Skeptical Query-Answering in Constrained Default Logic Torsten Schaub Theoretische Informatik, TH Darmstadt Alexanderstrae 10 D-64283 Darmstadt
[email protected]
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Michael Thielscher FG Intellektik, TH Darmstadt Alexanderstrae 10 D-64283 Darmstadt
[email protected]
Introduction
Nonmonotonic logics generally aim at extending an underlying classical logical system in order to provide conclusions that go beyond this system. In this way, approaches like Autoepistemic [4] or Default Logic [6] induce one or several so-called extensions of a given world-description, each of which represents a reasonable set of beliefs. This phenomenon of multiple extensions suggests two natural approaches to query-answering: A credulous one in which a query is said to be derivable if it belongs to a single extension, and a skeptical one in which one stipulates that a query lies in all extensions. In what follows, we develop a method for skeptical query-answering in Default Logics. This work builds on [10], where a general approach to skeptical reasoning in (semi-monotonic) Default Logics was proposed. There, we have described a high-level description of skeptical query-answering that abstracts from an underlying credulous reasoner. In this paper, we make the aforementioned meta-algorithm precise and employ it to extend an existing approach to credulous query-answering [9] based on the Connection Method [1]. At this point, the reader may wonder why we have chosen Constrained Default Logic rather than Reiter's original approach. In fact, Constrained Default Logic serves us as an exemplary Default Logic enjoying the property of semimonotonicity, which stipulates that the addition of default rules to a theory does not invalidate the application of previously applied default rules. As pointed out by [6], semi-monotonicity is indispensable for feasible query-answering in Default Logics. This is so because it allows for local proof procedures focusing on default rules relevant for answering a query; otherwise the whole set of default rules has to be taken into account. Now, semi-monotonicity is only enjoyed by so-called normal default theories in Reiter's Default Logic. However, since both aforementioned Default Logics coincide on normal default theories, our exposition applies to this part of Reiter's Default Logic, too. Of course, a similar question may arise concerning the choice of the Connection Method. Unlike resolution-based methods that decompose formulas in order to derive a contradiction, the Connection Method analyses the structure of formulas for proving their unsatis ability. In fact, we will see that skeptical query-answering requires numerous variants of similar subproofs. In such a case, it is advisable to reuse information gathered on similar structures. This approach is supported by the structure-sensitive nature of the Connection Method. As a result, we obtain a homogeneous characterization of skeptical default proofs at the level of the underlying deduction method. 2
Constrained Default Logic
We consider a very simple yet powerful extension of Constrained Default Logic [2], called Pre-Constrained Default Logic [8]. The idea is to supplement some initial consistency constraints that direct the subsequent reasoning process. This is a well-known technique, also used in Theorist [5], in which the \context of reasoning" is predetermined and subsequently dominated by some initial consistency requirements. These additional constraints play an important role in our approach to skeptical query-answering, as we will see below. A default theory (D; W ; C ) consists of a set of formulas W , a set of default rules D, and a set of formulas C representing some initial constraints. A default rule is any expression of the form : where ; ; are formulas. For convenience, we denote the prerequisite of a default rule by Prereq( ), its justi cation by Justif ( ), and its consequent by Conseq( ).1 A normal default theory is restricted to normal default rules whose justi cation is equivalent to the consequent. For simplicity, we deal with a propositional language over a nite alphabet and assume that W [ C is satis able. A constrained extension is de ned as follows. De nition 2.1Let (D; W ; C) be a default theory and let E and C be sets of formulas. De ne E0 = W and C0 = W [ C and for i 0 Ei+1 = Th(Ei) [ f j : 2 D; 2 Ei; C [ f g [ f g 6` ?g Ci+1 = Th(Ci) [ f ^ j : 2 D; 2 Ei; C [ f g [ f g 6` ?g S1 S (E; C ) is a constrained extension of (D; W ; C ) i (E; C ) = ( 1 i=0 Ei ; i=0 Ci): On 1
leave from IRISA, Campus de Beaulieu, 35042 Rennes Cedex, France These projections extend to sets and sequences of default rules in the obvious way.
