Kautz, H., Kearns, M.J., Selman, B.S., 1993. Reasoning with Characteristic Models, Proceedings of AAAI-93, 34-39.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....used for finding explanations. Given a Horn theory #, called the background theory, a formula q (called query) and a set of literals A Lit, an explanation of q w.r.t. A is a minimal set of literals E over A such that (i) # q, and (ii) # is satisfiable. As shown by Kautz et al. [31], it is possible to generate an explanation for a query q that is an atom, w.r.t. a given set A = S # p S of literals over a given subset S of the atoms, in polynomial time, if # is semantically represented by the set of its characteristic models, char(#) under syntax based ....

....the problem is intractable [42] A model m of # is characteristic, if m Cl# (mod(#) v ) where Cl# (S) denotes for any set of models S its closure under intersection, i.e. the least set of models S # s such that M,M # S # implies M S # . The polynomial abduction algorithm in [31] implicitly involves the computation of a minimal transversal of a hypergraph, which is the bulk of the computation e#ort. This result has been extended in [13] to the following result. Theorem 10. Given the characteristic set char(#) of a Horn theory #, a query literal q, and a subset A of all ....

H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proc. AAAI-93, pp. 34--39, 1993.

....is that deciding whether KB ff holds is intractable in already plain settings; e.g. in the propositional context, it is a well known co NP complete problem. More recently, model based reasoning has been proposed as an alternative form of representing and accessing a logical knowledge base, cf. [13, 24, 25, 26, 29, 30]. It can be seen as an approach towards Levesque s notion of vivid reasoning [31] which asks for a more straight representation of a knowledge base, from which common sense reasoning is easier and more suitable than from the traditional one. In model based reasoning, KB is represented by a ....

....than by a set of formulas. Reasoning from KB becomes then as easy as to test, given a query ff, whether ff is true in all models of S. For suitable ff, this can be decided efficiently. Moreover, it has also been shown that abduction from a KB represented by its characteristic models is polynomial [24, 25, 29], while this problem is intractable under formula representation [38, 15] This time speed up comes at the price of space; indeed, the formula based and the model based approach are orthogonal, in the sense that while a KB may have small representation in one formalism, it has an exponentially ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proc. AAAI-93, 1993.

....in polynomial time, as studied by Selman and Levesque [20] We contrast formula based (syntactic) with model based (semantic) representation of Horn theories. The latter form of representation, where a Horn theory is represented by the characteristic models, was advocated by Kautz et al. [12]. As they showed, important inference problems are tractable in the model based setting. Namely, whether a Horn theory logically implies a CNF , and whether a query q has an explanation w.r.t. an assumption set A, as well as computing one; note that the latter is intractable under formula based ....

....w.r.t. an assumption set A, as well as computing one; note that the latter is intractable under formula based representation. Similar results were shown for other theories by Khardon and Roth [14] We investigate the role of syntax for computing abductive queries. In the framework of [20, 12], the query is a positive letter q. However, it is of equal interest to consider negative queries as well, i.e. to explain the complement q of an atom q. Since the Horn property imposes semantic restrictions on theories, it is not straightforward to express such negative queries in terms of ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings AAAI-93, pages 34--39, 1993.

....problem inverse satisfiability. Besides its fundamental nature, there are many more factors that make inverse satisfiability a most interesting problem. A major motivation comes from AI (in fact, what we call here the inverse satisfiability problem is implicit in much of the recent AI literature [Ca93, DP92, KKS95, KKS93, KPS93]) A set of models such as those in Figure 1(b) can be seen as a state of knowledge. That is, it may mean that at present, for all we know, the state of our three variable world can be in any one of the three states indicated. In this context, formula OE is some kind of knowledge representation. ....

H. A. Kautz, M. J. Kearns, and B. Selman, Reasoning with characteristic models, in Proc. 11th National Conference on Artificial Intelligence, Washington, DC, AAAI Press, Menlo Park, CA, 1993, pp. 34--39.

.... in the latter setting, computing a maximum (in terms of the numbers of models) Horn core is co NP hard [15] In this paper, we consider the issue of computing Horn cores of the disjunction = S l i=1 i of Horn theories i , represented either by Horn CNFs, or by their characteristic models [13]. Characteristic models have been proposed as a model based alternative to the formula based theory representation. These two approaches are orthogonal with respect to space requirements, in the sense that one approach sometimes allows for an exponentially smaller representation than the other; ....

