M. Cadoli, T. Eiter, and G. Gottlob. Default logic as a query language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....techniques, we needed to run 2 hard problems on the DLP system at hand. Thus, we have compared the performance of the di#erent model checking methods by running various instances of the 2 complete problem Strategic Companies on the dlv system. The strategic companies problem is from [ Cadoli et al. 1997 ] it is, to the best of our knowledge, the only 2 complete KR problem from the business domain. No experimental results for any # 2 complete KR problems are known. Briefly, a holding owns companies, each of which produces some goods. Moreover, several companies may have joint control ....

.... by(P,C1,C2) strategic(C) controlled by(C,C1,C2,C3) strategic(C3) Here strategic(C)meansthatC is strategic, produced by(P, C1,C2) that product P is produced by companies C1andC2, and controlled by(C, C1, C2,C3) that C is jointly controlled by C1,C2andC3; we have adopted here from [ Cadoli et al. 1997 ] that each product is produced by at most two companies and each company is under joint control of at most three other companies. The problem is to find out the set of all strategic companies (i.e. under brave reasoning, for which C the fact strategic(C)istrue) Note that this problem cannot ....

M. Cadoli, T. Eiter, G. Gottlob. (1997), Default Logic as a Query Language. IEEETKDE, 9(3):448-463.

....then block b on top of a, and nally c on top of b. Due to space restrictions we refer to [Erd99,FLMP99] for complete encodings. initial: b c a c b a goal: Fig. 3. Simple BW Example Strategic Companies (STRATCOMP) nally, is a 2 complete problem, which has been rst described in [CEG97]: A holding owns companies C(1) C(c) each of which produces some goods. Some of these companies may jointly control another one. This is modeled by means of predicates prod(P; C1; C2) product P is produced by companies C1 and C2 and contr(C; C1; C2; C3) company C is jointly ....

....one to one to the strategic sets. Checking whether any given company C is strategic is done by brave reasoning: Is there any answer set containing C strategic(C1) strategic(C2) prod(P; C1; C2) strategic(C) contr(C; C1; C2; C3) strategic(C1) strategic(C2) strategic(C3) As in [CEG97] we assume that each product is produced by at most two companies and each company is jointly controlled by at most three companies. 7.2 Benchmark Data For 3SAT, we have randomly generated 3CNF formulas over n variables (where n denotes the size as plotted on the x axis of the graphs in Section ....

M. Cadoli, T. Eiter, and G. Gottlob. Default Logic as a Query Language. IEEE TKDE, 9(3):448-463, 1997.

....2: Simple BW Example Predicate arc is symmetric, since undirected arcs are bidirectional. Due to space restrictions we refer to [Erdem, 1999; Faber et al. 1999a] for a complete encoding. Strategic Companies (STRATCOMP) finally, is a 2 complete problem, which has been first described in [Cadoli et al. 1997] : A holding owns c companies, each of which produces some goods. Some of these companies may jointly control others. This is modelled by means of predicates prod(P; C1; C2) P is produced by C1 and C2) and cont(C; C1; C2; C3) company C is jointly controlled by C1, C2 and C3) Now, some ....

....of the following natural program correspond one to one to the strategic sets. Checking whether any given company C is strategic is done by brave reasoning: Is there any answer set containing C s c(C1) s c(C2) prod(P; C1; C2) s c(C) cont(C; C1; C2; C3) s c(C1) s c(C2) s c(C3) As in [Cadoli et al. 1997] we assume that each product is produced by at most two companies and each company is jointly controlled by at most three companies to allow for an easier representation. 6 Benchmark Data For 3SAT, we have randomly generated 3CNF formulas over n variables using a tool by Selman and Kautz [Selman ....

M. Cadoli, T. Eiter, and G. Gottlob. Default Logic as a Query Language. IEEE TKDE, 9(3):448--463, 1997.

....b on top of a, and finally c on top of b. initial: b c a c b a goal: Figure 2: Simple BW Example Due to space restrictions we refer to [Erd99, FLMP99] for a complete encoding. Strategic Companies (STRATCOMP) finally, is a P 2 complete problem, which has been first described in [CEG97] A holding owns companies C(1) C(c) each of which produces some goods. Some of these companies may jointly control another one. This is modelled by means of predicates produced by(P; C1; C2) product P is produced by companies C1 and C2 and controlled by(C; C1; C2; C3) ....

....to the strategic sets. Checking whether any given company C is strategic is done by brave reasoning: Is there any answer set containing C strategic(C1) strategic(C2) produced by(P; C1; C2) strategic(C) controlled by(C; C1; C2; C3) strategic(C1) strategic(C2) strategic(C3) As in [CEG97] we assume that each product is produced by at most two companies and each company is jointly controlled by at most three companies to allow for an easier representation. 6 Benchmark Data For 3SAT, we have randomly generated 3CNF formulas over n variables (where n denotes the size as plotted on ....

Marco Cadoli, Thomas Eiter, and Georg Gottlob. Default Logic as a Query Language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June 1997.

