Φ, letγ(S)be the least set satisfying the following conditions:
1. F ⊆γ(S)
2. Cn(γ(S)) =γ(S)(whereCnis closure under classical deduction)
3. If A:B1,...,Bn
A deductively closed setE ⊆Φis an extension forW (E ∈ext(W)) iffγ(E) =E, i.e.E is a fixed point of the operatorγ.
Reiter’s definition of an extension is not recursive since it appeals to a fixed point construction ofγ. In example (3.2.1), the extensions ofW = hD,F iandW0 =hD,F0i
are respectively:
ext(W) =
Cn({head light on, electrical system ok})
ext(W0) =
Cn({head light on, brake light fails})
However, the existence of an extension is not guaranteed in general for every de- fault theory. Multiple extensions for a given default theory are also possible. There are three basic decision problems associated with default reasoning:
Extension Existence: Given a default theoryW, isext(W)non-empty?
Intersection Membership: Given a default theoryWand a formulaA, isAa member ofT
ext(W)?
Union Membership: Given a default theoryW and a formulaA, is Aa member of
S
ext(W)?
The first order version of default theory is clearly not semi-decidable with respect to these decision problems. Even in the propositional case, the complexities of these problems are generally very high – ΣP2 and ΠP2 hard. Thus it is often desirable to identify subclasses of default theories with either guaranteed existence of extension(s) or with low computational complexity. A normal default theory for instance is one whose default rules are of the form:
A:B B
For the class N of normal propositional default theories the existence of an ex- tension is guaranteed but the complexity of determining whether a formula is in an extension isΣP2 complete. Analogously, the decision problem of determining whether a formula is in every extension isΠP2 complete (see chapter 4 of [43] for a summary of complexity results for default reasoning).
theories. A prerequisite free default theory is one whose default rules are of the form:
:B1, . . . , Bn C
Two default theoriesW andW0 are said to be extension equivalent if they have the same set of extensions, i.e. ext(W) = ext(W0). The class of prerequisite free default theories isrepresentationally completewith respect to the class of all default theories in the sense that every default theory is extension equivalent to some prerequisite free default theory. A default theoryW is said to be inconsistent if its only extension isΦ. In [151] Reiter shows that:
Theorem 3.2.1
(Reiter [151])
1. LetΓ be a set of formulae and W = hD,F ibe a default theory. DefineΓ0 = F and for eachi≥0,
Γi+1 =Cn(Γi)∪
C: A:B1, . . . , Bn
C ∈ D, A∈Γi, ¬B1, . . . ,¬Bn6∈Γ
ThenΓ is an extension forWiff
Γ =
∞
[
i=0 Γi
2. IfEandE0are extensions of a default theory andE0 ⊆ E, thenE0 =E
3. A default theoryW = hD,F iis inconsistent iff F is classically inconsistent iff
ext(W) ={Φ}.
4. Extensions exist for every normal default theory.
5. Extensions of a normal default theory are orthogonal in the sense that they are pairwise inconsistent.
We note that in (1) of theorem (3.2.1), the definition of Γi+1 makes essential ref- erence toΓ and hence it is not a recursive definition. Using Reiter’s results, we can show that there is a close relationship between consistent prerequisite free normal de- fault theories and reasoning from maximal consistent subsets. Note that by (3) of theorem (3.2.1), inconsistent default theories are extension equivalent trivially. By (4) of theorem (3.2.1), extension is guaranteed to exist for any default theory in the class
Theorem 3.2.2
For everyW =hD,F i ∈ P,ext(W)is of the form
{Cn(Σ∪ A) : A ∈MΣ(Γ)}
for someΓ and consistentΣ.
Proof:
LetW = hD,F i ∈ P be arbitrary but fixed. We setΣ = F. First we observe that by part (3) of theorem (3.2.1),Σis consistent and each extension ofW is also consistent. Define Γ = A: :A A ∈ D
LetE ∈ext(W)be an arbitrary but fixed extension, consider∆=Cn((E ∩Γ)∪Σ). We claim that
Claim:∆=E:
Proof of Claim: (∆⊆ E): SinceE ∈ext(W),γ(E) =Eand thus by properties (1) and (2) of definition (3.2.1) we have:
E ∩Γ ⊆ E =⇒ (E ∩Γ)∪Σ⊆ E
=⇒ Cn((E ∩Γ)∪Σ)⊆Cn(E)
=⇒ Cn((E ∩Γ)∪Σ)⊆ E
(∆ ⊇ E): Now we use part (1) of theorem (3.2.1). SinceE ∈ ext(W), we have E =
S∞
i=0Ei. We prove inductively thatEi⊆∆. For the basis it is trivial sinceE0 = Σ⊆∆. Now we assume the induction hypothesis that Ei ⊆ ∆and prove thatEi+1 ⊆ ∆. If A∈ Ei+1, thenA∈ EiorA∈{C: :CC ∈ D, ¬C6∈ E}. In the former case, we haveA∈∆ by the induction hypothesis. In the later case, we haveA∈ Γ andA∈ γ(E)by (3) of definition (3.2.1) henceA∈ E asγ(E) =E. HenceA∈ E ∩Γ and soA∈∆as required. This completes the proof thatE =S∞
i=0Ei⊆∆.
