T. Eiter and G. Gottlob. Propositional Circumscription and Extended Closed World Reasoning are \Pi  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Proof. Inclusion: We can rst guess M such that l 2 M and verify that M is an answer set. Then, by Theorem 2 we can use an NP oracle to verify that M is fi; pg preferred. Hardness : Given a 3 sat instance S and a literal l, it is P hard to decide whether l is true in a minimal model of S [3]. We construct a LPOD t(S) such that l is true in a fi; pg preferred answer set of t(S) i l is true in a minimal model of S. t(S) f :A 1 ; A 2 ; A 3 j A 1 A 2 A 3 2 Sg (10) f:a a j a 2 At(S)g (11) Now M is an answer set of t(S) i it is a model of S. We can see this by noting that ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed-world reasoning are  2 -complete. Theoretical Computer Science, 114:231-245, 1993.

....in practice, e.g. methods working for moderately sized instances of NP complete problems do not work for 2 complete problems. The complexity of the problems we are interested in has been extensively studied for existing logics for default reasoning under the standard, stability semantics [3, 15, 23, 19, 29, 2, 10]. Table 1 gives a partial summary of these results for the different logics. We note here that the semantics of circumscription has been originally proposed with respect to sceptical reasoning only. In this case, as shown in [1] reasoning in circumscription (restricted to Herbrand models) ....

Thomas Eiter and Georg Gottlob. Propositional Circumscription and Extended Closed-World Reasoning are  2 -Complete. Theoretical Computer Science 114(2): 231--245, 1993.

.... in the propositional case of several expressively rich non monotonic logical formalisms, including: circumscription (even minimizing predicates without priorities) Default Logic (even the normal case) Autoepistemic Logic, and several additional non monotonic modal logics and other formalisms [Eiter and Gottlob, 1993] [Stillman, 1992] Gottlob, 1992] There are several avenues to avoiding worst case complexity of default reasoning, generally. One is to employ approximations, perhaps sound but incomplete, e.g. as in [Cadoli and Schaerf, 1992] Another is to restrict expressive classes to those with ....

Tomas Eiter and Georg Gottlob. Propositional circumscription and extended closed world reasoning are 2 -complete. Theoretical Computer Science, 114(2):231--245, 1993. See also Addendum with two missing references in Theoretical Computer Science 118 (2) 1993 p. 315.

....of CQA is coNP complete for simple functional dependencies and existential queries. In this paper, wehave formulated the problem of CQA as a problem of nonmonotonic reasoning, more precisely of minimal entailment, whose complexity, even in the propositional case, can be at least # complete [19]. Having a problem of nonmonotonic reasoning with such complexity, it is not strange to try to use disjunctive logic programs with negation with a stable or answer set semantics to solve the problem of CQA, because such programs have nonmonotonic consequences and a # complete complexity ....

Eiter, T. and Gottlob, G. Propositional Circumscription and Extended Closed World Assumption are # # -complete. Theoretical Computer Science, 1993, 114, pp. 231-245.

....of CQA is coNP complete for simple functional dependencies and existential queries. Furthermore, in [8] the problem of CQA is formulated as a problem of non monotonic reasoning, more precisely of minimal entailment, whose complexity, even in the propositional case, can be P 2 complete [17]. Under those circumstances, it makes sense to apply techniques from logic programming, given its success in formalizing and implementing complex nommonotonic reasoning tasks [7] The problem then is to come up with the best logic programming speci cation and the best way to use them, so that ....

Eiter, T. and Gottlob, G. Propositional Circumscription and Extended Closed World Assumption are  p 2 -complete. Theoretical Computer Science, 1993, 114, pp. 231-245.