1
Observe that the initial constraints C enter merely the nal constraints C (at C0) but not the extension E . Thus, the initial constraints C direct the reasoning process without actually becoming a part of it. In particular, they usually decrease the number of applicable default rules. Let us guide the formal development of our approach by means of the following example. Consider the default statements \quakers are doves if they are no anti-paci sts," \republicans are hawks if they are no paci sts," \doves" as well as \republicans are conservative," along with the strict knowledge telling us that we have a \republican quaker". This is formalized in the following default theory: n o Q : P ; R : :P ; D : C ; H : C ; fQ; Rg; ; (1) D H C C The rst and second default cannot be combined in a single constrained extension due to their mutually exclusive justi cations. Hence, this theory has two constrained extensions, one containing D; :H; and C and another one containing H; :D; and C . In the sequel, we follow [9] in dealing with default theories in atomic format in the following sense: For a default theory = (D; W ; C ) in some language L ; let L be the language obtained by adding, for each 2 D; three new propositions, named ; ; : Then, is mapped into a default theory (D0 ; W 0 ; C 0) in L where n o : 2 D D0 =
W 0 = W [ fPrereq( ) ! ; ! Justif ( ); ! Conseq( ) j 2 Dg C0 = C : The resulting default theory (D0 ; W 0; C 0 ) is called the atomic format of the original default theory, . As shown in [7], this transformation is a conservative extension of the formalism. Hence, it does not a ect the computation of queries to the original default theory. We can therefore restrict our attention to atomic default rules (without losing generality). The advantages of atomic default rules over arbitrary ones are, rst, that their constituents are not spread over several clauses while transforming them into clausal format and, second, that these constituents, e.g. the consequents, are uniquely referable to. 0
0
3
Skeptical reasoning in constrained default logic
In classical logic, we can say that a formula is derivable from a set of facts W i it belongs to the deductive closure of W , that is if 2 Th(W ): Due to the possible existence of multiple extensions, this notion of derivability is not directly applicable to Default Logic. Rather we obtain two di erent notions of derivability: A formula is credulously derivable from (D; W ; C ) i 2 E for some constrained extension (E; C ) of (D; W ; C ). And a formula is skeptically derivable from (D; W ; C ) i belongs to all such extensions of (D; W ; C ). In our example, (1), D and H are only credulously derivable while C is also skeptically derivable. In order to furnish a corresponding proof-theory, we need the following concepts. A default proof segment in a default theory = (D; W ; C ), or -segment for short, is a ( nite) sequence of default rules hi ii2I such that W [ Conseq(f0; : : :; i?1g) ` Prereq(i ) for i 2 I and (2) W [ C [ fConseq(i ); Justif (i ) j i 2 I g is satis able. (3) A credulous default proof , or CDP for short, for a formula from is a -segment hi ii2I such that W [ fConseq(i ) j i 2 I g ` : Furthermore, we say that a formula is provable from a -segment hi ii2I i there is a CDP hj ij 2J for such that I J , i.e. if the segment is extendible to a CDP for . In our example, there are ve -segments, all of which are extendible to one of the two CDPs of C , namely E D
Q:P ; D:C and R :H:P ; HC: C : (4) D C Clearly, a formula is credulously derivable i it is provable from some -segment, since then it has a CDP. Accordingly, a formula is skeptically derivable i it is provable from all -segments.2 Now, an irresistible question is that on the concept of a default proof in skeptical reasoning. Since in general there is not a single CDP valid in each extension, it is however natural to view a skeptical default proof as being compound of multiple CDPs. In fact, we take a skeptical default proof of a formula to be a set P of CDPs such that P is complete : De nition 3.1 (Skeptical Default Proof)Let = (D; W ; C) be a default theory and a formula. A skeptical default proof of from is a set P of CDPs for such that for each constrained extension (E; C ) of there is some hi ii2I 2 P such that C [ fConseq(i ); Justif (i ) j i 2 I g is satis able. This view relies heavily on the notion of CDPs. In fact, we keep this fundamental idea and base our method for skeptical reasoning on credulous reasoning, too. The idea is to start with a CDP of a given query. Then, we determine in some way a representative selection of -segments incompatible with our initial CDP. These -segments indicate extensions in which our initial default proof is invalid. Intuitively, they can be thought of as putative counterarguments challenging our initial CDP. Next, we verify in turn whether our query is derivable from each such -segment. If this is indeed the case, then our initial query is skeptically derivable. 2
This is formally shown in [10].