To continue with the main proof we need to show thatE ∩Γ ∈ MΣ(Γ). Clearly,
E ∩Γ isΣ-consistent sinceE isΣ-consistent. To show maximal consistency, consider anyA ∈ Γ such that A 6∈ E ∩Γ. Sinceγ(E) = E by part (3) of definition (3.2.1), we haveA∈ Eor¬A∈ E. SinceA6∈ E ∩Γ, we haveA6∈ E. Thus,¬A∈ E. But∆= Eso
E ∩Γ ∈MΣ(Γ). SinceEis arbitrary, we conclude that every extension ofWmust be of the formCn(Σ∪ A)whereA ∈MΣ(Γ). Hence,ext(W)⊆{Cn(Σ∪ A) : A ∈MΣ(Γ)}.
Now to show that for everyA ∈MΣ(Γ),Cn(Σ∪ A)∈ext(W), we make use of part (1) of theorem (3.2.1). We consider an arbitraryA ∈ MΣ(Γ) and defineS0 = Σ = F
and for eachi≥0
Si+1 =Cn(Si)∪
A: :A
A ∈ D,¬A6∈Cn(Σ∪ A)
We note that fori≥2,Si=Si+1. Hence we only need to verify thatS2 =Cn(Σ∪ A), i.e. we need to verify that
Cn(Σ∪ A) = Cn Cn(Σ)∪ A: :A A ∈ D,¬A6∈Cn(Σ∪ A) (3.1)
We claim that in equation (3.1),
A: :A
A ∈ D,¬A6∈Cn(Σ∪ A)
=A
To verify our claim consider an arbitraryB∈ A. SinceA ⊆Γ, we haveB∈Γ and thus :B
B ∈ D. By the maximalΣ-consistency ofA, we have¬B 6∈ Cn(Σ∪ A). Conversely consider an arbitrary B ∈ {A : :AA ∈ D,¬A 6∈ Cn(Σ∪ A)}. It follows that B ∈ Γ and¬B 6∈ Cn(Σ∪ A). Suppose to the contrary thatB 6∈ A, then by the maximalΣ- consistency ofAwe have¬B∈ Cn(Σ∪ A)contradicting our previous claim. Hence B∈ Aas required.
Thus equation (3.1) can be rewritten as:
Cn(Σ∪ A) = Cn Cn(Σ)∪ A
(3.2)
To verify (3.2), we note that by reflexivityΣ∪ A ⊆Cn(Σ)∪ Aand thus by the mono- tonicity we getCn(Σ∪A)⊆Cn(Cn(Σ∪A)). Conversely by reflexivity and monotonic- ity,Cn(Σ)∪ A ⊆Cn(Σ∪ A)and thusCn Cn(Σ)∪ A)
⊆Cn Cn(Σ∪ A
=Cn(Σ∪ A)
by idempotence ofCn.
We note that for any given setΓ and any given consistent setΣ, the family of clo- sures ofΣ-maximal consistent subsets ofΓis precisely the family of extensions of some consistent prerequisite free normal default theory. The corresponding default theory is defined in the obvious way. It is easy to see that the following is an immediate
corollary of theorem (3.2.2). Corollary 3.2.1
LetW andΓ be defined as in theorem (3.2.2). ThenC∗UΣ(Γ) =T
ext(W)andC∗EΣ(Γ) =
S
ext(W).
More interestingly, Marek, Treur, and Truszczy ´nski [126] show that there is a close relationship between consistent normal default theories and reasoning from maximal consistent subsets. They also show that the class of consistent prerequisite free normal default theoriesPis representationally complete with respect to the class of consistent normal default theoriesN. Note that P is a proper subclass ofN. Thus the results of Marek, Treur, and Truszczy ´nski can be seen as a strengthening of theorem (3.2.2). The results of Marek, Treur, and Truszczy ´nski can be restated in terms ofrepresentation theory for default logic. A family of theories G is said to be trivial if Φ ∈ G, else
G is non-trivial. Furthermore G is said to be representable by a default theory W if ext(W) =G.
