T. Eiter, T. Ibaraki, and K. Makino. Computing intersections of Horn theories for reasoning with models. Artificial Intelligence, 110(1-2):57--101, 1999. Home/Search Document Details and Download
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On the Difference of Horn Theories (Extended Abstract) - Eiter, Ibaraki, Makino (Correct)
.... the familiar representation in terms of Horn CNFs, we also consider the model based representation of Horn theories through their sets of characteristic models [10] This alternative has also been studied repeatedly, since it offers advantages to formula based representation in certain cases; see [13, 10, 4] for more details. Our results on the complexity of these issues are summarized in Table 1, which gives a complete picture of the tractability intractability frontier of these problems. The table also shows results on the Horn envelope [11, 12] of the difference, i.e. the (unique) least Horn ....
.... Sigma 2 be Horn theories, and let S 1 as in (3.8) Then Cl (S 1 ) Sigma 2 = holds if and only if Sigma 1 n Sigma 2 is a Horn theory. Furthermore, if Sigma 1 n Sigma 2 is a Horn theory, then Cl (S 1 ) Sigma 1 n Sigma 2 (i.e. C (S 1 ) C ( Sigma 1 n Sigma 2 ) holds. It is known [4] that given Q 1 ; Q 2 f0; 1g n , deciding Cl (Q 1 ) Cl (Q 2 ) is possible in O(n(jQ 1 j jQ 1 j) time. Since Sigma 2 = Cl (C ( Sigma 2 ) we thus obtain from Lemma 2, turned into a straightforward algorithm, the following result. Theorem 2. Let Sigma 1 , Sigma 2 be Horn theories. ....
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T. Eiter, T. Ibaraki, and K. Makino. Computing Intersections of Horn Theories for Reasoning with Models. In: Proc. AAAI '98, pp. 292--297, 1998.
Abduction and the Dualization Problem - Eiter, Makino (2003) Self-citation (Eiter Makino) (Correct)
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T. Eiter, T. Ibaraki, and K. Makino. Computing intersections of Horn theories for reasoning with models. Artificial Intelligence, 110(1-2):57--101, 1999.
On Computing All Abductive Explanations - Eiter, Makino (2002) (2 citations) Self-citation (Eiter Makino) (Correct)
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T. Eiter, K. Makino, and T. Ibaraki. Computing intersections of Horn theories for reasoning with models. Artificial Intelligence, 110(1-2):57--101, 1999.
Disjunctions of Horn Theories and their Cores - Eiter, Ibaraki, Makino (2001) (1 citation) Self-citation (Eiter Ibaraki Makino) (Correct)
....(Horn envelope [15] which is the minimal Horn theory e such that logically implies e . Note that in general, di erent Horn cores may exist, while it is known that the Horn envelope is always unique (e.g. 21, 22, 15] Computing Horn envelopes and Horn cores has been investigated in [21, 22, 15, 2, 3, 4, 5, 10, 7, 1]. It has been shown that a Horn core of a theory , represented by a given CNF (conjunctive normal form) is computable in polynomial time with an oracle for NP [2, 3] and that all Horn cores can be generated with polynomial delay (i.e. the time between consecutive outputs is bounded by a ....
....holds. 2 Thus, computing a canonical Horn core is presumably dicult in general. We next point out that the diculty can be avoided if 2 is restricted in the following sense. Call a Horn theory sparse, if j j p(jC ( j) holds for some polynomial p. Then, based on the next lemma proved in [7], we have the following theorem. Lemma 4.3 Given the characteristic set C ( of a Horn theory f0; 1g n , the models of can be enumerated with O(n 2 jC ( j) time delay. Theorem 4.4 Given characteristic sets C ( 1 ) and C ( 2 ) of Horn theories 1 ; 2 f0; 1g n ....
[Article contains additional citation context not shown here]
T. Eiter, T. Ibaraki, and K. Makino. Computing Intersections of Horn Theories for Reasoning with Models. Articial Intelligence, 110:57-101, 1999.