....of CQA is coNP complete for simple functional dependencies and existential queries. In this paper, we have formulated the problem of CQA as a problem of nonmonotonic reasoning, more precisely of minimal entailment, whose complexity, even in the propositional case, can be at least 2 complete [19]. Having a problem of nonmonotonic reasoning with such complexity, it is not strange to try to use disjunctive logic programs with negation with a stable or answer set semantics to solve the problem of CQA, because such programs have nonmonotonic consequences and a 2 complete complexity ....

Eiter, T. and Gottlob, G. Propositional Circumscription and Extended Closed World Assumption are  p 2 -complete. Theoretical Computer Science, 1993, 114, pp. 231-245.

....Theorem 7.11 shows that parallel circumscription reduces to cautious autoepistemic reasoning. Hence, the problem of deciding whether a propo 106 sitional sentence is a circumscriptive consequence of a set of propositional sentence is in Pi 2 by Corollary 6.24. Eiter and Gottlob [20] have shown that in fact the problem is Pi complete. By Theorem 7.11 a decision method for cautious autoepistemic reasoning can be used for circumscriptive reasoning and a circumscriptive theorem prover can be employed as a decision method for cautious autoepistemic reasoning for a subclass ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are \Pi 2 -complete. Technical Report CD-TR 91/20, Christian Doppler Labor fur Expertensysteme, Institut fur Informationssysteme, Technische Universitat Wien, Vienna, Austria, May 1991.

....Proof. Inclusion: We can rst guess M such that l 2 M and verify that M is an answer set. Then by Theorem 2 we can use an NP oracle to verify that M is fi; pg preferred. Hardness : Given a 3 sat instance S and a literal l, it is 2 hard to decide whether l is true in a minimal model of S [3]. We construct a LPOD t(S) such that l is true in a fi; pg preferred answer set of t(S) i l is true in a minimal model of S. t(S) f A 1 ; A 2 ; A 3 j A 1 A 2 A 3 2 Sg (9) f:a a j a 2 At(S)g (10) Now M is a preferred answer set of t(S) i it is a minimal model of S. We can see ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed-world reasoning are  2 -complete. Theoretical Computer Science, 114:231-245, 1993.

....the problem of CQA is coNP complete for simple functional dependencies and existential queries. Furthermore, in [8] the problem of CQA is formulated as a problem of non monotonic reasoning, more precisely of minimal entailment, whose complexity, even in the propositional case, can be 2P complete [18]. Under those circumstances, it makes sense to apply techniques from logic programming, given its success in formalizing and implementing complex nora monotonic reasoning tasks [7] The problem then is to come up with the best logic programming specification and the best way to use them, so that ....

Eiter, T. and Gottlob, G. Propositional Circumscription and Extended Closed World Assumption are //-complete. Theoretical Computer Science, 1993, 114, pp. 231-245.

....in it. A polynomial case arises when A is Horn, 2CNF and with no clauses of the form :x :y. Further polynomial cases were found for other restricted versions, for example, when A is Horn and Q = or when A is 2CNF, fl is a literal and Z = 8] An interesting result is the one presented in [12], where the inference of the propositional version of Circumscription (formula (5) with Q = and Z = or not) is proved to be complete for the class Pi 2 of the polynomial hierarchy. Propositional circumscription is a special case of formula (5) defined as follows: Definition 3 ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world assumption are \Pi 2 -complete. Theoretical Computer Science, 114:231--245, 1993.

....tableaux, a kind of hyper type rule, and a kind of cut rule. The closure conditions are as in the classical case, but a new closure condition is added, to close branches that do not lead to minimal models. This is achieved by means of a local minireality test, that can also be found in [29, 17]. We can adapt and adopt such a test in our framework on the basis of the definition of grounded model given in [30] and our circumscriptive characterization of CQA given above. Let B be a data closed branch in TP(IC [J r) with op(B) r L) J K. We associate to B a Herbrand structure A (B) ....

Eiter, T. and Gottlob, G. Propositional Circumscription and Extended Closed World Assumption are H-complete. Theoretical Computer Science, 114 (1993) 231-245.