....of some Horn core. In fact, we can interrupt the algorithm at any step, and the contents of is a Horn theory that implies . The approximation improves with iterations, and eventually yields a Horn core. 2 13 4 Characteristic Models For a Horn theory , a model v 2 is called characteristic [13], if v 62 Cl ( nfvg) holds. The set of all characteristic models of , the characteristic set of , is denoted by C ( Note that every Horn theory has the unique characteristic set C ( and that max( C ( For example, a Horn theory = f(0101) 1001) 1000) 0001) ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proceedings AAAI93, pages 34-39, 1993.

....prove intractability (unless P=NP) for each of the problems mentioned above. For a concise overview, see Table 1 in Section 6. Besides the 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, ....

....Cl ( holds, where Cl (S) is the closure of S f0; 1g n under bitwise AND (i.e. intersection) of models v and w, denoted by v V w. Observe that any Horn theory has the least (unique smallest) model, which is given by V v2 v. For a Horn theory , a model v 2 is called characteristic [14], if v 62 Cl ( n fvg) holds. The set of all characteristic models of , the characteristic set of , is denoted by C ( Note that every Horn theory has the unique characteristic set C ( For example, the theory = f(0101) 1001) 1000) 0001) 0000)g is Horn, and has C ( ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proceedings AAAI-93, pages 34-39, 1993.

....or prove intractability (unless P=NP) for each of the problems mentioned above. For a concise overview, see Table 1. ffl Besides 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 ....

.... 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 ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proc. AAAI '93, 1993.

....P. We conjecture that (b) is coNP complete and that (c) is Sigma p 2 complete. Another interesting open problem is this: Given a Horn formula, how hard is it to generate its characteristic models, that is, a minimal set of models M such that sat(OE) M . Characteristic models were shown in [KKS2] to be important alternative representations of a Horn formula. We suspect that this dual problem is also equivalent to the transversal problem. Finally, it would be interesting if the insights into Horn formulae presented in this paper could lead to improved algorithms for learning Horn ....

H. A. Kautz, M. J. Kearns, B. Selman "Reasoning with characteristic models," to appear in AAAI, 1993.

....is a very interesting open problem. The most satisfying (and, in our view, likely) way of overcoming it is by developing a polynomial time algorithm for updating Horn formulas by clauses in Winslett s formalism. 32 Incremental Recompilation of Knowledge 4. 2 Characteristic Model Approximation [16] introduced an interesting alternative way of representing Horn formulae, namely, characteristic models. Let Gamma be a Horn formula, and let H be its set of models. It is easy to see that H = H , where H is the smallest set that contains H and is closed under componentwise multiplication ....

....is closed under componentwise multiplication (AND) of its models; that is, iff h 1 ; h 2 2 H implies h 1 AND h 2 2 H . This raises the possibility of the following alternative representation of H : We represent it by a minimal set of models C such that C = H( H ) This was first proposed in [16]; they called this set C the set of characteristic models of H , and they showed that it is exactly the set of all elements of H that cannot be represented as the AND of any subset of H . There are Horn sets that can be represented much more succinctly by characteristic models than by formulae, ....

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H. A. Kautz, M. J. Kearns, B. Selman "Reasoning with Characteristic Models," Proceedings of 1993 AAAI, pp. 34--39, 1993.

....conversion problem arises in the context of reasoning. Given a knowledge base that can be represented as a conjunction of propositional Horn clauses with empty consequents, an efficient solution to the conversion problem could be used to efficiently generate a collection of characteristic models [24, 28] to use in various reasoning tasks (for example, determining whether a query is entailed by a knowledge base) 23, 26] 5 The conversion problem is also related to the problem of determining if a version space has converged. For a concept class C the version space [33] induced by positive example ....

H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the 11th National Conference on Artificial Intelligence, pages 34--39, Washington, DC, 1993. AAAI Press.

....transformations as this will usually find an acceptable theory, even when it is intractable to find an optimal one. Our negative results may inspire future researchers and developers to look for other techniques to modify existing theories, perhaps by changing the underlying representation [KKS93, KR94b] or by exploiting other information that may be available, such as the assumption (if true) that each training example includes only the information required to classify that instance [GGK97] The appendix supplies the relevant proofs. We close this section by describing related research. Related ....

Henry Kautz, Michael Kearns, and Bart Selman. Reasoning with characteristic models. In AAAI-93, pages 34--39, 1993.

.... in AI [11] However, it is known that deduction from a set of propositional clauses is co NP complete and abduction is NP complete [13] Recently, an alternative way of representing knowledge, i.e. by a subset of its models, which are called characteristic models, has been proposed (see e.g. [6, 7, 8, 9]) Deduction from a knowledge base in this model based approach can be performed in linear time, and abduction is also performed in polynomial time [6] In this paper, we propose yet another method of knowledge representation, i.e. the use of ordered binary decision diagrams (OBDDs) 1, 2, 12] ....