2
In order to illustrate this approach, let us verify that C is skeptically derivable in our example. We start with an arbitrary CDP of C from Default Theory (1). Consider the second CDP in (4): R :H:P ; HC: C : This proof takes place in the extension of (1) containing H; :D; and C . Next, we regard all -segments \challenging" default rules in
R : :P ; H : C : This notion is captured formally by the property of orthogonality:3 Two default proof segments h i i i2I H C and hj ij 2J in a default theory (D; W ; C ) are called C -orthogonal i W [ C [ fConseq(k ); Justif (k ) j k 2 I [ J g is unsatis able. That is, a -segment is orthogonal to another one if its induced constraints, i.e. the set of justi cations and consequents of its default rules, are incompatible with the same constraints of the other -segment. Observe that each -segment orthogonal to a given CDP indicates one or more extensions in which our CDP is not valid. The formal basis for our approach is laid in the following theorem. Theorem 3.1Let (D; W ; C) be a default theory and a formula. Then, is skeptically derivable from (D; W ; C) i there is a CDP hi ii2I for and is provable from all default proof segments which are C -orthogonal to hi ii2I .
In our example, there are two -segments orthogonal to R :H:P ; HC: C ; namely E D E D Q : P and Q : P ; D : C : D D C This is so because the justi cation of the rst default rule R :H:P in our CDP is contradictory to the justi cation of the default rule QD: P : There is no -segment orthogonal to the second default rule in our default proof. In fact, we can restrict our attention to minimal orthogonal -segments, as will be shown in Theorem 3.2. Accordingly, it is sucient E D Q : P to consider the -segment D : E D Intuitively, we then focus on all extensions of the initial default theory to which the -segment QD: D contributes. Then, we check whether our initial query C belongs to these extensions. Importantly, this is accomplished by using only default rules relevant for deriving C and hence without computing any extensions. We achieve this by checking whether C is skeptically derivable from the default theory obtained by \applying" the default rules in our orthogonal -segment. In this way, we try to prove our query under the restrictions imposed by the -segment contesting our initial CDP. For this, we add the consequence of the default rule QD: P to the facts of Default Theory (1). Furthermore, we have to add its justi cation to the set of initial constraints. This yields the following default theory: o n Q : P ; R : :P ; D : C ; H : C ; fQ; R; Dg; fP g (5) D H C C Now, it remains to be shown that C is skeptically derivable from Default Theory (5). Proceeding recursively, we
check rst whether C is credulously derivable from this default theory. In fact, C can be proven by means of DC: C : Next, we have to nd all minimal -segments orthogonal to DC: C : However, there are no such segments since Default Theory (5) has a single extension containing D and C , in which, moreover, the default rule R :H:P is blocked due to the initial constraint P . Importantly, the previous step supplies Dus withE an alternative CDP of C from our original default theory in (1). This Q : P D : C is obtained by appending the -segment D and the CDP of C from default theory (5), viz. C . This results
in the rst CDP given in (4). Since there are no other -segments orthogonal to our initial default proof R :H:P ; HC: C we are done. As a result, we have obtained a skeptical default proof consisting of two CDPs supporting the skeptical conclusion C . This approach is justi ed by the following theorem: Theorem 3.2Let hi ii2I be a default proof segment in a default theory (D; W ; C) and a formula. Then, is provable from all default proof segments hj ij 2J where I J i is skeptically derivable from (D n fi j i 2 I g; W [ fConseq(i ) j i 2 I g; C [ fJustif (i ) j i 2 I g): Let us summarize our approach to checking whether a query ' is skeptically derivable: We start with a CDP of ': Then, we determine all minimal orthogonal -segments contesting our initial CDP. Next, we check in turn whether ' is skeptically derivable under the restrictions imposed by each such -segment. In this way, we check whether ' belongs to all extensions orthogonal to the one containing our initial CDP of ': It is important to note that the choice of the initial CDP is a \don't care"-choice in so far that deciding whether ' is skeptically entailed is independent of which CDP is initially chosen. However, the composition of a resulting skeptical default proof depends on the latter choice. The approach we have outlined so far can be put together to a concrete algorithm as follows. Let (D; W ; C ) be a default theory and ' be a formula. We assume that there is a function cred(D; W ; C ; ') such that hi ii2I if hi ii2I is a CDP of ' from (D; W ; C ) cred(D; W ; C ; ') = ? otherwise Moreover, given a CDP hi ii2I , let orth(D; W ; C ; hi ii2I ) yield the set of all minimal -segments C -orthogonal to hi ii2I : These two functions yield the following algorithm for skeptical query-answering. Similar to its credulous source, it returns ? if ' is not skeptically derivable; otherwise it returns a set of CDPs forming a skeptical default proof. The function concatenates two -segments. 3 Orthogonality usually refers to distinct extensions (c.f. [6]). In constrained default logic [2], two constrained extensions (E; C ) and (E 0 ; C 0 ) are orthogonal i C [ C 0 is unsatis able.