Theorem 3.2.3
(Marek, Treur, and Truszczy ´nski [126])
1. If a family of non-trivial theoriesGis representable by a default theoryW ∈ N, thenGis representable by a default theoryW0 ∈ P, i.e. P is representationally complete with respect toN.
2. A family of non-trivial theoriesGis representable by a default theoryW ∈ N iff there is a set of formulaeΓ such that
G={Cn(A) :A ∈M∅(Γ)}
Theorem (3.2.3) shows that each consistent normal default theory W is expres- sively equivalent to a family of maximal consistent subsets of some set Γ, i.e. the family of closures of these maximal consistent subsets ofΓ is precisely the family of extensions ofW. But note thatΓ need not be unique in general, i.e. it is possible that for someΓ0 6= Γ,{Cn(A) : A ∈ M∅(Γ)} = {Cn(B) : B ∈ M∅(Γ0)}. The following are immediate corollaries of theorem (3.2.3).
Corollary 3.2.2
1. For any Γ ⊂ Φwith M∅(Γ) 6= ∅, CE∅(Γ) = C∗E∅(Γ) = Sext(W) andCU∅(Γ) = C∗U∅(Γ) =T
2. A family of non-trivial theoriesGis representable by a default theoryW ∈ P iff there is a set of formulaeΓsuch that
G={Cn(A) :A ∈M∅(Γ)}
Proposition 3.2.1
For everyW ∈ N,ext(W)can be expressed in two equivalent forms:
{Cn(A) : A ∈M∅(Γ0)} = {Cn(Σ∪ B) : B ∈MΣ(Γ)} (3.3)
for some consistentΣand someΓ andΓ0.
Proof:
Consider an arbitraryW ∈ N. By (2) of theorem (3.2.3), there exitsΓ0such thatext(W)
can be expressed as the LHS of equation (3.3). By (1) of theorem (3.2.3),P is represen- tationally complete with respect to N. Hence, there exists a W0 ∈ P where W0 is extensionally equivalent toW. By theorem (3.2.2), there exitsΓ such thatW0 can be expressed as the RHS of equation (3.3).
We can give a representational completeness result similar to (1) of theorem (3.2.3). SetS0 = {(Γ,∅) : Γ ⊆ Φ}andS = {(Γ, Σ) : Γ, Σ⊆ Φ, Σ 6` ⊥}. We say that a family of theoriesGis representable by a(Γ, Σ)∈SiffG={Cn(Σ∪ A) : A ∈MΣ(Γ)}.
Theorem 3.2.4
If a family of theoriesGis representable by some(Γ, Σ)∈S, thenGis representable by some(Γ0,∅)∈S0
Proof:
LetGbe a family of theories representable by some(Γ, Σ)∈S. ThenGis representable by someW0 ∈ P. Since P is a subclass ofN, we can apply proposition (3.2.1) toW0
and thereby yielding the existence of aΓ0such thatext(W0) ={Cn(A) :A ∈ M ∅(Γ0)}. This suffices to show thatGis representable by some(Γ0,∅)∈S0.
We can summarise our results with figure (3.1). P,N,S0 andSare all expressively equivalent in the sense that a family of theories G is representable in one of these classes iffGis representable in all the other classes.
Of course these results show that the formalism of default reasoning is equipped with capabilities for handling inconsistencies. Taking union (intersection) over exten- sions of a consistent normal default theory corresponds to existential (universal) con-
ext(W0) =ext(W) r ' & $ % r r @ @ @ @ @ R S0 S (Γ0,∅) (Γ, Σ) ' & $ % r @ @ @ @ @ I r P N W0 W
Figure 3.1: Expressive equivalence ofP,N,S0andS
sequence in Rescher’s sense. But there are also limitations to default reasoning – in- consistencies in the set of facts in a default theory still trivialises the extension. Clearly there is room for improvement here. One possible solution suggested by Hunter in [91] is to replace the underlying classical logic with a weaker paraconsistent logic as the underlying deduction mechanism. Indeed, we can keep most of definition (3.2.1) intact. The only modification we need to make is the replacement of the closure con- dition in clause 2 of definition (3.2.1) with the deductive closure of a weaker paracon- sistent logic. As to which paraconsistent logic should be used, we need not decide
a priorihere. In fact it would seem to be more useful to study and compare the be- haviour of the resulting mechanisms obtained by plugging in various paraconsistent logics. We’ll leave this analysis and study for future work.
From a preservational point of view, the upshot of theorem (3.2.2) and theo- rem (3.2.3) is that the kind of analyses and accounts presented in the last chapter has direct counterparts for default theories inP andN.