....is an co NP complete problem [2] Notice that in the classical propositional logic (PL) a formula F simply represents all of its models, so model checking for PL is clearly in P. Similarly, deciding whether a formula logically follows from CIRC(T ) inference) is a Q p 2 complete problem [9], while inference for PL is co NP complete. A natural question is therefore: is it possible to itranslatej CIRC(T ) into a propositional formula F and then use F (instead of T ) to solve the model checking problem in time polynomial in jT j Clearly such translation cannot be performed in ....

....several belief revision operators, both model and syntax based (Ginsberg s and WIDTIO) It is interesting to observe that some revision operators and CIRC have similar properties. For instance, Ginsberg s operator and CIRC have the same time complexity and the same compactability properties (see [2, 9, 6] for the results on CIRC) It is therefore natural to ask whether this is a chance or not. A rst study of relationships between belief revision and CIRC has been done in [23] where the author remarked similarities between CIRC and her operator. Subsequently, in [14] the authors pointed out ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are Q p 2 -complete. Theoretical Computer Science, 114:231245, 1993.

....write j= circ to denote that is true in every minimal model of . It has been shown that the model checking problem for propositional circumscription is coNP complete (Cadoli [Cad92] whereas the inference problem for propositional circumscription is P 2 complete 1 (Eiter and Gottlob [EG93]) In fact, the model checking problem for propositional circumscription remains coNP complete even when restricted to 3CNF formulas, while the inference problem j= circ for propositional circumscription remains P 2 complete even when is a 3CNF formula and is a negative literal :u. ....

....in coNP and Inf Circ(S) is in P 2 . There are natural sets S of logical relations such that MC Circ(S) is coNP complete and Inf Circ(S) is P 2 complete. In particular, this holds true for the set S = fR 0 ; R 1 ; R 2 ; R 3 g of logical relations in Example 3 that give rise to 3 Sat (see [Cad92,Cad93,EG93]) In contrast, as pointed out in [Cad92,Cad93,CL94] if S is a bijunctive or a dual Horn set of logical relations, then MC Circ(S) in P and Inf Circ(S) is in coNP. Moreover, if S is a Horn set of logical relations, then both MC Circ(S) and Inf Circ(S) are in P; this is so because every satis able ....

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Th. Eiter and G. Gottlob. Propositional circumscription and extended closedworld reasoning are  P 2 -complete. Theoretical Computer Science, 114:231-245, 1993.

....traditional circumscriptions We apply now our results to traditional circumscriptions (traditional except that we are in the finite propositional case) Predicate circumscription (see e.g [McC80, PM86, Lif94] has already been simplified into propositional circumscription by many authors (e. g [EG93]) We assume familiarity with this now well known notion. Definition 6.1 A circumscription is defined once we know the set A of the circumscribed propositional symbols, the set A 0 of the varying ones, the remaining propositional symbols of V(LL) in A 00 , being fixed. V(LL) A [ A 0 [ ....

Thomas Eiter and Georg Gottlob. Propositional circumscription and extended closed-world reasonning are \Pi P 2 -complete. Theoretical Computer Science, 114:231--245, 1993.

.... and to rank these decision problems in the polynomial time hierarchy (PH) 1] In fact, complexity issues have received much attention in the NMR community recently, and the decision problems of default logic, strong) autoepistemic logic and circumscription have been systematically analyzed [4, 5, 8, 18, 21, 26]. To summarize these results in the propositional case, the major decision problems of these four nonmonotonic logics are complete problems on the second level of PH. These complexity results suggest that (i) the expressive powers of non monotonic logics exceed that of classical monotonic logic ....

....given in Proposition 1 is straightforward to establish. The notion of hP; F i extensions proposed is also appropriate if the brave 3 reasoning strategy is used in conjunction with parallel circumscription. 2 Gottlob s B proofs [8] capture this closure. 3 Note, e.g. that Eiter and Gottlob [5] consider the complexity of propositional circumscription according to the cautious strategy only. Proposition 1. Given a minimal model theory hP; F; T i, the intersection of hP; F i extensions of the theory T L coincide with the intersection of the theories True(M) for all hP; F i minimal ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are \Pi p 2 -complete. Theoretical Computer Science, 114:231245, 1993.