3
skep(D; W ; C ; ') = if cred(D; W ; C; ') =?
then return ? else let hiii2I = cred(D; W ; C; ') in let O = orth(D; W ; C; hiii2I ) in P := ; ; while O =6 ; do select hj ij2J 2 O ; let W0 0 = W [ fConseq(j ) j j 2 J g in let C = C [ fJustif (j ) j j 2 J g in if skep(D; W 0 ; C0; ') =? then return ? else P := P [ fhj ij2J hl il2L j hl il2L 2 skep(D; W 0; C0; ')g ; O := O n fhj ij 2J g od ; return fhiii2I g [ P
The variable P accumulates the set of resulting CDPs. Whenever the procedure ends up with success, the CDPs in P form a skeptical default proof of ' from (D; W ; C ). Finally, we have to address the determination of minimal -segments orthogonal to a CDP at hand. Again, this is accomplishable by appeal to credulous reasoning. Note that the notion of orthogonality refers to the consistency constraints induced by a CDP. For this, we have to consider additionally a default rule's justi cation any time the default rule applies: For a set of atomic default rules D, we de ne the set of normalization rules as N (D) = f ! j : 2 Dg: Intuitively, the addition of normalization rules to the facts of a default theory in atomic format turns each atomic default rule : into a default rule of the form ^: : With this, the following theorem tells us how to determine -segments orthogonal to a CDP at hand: Theorem 3.3Let hi ii2I and hj ij2J be default proof segments in a default theory (D; W ; C) and a formula. Then, hi ii2I and hj ij 2J are C -orthogonal i there is i 2 I such that hj ij 2J is a CDP for :Conseq(i ) _ :Justif (i ) in default theory (D n fk j k < ig; W [ N (D n fk j k < ig) [ fConseq(k ); Justif (k ) j k < ig; C ): (6)
R : :P H : C 4 That is, in order to nd -segments orthogonal to our CDP H ; C ; we consider the default theories n
o
Q : P ; R : :P ; D : C ; H : C ; fQ; Rg [ fD ! P; H ! :P g; ; D H C C n o Q : P ; D : C ; H : C ; fQ; Rg [ fD ! P g [ fH; :P g; ; D C C
(7) (8)
In turn, we must determine all minimal CDPs of the negated consequences or the negated justi cations of R :H:P and H : C : That is, rst we search for a proof for :H _ P from Theory (7) and then for a proof of :C from Theory (8). C D E While :C is not provable from (8), :H _ P is provable from (7) yielding a single orthogonal -segment, QD: P : This leads to the following algorithm for the function orth: orth(D; W ; C ; hi ii2I ) = let W 0 = W [ N (D) [ fConseq(k ) j k < ig in let O = ; in
for i 2 I do let ' = :Conseq(i ) _ :Justif (i) in O := O [ fhj ij 2J j hj ij 2J = cred(D; W 0; C ; ') and there is no J 0 J such that hj ij 2J = cred(D; W 0; C ; ') g od ; return O 0
0
0
This procedure yields a set containing all minimal -segments orthogonal to a given CDP hi ii2I : In all, our approach has the following advantages. First, it avoids the computation of entire extensions. Second, it is goal-directed and thus restricted to default rules relevant to proving a given query. Third, it takes advantage of basic techniques developed for credulous reasoning. Clearly, it is necessary to consider all mutually orthogonal CDPs belonging to distinct extensions. In this way, we cannot get around the exponential factor present in the worst-case, where there is an exponential number of extensions each of which comprises a CDP orthogonal to all CDPs in all other 4 For clarity, we refrain from presenting Theory (7) and (8) in atomic format. Moreover, we discard normalization rules for normal default rules since they are tautological.