.... way of representing knowledge, i.e. by a subset of its models, which are called characteristic models, has been proposed (see e.g. 6, 7, 8, 9] Deduction from a knowledge base in this model based approach can be performed in linear time, and abduction is also performed in polynomial time [6]. In this paper, we propose yet another method of knowledge representation, i.e. the use of ordered binary decision diagrams (OBDDs) 1, 2, 12] An OBDD is a directed acyclic graph representing a Boolean function, and can be considered as a variant of decision trees. By restricting the order of ....

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H.A. Kautz, M.J. Kearns, and B. Selman, \Reasoning with Characteristic Models," in Proc. of AAAI-93, pp.34-39, 1993.

....1 1 Introduction The traditional form of representing knowledge in AI is through logical formulas [McC58, MH69] where all the logical conclusions of a given formula are assumed to be accessible to an agent. Recently, an alternative way of capturing such information has been developed [KKS93, KR94]. Instead of using a logical formula, the knowledge representation is composed of a particular subset of its models, the set of characteristic models. This set retains all the information about the formula, and is useful for various reasoning tasks. In particular, using model evaluation with the ....

....which are in Horn form and to their representation as characteristic models. The characteristic models of Horn formulas have been shown to be useful. There is a linear time deduction algorithm using this set, and abduction can be performed in polynomial time, while using formulas it is NP Hard [KKS93]. Furthermore, an algorithm for default reasoning using characteristic models has been developed, for cases where formula based algorithms are not known [KR95] Hence, the question arises, whether one can efficiently translate a Horn formula into its set of characteristic models and then use this ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39, 1993.

....form of the database, e.g. clausal, Horn, etc. 3 Nothing in the foundational approach requires however basic beliefs to be specified in this way. Why should the notion of basic beliefs, and foundational revision more generally, be incompatible with, say, a model based representation of beliefs [KKS93] which dispenses with axiom sets altogether syntax independence. Rather, different agents will be seen as using different foundational operators at various times. This is the approach implicitly followed in the mental situation calculus of [dVS94b] mentioned in the introduction, a language ....

Henry Kautz, Michael Kearns, and Bart Selman. Reasoning with characteristic models. In Proceedings of the Eleventh American National Conference on Artificial Intelligence, pages 34--39, 1993.

.... setting, computing a maximum (in terms of the numbers of models) Horn core is co NP hard [15] In this paper, we consider the issue of computing Horn cores of the disjunction Sigma = S l i=1 Sigma i of Horn theories Sigma i , represented either by Horn CNFs, or by their characteristic models [12]. Characteristic models have been proposed as a model based alternative to the formula based theory representation. These two approaches are orthogonal with respect to space requirements, in the sense that one approach sometimes allows for an exponentially smaller representation than the other; ....

....core. In fact, we can interrupt the algorithm at any step, and the contents of is a Horn theory which implies . The approximation improves with computation time, and eventually yields a Horn core. 2 4 Characteristic Models For a Horn theory Sigma, a model v 2 Sigma is called characteristic [12], if v 62 Cl ( Sigma n fvg) holds. The set of all characteristic models of Sigma, the characteristic set of Sigma, is denoted by C ( Sigma) Note that every Horn theory Sigma has the unique characteristic set C ( Sigma) and that max( Sigma) C ( Sigma) For example, a Horn theory ....

H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proceedings AAAI-93, pages 34--39, 1993.

....for all OE 2 L we have that Sigma implies OE if and only if r Sigma satisfies OE. That is, the semantic implication relation for Sigma reduces to the truth in a single example relation r Sigma . The second area is in automated reasoning, where reasoning with models has been recently studied [KKS93, KR94b] Assume the language consists of propositional formulae, and the objects are models (i.e. truth assignments) A set of models M O is a sufficient set of examples for Sigma and L, if for all OE 2 L we have: if for all m 2 M OE is true in m, then Sigma implies OE. That is, instead of ....

....has this property. In these applications it is of course important that the example relation or the set of examples is small; otherwise there is no sense in using example based deduction. The size issue has been treated for database dependencies in [BDFS84, MR86] and for propositional formulae in [KKS93, KR94b] In this paper we show that these two application areas of the general idea of example based reasoning are actually very closely related. Namely, we show that the characteristic models of [KKS93] are essentially the intersection generators for sets of functional dependencies in [BDFS84] ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39, 1993.