....that satisfies . Consequently, inference in classical propositional logic is a coNPcomplete problem and thus considered to be intractable. In the case of propositional circumscription, inference turns out to have even higher inherent computational complexity. Indeed, as shown by Eiter and Gottlob [EG93], the inference problem for propositional circumscription is P 2 complete. Recall that the class P 2 constitutes the second level of the polynomial hierarchy PH and thus contains both NP and coNP as subclasses. Moreover, the prototypical P 2 complete problem is P 2 SAT, i.e. ....

....reflection reveals that this problem is polynomial time equivalent to the special case in which is simply a clause (i.e. a disjunction of literals) since can be inferred from under propositional circumscription if and only if each clause of can be so inferred. Moreover, Eiter and Gottlob [EG93] established that 1 the inference problem for propositional circumscription remains P 2 complete even when is a 3CNF formula and the clause consists of a single negated variable. Are there restricted classes of propositional formulas on which the inference problem for propositional ....

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Th. Eiter and G. Gottlob. Propositional circumscription and extended closed-world reasoning are  P 2 -complete. Theoretical Computer Science, 114:231--245, 1993.

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T. Eiter and G. Gottlob. Propositional Circumscription and Extended Closed World Reasoning are \Pi

....Reiter [37] has introduced the closed world assumption (CWA) as a principle for inferring negative information from a logical database. Formally, CWA(DB) ground(DB) f:A j A 2 HB DB and DB 6j= Ag: For example, CWA(fP (a) Q(b)g) fP (a) Q(b) P (b) Q(a)g. It follows from results in [12] that computing CWA(DB) has propositional complexity k . Observe that CWA(DB) may not be classically consistent (under Herbrand interpretations) for example, CWA(fP Qg) fP Q, P , Qg which has no model. As shown in [12] deciding whether CWA(DB) is consistent is in 2 and coNP hard in ....

....Q(b)g) fP (a) Q(b) P (b) Q(a)g. It follows from results in [12] that computing CWA(DB) has propositional complexity k . Observe that CWA(DB) may not be classically consistent (under Herbrand interpretations) for example, CWA(fP Qg) fP Q, P , Qg which has no model. As shown in [12], deciding whether CWA(DB) is consistent is in 2 and coNP hard in the propositional case; the precise complexity of this problem is open. In a re ned notion of partial CWA (cf. 17] which is in the spirit of protected circumscription [35] only atoms A involving a particular predicate P or, ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are  2 -complete. Theoretical Computer Science, 114(2):231{ 245, 1993. Addendum 118:315.

....Reiter [36] has introduced the closed world assumption (CWA) as a principle for inferring negative information from a logical database. Formally, CWA(DB) ground(DB) f:A j A 2 HB DB and DB 6j= Ag: For example, CWA(fP (a) Q(b)g) fP (a) Q(b) P (b) P (a)g. It follows from results in [11] that computing CWA(DB) has propositional complexity FP . Observe that CWA(DB) may not be classically consistent (under Herbrand interpretations) for example, CWA(fP Qg) fP Q, P , Qg which has no model. As shown in [11] deciding whether CWA(DB) is consistent is in Theta 2 and ....

....Q(b)g) fP (a) Q(b) P (b) P (a)g. It follows from results in [11] that computing CWA(DB) has propositional complexity FP . Observe that CWA(DB) may not be classically consistent (under Herbrand interpretations) for example, CWA(fP Qg) fP Q, P , Qg which has no model. As shown in [11], deciding whether CWA(DB) is consistent is in Theta 2 and coNP hard in the propositional case; the precise complexity of this problem is open. In a refined notion of partial CWA (cf. 16] which is in the spirit of protected circumscription [34] only atoms A involving a particular predicate ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are \Pi 2 -complete. Theoretical Computer Science, 114(2):231--245, 1993. Addendum 118:315.