....c language is not o ered. The example demonstrates the ability of answer set semantics to represent reasoning by cases and to nicely combine disjunction with defaults. For comparison of these properties with the use of disjunction in Reiter s default logic see [30] 20 The next example from [11] demonstrates the expressive power of the language. Example 5.2 Suppose a holding owns some companies producing a set of products. Each product is produced by at most two companies. We will use a relation produced by(P; C 1 ; C 2 ) which holds if a product P produced by companies C 1 and C 2 . ....

....deciding whether a program of FA Prolog has an answer set is NP complete [47] 2. A decision problem P can be solved by a uniform program of FA Prolog i it is in the class NP [68] 3. A decision problem P can be solved by a uniform program of FDA Prolog i it is in the complexity class P 2 [11] It is interesting to note that the problem from example 5.2 is P 2 complete [16] and therefore the use of disjunction is essential. The above theorem shows that for decision problems we have a complete answer. The problem remains open for arbitrary search problems but it is clear that both, ....

M. Cadoli, T. Eiter, and G. Gottlob. Default logic as a query language. IEEE Transactions on Knowledge and Data Engineering, 9(3), pages 448-463.

....we needed to run # P 2 hard problems on the DLP system at hand. Thus, we have compared the performance of the di#erent model checking methods by running various instances of the # P 2 complete problem Strategic Companies on the dlv system. The strategic companies problem is from [ Cadoli et al. 1997 ] it is, to the best of our knowledge, the only # P 2 complete KR problem from the business domain. No experimental results for any # P 2 complete KR problems are known. Briefly, a holding owns companies, each of which produces some goods. Moreover, several companies may have joint ....

.... by(C,C1,C2,C3) # strategic(C1) # strategic(C2) # strategic(C3) Here strategic(C) means that C is strategic, produced by(P, C1, C2) that product P is produced by companies C1 and C2, and controlled by(C, C1, C2, C3) that C is jointly controlled by C1, C2 and C3; we have adopted here from [ Cadoli et al. 1997 ] that each product is produced by at most two companies and each company is under joint control of at most three other companies. The problem is to find out the set of all strategic companies (i.e. under brave reasoning, for which C the fact strategic(C) is true) Note that this problem cannot ....

M. Cadoli, T. Eiter, G. Gottlob. (1997), Default Logic as a Query Language. IEEETKDE, 9(3):448-463.

.... has been pointed out as a necessary requirement for knowledge representation and commonsense reasoning [2,18,24] This view has been confirmed by results which prove that without disjunction, the expressive capability of logic programming is limited such that relevant problems can not be expressed [15,7]. Disjunctive logic programming (DLP) has a very high expressive power. In [16] it is proved that, under stable model semantics, disjunctive programs capture the complexity class P 2 , Work partially supported by the Austrian Science Fund (FWF) under grants N Z29 INF, P12344 INF and ....

M. Cadoli and T. Eiter and G. Gottlob. Default Logic as a Query Language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June, 1997.

....classes in which they lie. This last result also abstracts from the previously used graph theoretic setting, since the Conversion Lemma works on binary encodings of arbitrary data types. Subsequently, the notion of succinct representation and variants of the results from [2] have been employed in [4] and in [7] to characterize the computational complexity of a number of problems from Logic Programming and Databases; specifically, the combined complexity of queries based on Default Logic and certain issues arising from disjunctive Datalog queries under various semantics, respectively. Similar ....

M. Cadoli, T. Eiter, G. Gottlob: "Default logic as a query language". In: Proc. 4th Int. Conf. Principles of Knowledge Representation and Reasoning, 1994.

....that has been developed for representing and reasoning with default knowledge [Rei80] to formalise the use of general rules in deductive databases, to give what we call default databases. Previously, Cadoli et al. have proposed default logic as a query language for finite relational databases [CEG94] We will adopt the same language, which they call default query language (DQL) for our default databases. In this paper, we provide an overview of default logic, motivate its applicability to capturing general rules in databases, for generating default facts, and then develop a framework for ....

....that contain no quantifiers or functions, though they may contain unquantified variables. Each default rule is a scheme where variables can be grounded only with constants. The set of constants is the set of attribute values used in the EDB. This language is the query language proposed in [CEG94] Now we have defined the components, we can define the notion of a default database. A default database is a default theory (D; W ) where D is a default rules database, and W is the union of an extensional database and an intentional database. Consider the following example where the extensional ....

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M Cadoli, T Eiter, and G Gottlob. Default logic as a query language. In J Doyle, E Sandewall, and P Torasso, editors, Proceedings of the Fourth International Conference Principles of Knowledge Representation and Reasoning, pages 473--484. Morgan Kaufmann, 1994.

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M. Cadoli, T. Eiter, and G. Gottlob. Default logic as a query language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June 1997.

....we associate with the first level) We next show that also harder problems, located at the second level of the Polynomial Hierarchy, can be encoded by the GCO technique. To this end, we consider the following problem, which is known under the name Strategic Companies. Definition 3. 5 (STRATCOMP [15]) Suppose there is a collection C = fc 1 ; c m g of companies c i owned by a holding, a set G = fg 1 ; g n g of goods, for each c i a set G i G of goods produced by c i , and a set O i C of companies controlling (owning) c i . This control can be thought of as a majority in ....