4
extensions. In fact, skeptical reasoning is P2 -complete [3]. This theoretical threshold however is both inevitable in the worst-case and (probably) irrelevant for practical applications. The aforementioned advantages become obvious by looking at some more examples. Suppose that we extend Default Theory (1) with a default rule like RW: W saying that \republicans are Western fans". Proving that W is skeptically derivable is doable by a single CDP, since this CDP is not contested by any orthogonal -segments. For another example, suppose that we extend Default Theory (1) with default rules like QV: V and R:: :V V saying that \quakers are vegetarians" and \republicans aren't vegetarians". This leads to four distinct extensions. Proving that C is skeptically derivable however is doable with the same steps as described above. That is, the two additional extensions do not increase computational e orts. This is so because the new default rules are irrelevant to proving C . 4
Credulous query-answering
In what follows, we extend the approach for query-answering in Default Logics developed in [9] to Pre-Constrained Default Logic. This approach is based on the Connection Method [1], which allows for testing the unsatis ability of formulas in conjunctive normal form (CNF). Unlike resolution-based methods that decompose formulas in order to derive a contradiction, the Connection Method analyses the structure of formulas for proving their unsatis ability. This structure-sensitive nature allows for an elegant characterization of proofs in Default Logic, as we see below. In the Connection Method, formulas in CNF are displayed two-dimensionally in the form of matrices (see (9) for an exemplar). A matrix is a set of sets of literals (literal occurrences, to be precise).5 Each column of a matrix represents a clause of the CNF of a formula. In order to show that a sentence ' is entailed by a sentence W , we prove that W ^ :' is unsatis able. In the Connection Method this is accomplished by path checking: A path through a matrix is a set of literals, one from each clause. A connection is an unordered pair of literals which are identical except for the negation sign (and possible indices). A mating is a set of connections. A mating spans a matrix if each path through the matrix contains a connection from the mating. Finally, a formula, like W ^ :'; is unsatis able i there is a spanning mating for its matrix. The approach of [9] relies on the idea that a default rule can be decomposed into a classical implication along with two qualifying conditions, one accounting for the character of an inference rule and another one enforcing the respective consistency conditions. The computational counterparts of these qualifying conditions are given by the proof-oriented concepts of admissibility and compatibility , which we introduce in the sequel. In order to nd out whether a formula ' is in some extension of a default theory (D; W ; C ), we proceed as follows: First, we transform the default rules in D into their sentential counterparts. This yields a set of indexed implications: o n : WD = ! 2 D Second, we transform both W and WD into their clausal forms, CW and CD . The clauses in CD , like f: ; g; are called -clauses; all other clauses like those in CW are referred to as !-clauses. Finally, a query ' is derivable from (D; W ; C ) i there is a spanning mating for the matrix CW [ CD [ f:'g agreeing with the concepts of admissibility and compatibility.6 A useful concept is that of a core of a matrix M wrt a mating , which allows for isolating the clauses relevant to the underlying proof. We de ne the core of M wrt as7 (M; ) = fc 2 M j 9 2 : c \ 6= ;g : For instance, the core of (9) wrt the drawn mating is given by all clauses connected by arcs. Then, the proof-theoretic counterpart of groundedness as expressed in (2) can be captured as follows [9]: De nition 4.1 (Admissibility)Let CW be a set of !-clauses and CD be a set of -clauses and let be a mating for CW [ CD : Then, (CW [ CD ; ) is admissible S i there is anenumeration hf: i ; i gii2I of (CD ; ) such that for i 2 I; is a spanning mating for CW [ ij?=01 ff: j ; j gg [ ff: i gg: Note that sometimes not all connections in are needed for showing the unsatis ability of the previous submatrices. In fact, we have to extend the notion of compatibility found in [9] in order to deal with a set of constraints C . In what follows, let us agree on abbreviating Justif (i ) by i (by appeal to atomic default rules). De nition 4.2 (C-compatibility)Let CW and CC be sets of !-clauses and let CD be a set of -clauses. Let be a mating for CW [ CD and let hf: i ; i gii2I be an?Senumeration of (CD ; ). Then, (CW [ CD ; ) is C -compatible wrt I i there is no spanning mating for CW [ CC [ i2I ff: i ; i g; f i gg . The following theorem shows that our extended method is sound and complete: Theorem 4.1Let (D; W ; C) be a default theory in atomic format and ' an atomic formula. Then, ' 2 E for some constrained extension (E; C ) of (D; W ; C ) i there is a spanning mating for the matrix M = CW [ CD [ ff:'gg of W [WD [f:'g and an enumeration hf: i ; i gii2I of (CD ; ) such that (M; ) is admissible wrt I and C -compatible wrt I .
5 In the sequel, we simply say literal instead of literal occurrences; the latter allow for distinguishing between identical literals in di erent clauses. 6 Without loss of generality, we deal with atomic queries only, since any query can be transformed into \atomic format". 7 Recall that we deal with literal occurrences.