....general. Proof: By an inductive argument, we can see that for any circumscriptive atom = CIRC( P ; Z) such that nd( k for constant k, deciding M j= is in . Indeed, if k = 1, then is an ordinary circumscription, for which deciding M j= is well known to be in coNP = 1 , cf. [18]. Assume the statement holds for k 1, and consider k = k 1. Note that M 6j= iff either (a) M 6j= or (b) some model N exists such that N P ;Z M and N j= By the induction hypothesis, we can guess N and check whether either (a) or (b) holds for this N in polynomial time using a ....

....the lack of negation applied to blocks, a simple and appealing polynomial time embedding of L CIRC into NATs seems not straightforward. 4.2 Complexity of NATs Ordinary circumscription can express a QBF sentence = 8X9Y (where 2 L) as follows. Let u be a fresh atom. Proposition 4. 5 (cf. [18]) is true if and only if CIRC( u; Y ) j= u, where = u. This circumscription can be easily stated as a NAT. Set T 1 = fY; u : u abg: Then Proposition 4.5 implies that T 1 j= u iff is true. Recall that M [S] denotes the assignment to the atoms in S as given by M . Then, every ....

T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are  2 -complete. Theoretical Computer Science, 114(2):231--245, 1993. Addendum 118:315.

.... at the low end of the polynomial hierarchy, which range from co NP ( entailment) to (Geffner s conditional entailment) In general, they have lower complexity than well known logical formalizations of nonmonotonic reasoning such as default logic, circumscription, or autoepistemic logic [88, 50, 33]. The off line computation of rankings does in general not pay off with respect to worst case complexity, and in particular does not buy tractability. Furthermore, computing the ranking associated with a knowledge base is as difficult as solving the reasoning problem. Horn constraints have ....

.... common sense reasoning, including major formalisms such as circumscription [67, 65] default logic [82] Doyle and McDermott s nonmonotonic logics [69, 68] and Moore s autoepistemic logic [70] see [66] The computational complexity of nonmonotonic logics has been studied in many papers, e.g. [56, 87, 50, 74, 33, 21, 77] to mention a few comprehensive studies, and is quite well understood. As in the case of belief revision, the complexity of most of these logics resides at the second level of the polynomial hierarchy. More precisely, the problem of deciding whether a given classical formula is a consequence of ....

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T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are  2 -complete. Theoretical Computer Science, 114(2):231--245, 1993. Addendum 118:315.

....formula G, decide whether Curb(T ; p; z) j= G. The precise complexity of curbing, for both model checking and inferencing, was left open in [8] Note that model checking for propositional circumscription is coNP complete [3] and inferencing under propositional circumscription is P 2 complete [7]. It was conjectured in [21, 11] that curbing is of higher complexity than circumscription. This is intuitively supported by a result of Bodenstorfer [2] stating that in an explicitly given set of models, witnessing that some particular model is good may involve an exponential number of smaller ....

T. Eiter and G. Gottlob. Propositional Circumscription and Extended Closed World Reasoning are  P 2 -complete. Theoretical Computer Science, 114(2):231--245, 1993. Addendum 118:315.

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T. Eiter and G. Gottlob. Propositional circumscription and extended closed-world reasoning are  2 -complete. Theoretical Computer Science, 114:231--245, 1993.

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T. Eiter and G. Gottlob. Propositional circumscription and extended closed world reasoning are \Pi 2 -complete. Theoretical Computer Science, 114:231--245, 1993.

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Thomas Eiter and Georg Gottlob. Propositional Circumscription and Extended ClosedWorld Reasoning are  2 -Complete. Theoretical Computer Science 114(2): 231-245, 1993.

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