....strategic, if it belongs to some strategic set of C . Computing the set of all strategic companies is relevant when companies should be sold, as selling any company which is strategic would for sure lead to a violation of any of conditions (1) and (2) This problem is 2 hard in general [15]; reformulated as a decision problem ( Given a particular company c in the input, is c strategic ) it is 2 complete. To our knowledge, it is one of the rare KR problems from the business domain of this complexity that have been considered so far. In the following, we adopt the setting ....

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M. Cadoli, T. Eiter, and G. Gottlob. Default Logic as a Query Language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June 1997.

....all goods can be still produced, and that no company is sold which would still be controlled by the holding after the transaction. A company is strategic, if it belongs to a strategic set, which is a minimal set of companies satisfying these constraints. This problem is P 2 hard in general (Cadoli, Eiter, Gottlob 1997); reformulated as a decision problem ( Given a further company c in the input, is c strategic ) it is P 2 complete. To our knowledge, it is the only KR problem from the business domain of this complexity that has been considered so far. In the following encoding, strat(X) means that X is ....

....been considered so far. In the following encoding, strat(X) means that X is strategic, company(X) that X is a company, produced by(X; Y; Z) that product X is produced by companies Y and Z, and controlled by(W; X; Y; Z) that W is jointly controlled by X; Y and Z. We have adopted the setting from (Cadoli, Eiter, Gottlob 1997) where each product is produced by at most two companies and each company is jointly controlled by at most three other companies. Given the facts F for company, controlled by and produced by, the answer sets of the following program P1 (actually P1 [ F ) correspond one to one to the strategic ....

Cadoli, M.; Eiter, T.; and Gottlob, G. 1997. Default Logic as a Query Language. IEEE Transactions on Knowledge and Data Engineering 9(3):448-463.

....2 . The proof in [29] establishes that these results hold even if W is a universal theory and no quanti ers occur in D. Moreover, like in the propositional case, they hold if in addition the default theory is normal and W is a consistent set of literals (this is immediate from [29] and results in [8]) The polynomial time transformations of default reasoning to the main credulous default abduction tasks in 1 3 established the hardness parts of Theorem 4.6. Clearly, these transformations also work for rst order theories. Therefore, from the complexity results for rst order default ....

M. Cadoli, T. Eiter, and G. Gottlob. Default Logic as a Query Language. In Proc. KR-94, pp. 99-108, 1994. Full paper to appear in IEEE Transactions on Knowledge and Data Engineering.

.... and true negation has been pointed out as a necessary requirement for knowledge representation [16, 2] This view has been confirmed by results which prove that without disjunction, the expressive capability of logic programming is limited such that relevant problems can not be expressed [12, 7]. Moreover, disjunction has been recognized as an important feature for a declarative KR language, which allows to express knowledge in a simple and natural way. For example, the rule genotype(P,T1) v genotype(P,T2) parent(C,P) heterozygot(C,T1,T2) from a blood group knowledge base 1 may ....

....for the Smodels program, which is then translated into default rules using the translation which maps a rule A : GammaB 1 ; B n ; not C 1 ; not C k into the default B1 Delta Delta DeltaB n : notC1 ; notC k A . Strategic Companies. The strategic companies problem is from [7]; it is, to the best of our knowledge, the only Sigma P 2 complete KR problem from the business domain. No experimental results for any Sigma P 2 complete KR problems are known. Briefly, a holding owns companies, each of which produces some goods. Moreover, several companies may have ....

[Article contains additional citation context not shown here]

M. Cadoli, T. Eiter, and G. Gottlob. Default Logic as a Query Language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June 1997.

....a key role. We may use a literal translation of Phi into a dlv program P : d 11 : d 1c 1 : d n1 : d nc n : It is easy to see that the stable models of P correspond 1 1 to the prime implicants of Phi. Example 4 (Strategic Companies) The strategic companies problem is from [3]; it is, to the best of our knowledge, the only Sigma P 2 complete KR problem from the business domain. No experimental results for any Sigma P 2 complete KR problems are known. Briefly, a holding owns companies, each of which produces some goods. Moreover, several companies may have ....

.... strategic(C1) strategic(C2) strategic(C3) Here strategic(C) means that C is strategic, produced by(P; C1; C2) that product P is produced by companies C1 and C2, and controlled by(C; C1; C2; C3) that C is jointly controlled by C1; C2 and C3; we have adopted here from [3] that each product is produced by at most two companies and each company may be controlled by consortia of at most three other companies each. The problem is to find out the set of all strategic companies (i.e. under brave reasoning, for which C the fact strategic(C) is true) Note that this ....

M. Cadoli, T. Eiter, and G. Gottlob. Default Logic as a Query Language. IEEE Transactions on Knowledge and Data Engineering, 9(3):448--463, May/June 1997.

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M. Cadoli, T. Eiter, and G. Gottlob. Default logic as a query language. IEEE Transactions on Knowledge and Data Engineering, 9(3), pages 448-463.

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