On the Difference of Horn Theories - Eiter, Ibaraki, Makino (2000) Self-citation (Eiter Ibaraki Makino) (Correct)
.... familiar representation in terms of Horn CNFs, we also consider the modelbased representation of Horn theories through their sets of characteristic models [14, 15] This alternative has also been studied repeatedly, since it o ers advantages to formula based representation in certain cases; see [19, 18, 15, 6] for more details. Both formula based and model based representations allow polynomial time algorithms for many problems. In some cases, however, formula based representation is polynomial while model based representation is intractable, and vice versa. Thus, like with many other problems [15, ....
....15, 6] for more details. Both formula based and model based representations allow polynomial time algorithms for many problems. In some cases, however, formula based representation is polynomial while model based representation is intractable, and vice versa. Thus, like with many other problems [15, 19, 6], formula based and model based representations complement each other with respect to their tractability intractability pro le. Our results on the di erence of Horn theories have applications in di erent contexts. On one hand, we extend the results on propositional knowledge base (KB) ....
[Article contains additional citation context not shown here]
T. Eiter, T. Ibaraki, and K. Makino. Computing Intersections of Horn Theories for Reasoning with Models. Articial Intelligence, 110: 57-101, 1999.
Disjunctions of Horn Theories and their Cores - Eiter, Ibaraki, al. (1998) (1 citation) Self-citation (Eiter Ibaraki Makino) (Correct)
....a maximal Horn theory c , if we view theories as sets of models, and the least Horn upper bound (Horn envelope [11] which is the minimal Horn theory e , i.e. e 0 for any Horn theory 0 such that 0 . Computing Horn envelopes and Horn cores has been investigated in [10, 11, 2, 4, 8, 6, 1]. It has been shown that a Horn core of a given CNF formula is computable in polynomial time with an oracle for NP [2] and that computing a maximum (in terms of the numbers of models) Horn core is co NP hard [11] if the theory is given by the set of its models. In this paper, we consider ....
T. Eiter, T. Ibaraki and K. Makino. Computing Intersections of Horn Theories for Reasoning with Models. In Proc. AAAI '98, pp. 292--297, 1998.
Disjunction of Horn theories and their cores - Eiter, Ibaraki, Makino (1999) (1 citation) Self-citation (Eiter Makino Ibaraki) (Correct)
....(Horn envelope [15] which is the minimal Horn theory Sigma e Sigma such that Sigma logically implies Sigma e . Note that in general, different Horn cores may exist, while it is known that the Horn envelope is always unique. Computing Horn envelopes and Horn cores has been investigated in [14, 15, 2, 4, 5, 9, 7, 1]. It has been shown that a Horn core of a given CNF (conjunctive normal form) is computable in polynomial time with an oracle for NP [2] and that all Horn cores can be generated with polynomial delay, if the theory Sigma is given by the set of its models. However, in the latter setting, ....
....Horn core is presumably difficult in general. We next point out that the difficulty can be avoided if Sigma 2 is restricted in the following sense. Call a Horn theory Sigma sparse, if j Sigmaj p(jC ( Sigma)j) holds for some polynomial p( Delta) Then, based on the next lemma proved in [7], we have the following theorem. Lemma 4.7 Given the characteristic set C ( Sigma) of a Horn theory Sigma f0; 1g n , the models of Sigma can be enumerated with O(n 2 jC ( Sigma)j) time delay. Theorem 4.8 Given characteristic sets C ( Sigma 1 ) and C ( Sigma 2 ) of Horn ....
[Article contains additional citation context not shown here]
T. Eiter, K. Makino, and T. Ibaraki. Computing Intersections of Horn Theories for Reasoning with Models. Technical Report 9803, Institut fur Informatik, Universitat Gießen, Germany, April 1998. Abstract in: Proc. AAAI-98, pages 292--297, 1998.
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