5
Finally, (M; ) represents the CDP hi ii2I for ' from (D; W ; C ). As an illustration, let us verify that C is credulously derivable according to the recipe given above. The encoding of the set of default rules yields the set WD of implications: fQ1 ! D1 ; R2 ! H2 ; D3 ! C3 ; H4 ! C4 g: The indexes denote the respective default rules in (1) from left to right. In order to verify that a republican quaker is conservative, C , we rst transform the facts in Default Theory (1) and the implications in WD into their clausal form. The resulting clauses are given two-dimensionally as the rst six columns of the matrix in (9). The full matrix is obtained by adding the clause containing the negated query, :C . In fact, the matrix has a spanning mating, viz ffR; :R2 g; fH2 ; :H4 g; fC4 ; :C gg: We have indicated these connections in (9) as arcs linking the respective literals. :Q1
Q
R
D1
:H4
:D3
:R2
C4
C3
H2
:C
(9)
This proof corresponds to the second one in (4) and yields the following enumeration: hf:R2 ; H2 g; f:H4 ; C4 gi
For admissibility, we must therefore consider the following two submatrices of Matrix (9): :R2
Q
:H4
:R2
Q
R
R
H2
(10) (11)
Observe that each of these submatrices has a spanning mating, so that the original matrix and its mating, given in (9), constitute an admissible proof. For compatibility, we have to verify that the following matrix has no spanning mating:8 :R2
Q
R
H2
:H4
C4
(12)
:P2
In fact, this is the case since the matrix contains a non-complementary path, viz. fQ; R; H2 ; C4 ; :P2 g: We thus obtain an admissible and compatible proof for the original query, C , asking whether a republican quaker is conservative. 5
Skeptical query-answering
Now, let us return to skeptical query-answering. The basic idea is to extend the given method for credulous queryanswering by adding another condition on proofs ensuring that a query is skeptically derivable. This extra condition is motivated by the general idea described in Section 3. At rst, we account for the proof-theoretic counterpart of -segments orthogonal to a given CDP. Let CN (D) be the clausal representation of N (D), i.e. CN (D) = ff: ; g j 2 Dg: These clauses are needed for adding the justi cations of default rules while determining -segments orthogonal to a CDP at hand. Such an addition is obsolete for normal default theories. De nition 5.1 (Challenge)Let CW and CC be sets of !-clauses and let CD be a set of -clauses. Let be a mating for CW [ CD and let hf: i ; i gii2I be an enumeration of (CD ; ). Then, a challenge at i 2 I is a minimal set of default rules = f j f: ; g 2 (CD ; i)g for some spanning mating i for the matrix (13) Mi = CW [ CN (D) [ ff k g; f j g j k < ig [ CD [ ff i g; f i gg such that (Mi ; i ) is admissible and C -compatible (wrt some index set Ii ). Observe that the matrix representation allows us to simplify (13) by replacing CD and CN (D) by (CD nff: k ; k g j k ig) and (CN (D) nff: k ; j g j k ig); respectively, since the subtracted clauses are subsumed by ff k g; f j g j k < ig and the query clauses ff i g; f i gg: Now, the overall idea is that (CW [ CD ; ) represents a CDP hi ii2I for the considered query. Then, each default proof (Mi ; i) induces a challenge that corresponds to a minimal -segment C -orthogonal to hi ii2I : This is so because (Mi ; i) represents a CDP of :Conseq(i ) _ :Justif (i ) from Default Theory (6) (c.f. Theorem 3.3). Of course, any matrix Mi may have several spanning matings i and hence may induce di erent challenges. Now, we are ready to formulate our additional condition on default proofs for skeptical reasoning: De nition 5.2 (Protection & Stability)Let CW and CC be sets of !-clauses, CD be a set of -clauses and ' an atomic formula. Let be a mating for CW [ CD and let hf: i ; i gii2I be an enumeration of (CD ; ). Let hij ij 2Ji be the family of all challenges at i 2 I: We say that (CW [ CD ; ) is protected under C against ij by (Mij ; ij ) i ij is a spanning mating for the matrix Mij = CW [ ff g j 2 ij g [ CD [ ff:'gg such that (Mij ; ij ) is admissible, C [ Justif (ij )-compatible, and stable for ' under C [ Justif (ij ). We say that (CW [ CD ; ) is stable for ' under C i (CW [ CD ; ) is protected under C against all challenges ij .
8 The clause f:P g represents the justi cation of . Note that it is unnecessary to add clauses for justi cations of normal default 2 2 rules, as in the case of 4 .
6
The idea is that (CW [ CD ; ) represents a CDP for the query ': In order to verify whether ' is skeptically derivable, we must show that (CW [ CD ; ) satis es in addition the stability criterion. For this, we intuitively proceed as follows. First, we isolate all challenges against our CDP (CW [ CD ; ): In turn, we then verify whether ' is also skeptically derivable from the matrices obtained by adding the consequents of the default rules in to CW and adding the justi cations of the default rules in as additional constraints on the compatibility check. This amounts to verifying whether ' is in all extensions to which the default rules in contribute. Accordingly, a skeptical default proof is given by a stable credulous default proof (CW [ CD ; ) along with all its protecting default proofs (Mij ; ij ). Of course, there may be several such skeptical proofs depending on the initial choice. Now, let us examine whether our default proof in (9) is stable and thus renders C a skeptical conclusion of Default Theory (1). For this, we consider the obtained enumeration hf:R2 ; H2 g; f:H4 ; C4 gi: In turn, we determine all emerging challenges. That is, we consider all minimal default proofs of :H2 _ P2 and :C4 : These formulas represent the negated consequents (and justi cations) of the used default rules, 2 and 4 , respectively. In the rst case, we consider the matrix obtained from our original matrix, (9), by replacing the query clause f:C g by the clauses fH2 g and f:P2 g: This allows us moreover to eliminate the -clause f:R2 ; H2 g from (9) since it is subsumed by fH2 g: Analogously, we can omit the normalization clause f:H2 ; :P2 g due to the presence of f:P2 g: Hence, we have to add only the normalization clause f:D1 ; P1 g since 3 and 4 are normal default rules. The modi cations to our initial matrix in (9) are indicated as dashed boxes.9 This results in the following derivative of Matrix (9):10 M2 =
:D3
:Q1
Q
R
C3
D1
:H4
C4
:D1
P1
H2
:P2
(14)
By discarding the two query clauses, Matrix11 M2 can be seen as the proof-theoretic counterpart of Default Theory (7). In fact, M2 has the spanning mating 2 = ffQ; :Q1 g; fD1 ; :D1 g; fP1 ; :P2 gg: This CDP involves a single default rule, namely 1 : We thus obtain the singular enumeration hf:Q1 ; D1 gi inducing the following two matrices for verifying admissibility and compatibility, respectively. :Q1
Q
R
:D1
P1
:H2
:Q1
:P2
Q
D1
R
:D1
P1
:H2 :P2
(15)
Both matrices contain the normalization clauses f:D1 ; P1 g and f:H2 ; :P2 g: The left matrix is complementary and thus con rms admissibility, while the right matrix has the open path fQ; R; D1 ; P1 ; :H2 g establishing compatibility. Consequently, (M2 ; 2) provides us with a CDP of :H2 _ P2 : This CDP is orthogonal to our initial proof in (9). As a result, (M2 ; 2) induces the challenge 21 = f1 g: There is no other challenge induced by M2 : Now, let us rst verify whether our CDP in (9) is protected against 21 by some other CDP before we determine more challenges: For establishing stability, intuitively, we verify whether our initial query C belongs to the extensions formed (among others) by the default rules in 21 = f1 g: For this, we consider the matrix M21 obtained from our initial matrix in (9) by adding the consequents of all default rules in 21: This amounts to replacing the -clause f:Q1 ; D1 g in (9) by the !-clause fD1 g : M21 =
Q
R
D1
:R2
H2
:D3
C3
:H4
C4
:C
(16)
Then, we verify according to De nition 5.2 whether there is a spanning mating 21 for M21 such that (M21 ; 21) is admissible, fP1 g-compatible (since Justif (21) = fP1 g), and stable. Admissibility and compatibility of (M21 ; 21) are easily veri ed by regarding the two following matrices induced by the only -clause used in (16), f:D3 ; C3 g: Q
R
D1
:D3
Q
D1
R
:D3
C3
P1
The complementarity of the left matrix establishes admissibility while the non-complementarity (initiated by the open path fQ; R; D1 ; C3 ; P1 g) of the right matrix con rms fP1 g-compatibility. Now, it remains to be shown that (M21; 21) along with its induced enumeration hf:D3 ; C3 gi is stable for C under fP1 g. However, there is no challenge to f:D3 ; C3 g since :C3 is not provable from the matrix obtained by replacing the query clause f:C g in (16) by fC3 g:12 As a result, we obtain that our CDP in (9) is protected against 21 by (M21 ; 21): This is done to underline the utility of structure-oriented theorem proving. For simplicity, we have refrained from turning the two query clauses fH2 g and f:P2 g into fH2 ; 'g; f:P2 ; 'g along with the single atomic query clause f:'g (as stipulated in Theorem 4.1). 11 The index 2 of M and is taken due to the index of the query :H _ P . 2 2 2 2 12 In fact, there is no compatible default proof. 9 10
7
Next, we must consider all challenges of the second -clause in (10). For this, we determine all CDPs of :C4 (the consequent of 4 ) from the matrix obtained in the following way. First, we add the clauses fH2 g and f:P2 g representing the \consequent" and the \justi cation" of the rst -clause in (10) to our initial matrix, (9). The rst addition is accomplished by replacing the -clause f:R2 ; H2 g by fH2 g: Second, we add the normalization clause f:D1 ; P1 g: As with Matrix (14), the second normalization clause f:H2 ; :P2 g can be omitted since it is subsumed by f:P2 g: Again, no normalization clauses are added for the normal default rules 3 and 4 . Finally, we replace the original query clause f:C g of (9) by fC4 g: This allows us to eliminate the -clause f:H4 ; C4 g since it is subsumed by fC4 g: :Q1
Q
R
D1
H2
:P2
:D3
C3
:D1
P1
C4
Observe that this matrix can be regarded as the proof-theoretic counterpart of Default Theory (8) if we discard the query clause. The above matrix has a spanning mating inducing the enumeration hf:Q1 ; D1 gi: The admissibility of the previous proof can be veri ed in a straightforward way, so that compatibility remains to be shown. For this, we have to consider the following matrix: :Q1
Q
R
D1
H2
:P2
:D1
P1
Obviously, this matrix has no open path so that our proof is not compatible. Accordingly, :C4 is not derivable (c.f. Default Theory (8)) and therefore there are no more challenges, apart from 21: In this way, we have shown that our initial CDP in (9) is stable for C (in addition to its admissibility and compatibility veri ed in Section 4). This is so because it is protected against its only challenge 21 by (M21 ; 21): This tells us that C is skeptically derivable from Default Theory (9). In general, we have the following result. Theorem 5.1Let (D; W ; C) be a default theory in atomic format and ' an atomic formula. Then, ' 2 E for all constrained extension (E; C ) of (D; W ; C ) i there is a spanning mating for the matrix M = CW [ CD [ ff:'gg of W [ WD [ f:'g and an enumeration hf: i ; i gii2I of (CD ; ) such that (M; ) is admissible wrt I , C -compatible wrt I , and stable for ' under C . 6
Conclusion
We have developed an approach to skeptical query-answering in Constrained Default Logic based on the Connection Method. This has been accomplished by elaborating on a recently proposed, general idea for skeptical reasoning in (semi-monotonic) Default Logics [10]. As a result, we obtained a precise algorithm that returns a skeptical default proof if the query is contained in all extensions of the underlying default theory. The approach has then been combined with a method for credulous query-answering based on the Connection Method. This was accomplished by adding another restriction on credulous default proofs, expressed by the stability criterion. As a result, we obtained a homogeneous characterization of skeptical default proofs at the level of the underlying deduction method. This approach was supported by the structure-sensitive nature of the Connection Method. The value of this for structure-sharing among the diverse subproofs involved is detailed for credulous query-answering in [9]. Even though we have not discussed it here, it should be obvious that the utility of structure-sharing applies to skeptical query-answering, too. We tried to indicate this by stressing the common structures involved in the skeptical default proof carried out in the previous section. References
[1]W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, second edition, 1987. [2]J. Delgrande, T. Schaub, and W. Jackson. Alternative approaches to default logic. Arti cial Intelligence, 70:167{237, 1994. [3]G. Gottlob. Complexity results for nonmonotonic logics. Journal of Logic and Computation, 2(3):397{425, June 1992. [4]R. Moore. Semantical considerations on nonmonotonic logics. Arti cial Intelligence, 25:75{94, 1985. [5]D. Poole. A logical framework for default reasoning. Arti cial Intelligence, 36:27{47, 1988. [6]R. Reiter. A logic for default reasoning. Arti cial Intelligence, 13(1{2):81{132, 1980. [7]A. Rothschild. Algorithmische Untersuchungen zu Defaultlogiken. Master's thesis, FG Intellektik, FB Informatik, TH Darmstadt, Alexanderstrae 10, D{64283 Darmstadt, Germany, 1993. [8]T. Schaub. Variations of constrained default logic. In M. Clarke, R. Kruse, and S. Moral, editors, Proceedings of European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, pages 312{317. Springer Verlag, 1993. [9]T. Schaub. A new methodology for query-answering in default logics via structure-oriented theorem proving. Journal of Automated Reasoning, 1995. Forthcoming. [10]M. Thielscher and T. Schaub. Default reasoning by deductive planning. Journal of Automated Reasoning, 1995. Forthcoming.
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