....tree and in some cases the set of models is much too large to be represented explicitly. The interest of this, is that for certain kind of information the model based representation is much more compact and enable much faster reasoning than the traditional representation using logical formulas[6,7]. In this paper, we show how symmetries can be used to represent the set of models by a subset of characteristic models(non symmetric models) from which all others can be generated. We present an algorithm for enumerating all non symmetric models, and we show results obtained on some known ....

H. A. Kautz, M. J. Kearns, and B. Selman. Reasoning with characteristic models. In procedings of AAAI-93, pages 34--39, 1993.

....tutorial relatively small, we have had to skip many other large bodies of techniques. For example, we have not considered techniques that radically change the representation, perhaps by re expressing the information as a neural net [TS93] or by considering the characteristic models of the theory [KKS93, KR94]. We also chose to focus on techniques specific to reasoning, and so by pass the huge inventory of techniques associated with improving the efficiency of computations, in general including clever compilation techniques, and exploiting parallel algorithms. Needless to say, these techniques are ....

Henry Kautz, Michael Kearns, and Bart Selman. Reasoning with characteristic models. In AAAI-93, pages 34--39, 1993.

....major weakness of our scheme. Overcoming it is a very interesting open problem. The most satisfying (and, in our view, likely) way of overcoming it is by developing a polynomial time algorithm for updating Horn formulas by clauses in Winslett s formalism. 4. 2 Characteristic Model Approximation Kautz, Kearns, and Selman (1993, 1995) introduced an interesting alternative way of representing Horn formulae, namely, characteristic models. Let Gamma be a Horn formula, and let H be its set of models. It is easy to see that H = H , where H is the smallest set that contains H and is closed under component wise ....

....under component wise multiplication (AND) of its models; that is, iff h 1 ; h 2 2 H implies h 1 AND h 2 2 H . This raises the possibility of the following alternative representation of H : We represent it by a minimal set of models C such that C = H( H ) This was first proposed by Kautz, Kearns, and Selman (1993); they called this set C the set of characteristic models of H , and they showed that it is exactly the set of all elements of H that cannot be represented as the AND of any subset of H . There are Horn sets that can be represented much more succinctly by characteristic models than by formulae, ....

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Kautz, H. A., Kearns, M. J., & Selman. B (1993). Reasoning with Characteristic Models.

....size of the original knowledge base. For each entry of Tables 1 and 2 marked non compilable , we proved that not only no efficient basis exists but also that no other compilation is possible. The possibility of rewriting off line a propositional theory, was addressed by Kautz, Kearns Selman in [21]. In particular, they investigate the reformulation of a Horn formula into the set of its characteristic models, where characteristic models are independent models that cannot be obtained as intersection of the others. As it turns out, the compactness of the representation using Horn clauses and ....

H. A. Kautz, M. J. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the Eleventh National Conference on Artificial Intelligence (AAAI-93), pages 34--39, 1993.

....and mistake bound learning algorithms with (exact and approximate) reasoning algorithms. We then consider Learning to Reason algorithms that use models (satisfying assignments) as the knowledge representation language. A characterization of reasoning with models for Horn theories was given in [KKS93] and for general theories in [KR94c] We build on these results to exemplify the usefulness of the new approach: ffl Consider the reasoning problem W j= ff, where W is some CNF formula and ff is a log nCNF (i.e. a CNF formula with at most log n literals in each clause) Then, when W has a ....

....using the set ae. Proof: i) is immediate from Theorem 6.3, since g = ae H lub . For (ii) notice that if the query ff is a Horn query, then model based reasoning with ae is correct, by Theorem 6.4. For correct reasoning with general queries, it is possible to use the reasoning algorithm from [KKS93]. 7 Learning to Reason with Horn Queries In previous sections (e.g, Theorem 6.5, Theorem 6.6) we have shown that for various worlds we can learn to reason with respect to all common queries. We give here an orthogonal result that shows that we can learn to reason from every Boolean function ....

H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39, 1993.

....problem arises in the context of reasoning. Given a knowledge base that can be represented as a conjunction of propositional Horn clauses (with empty consequents) an efficient solution to the conversion problem could be used to efficiently generate a collection of characteristic models [KKS93, KR94] to use in various reasoning tasks (for example, determining whether a query is entailed by a knowledge base) Kha95, KMR95] The conversion problem is also related to the problem of determining if a version space has converged. For a concept class C the version space [Mit82] induced by ....

H. A. Kautz, M. J. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the 11th National Conference on Artificial Intelligence, pages 34--39, Washington, DC, July 1993. AAAI Press.

.... to reasoning from a cognitive point of view and indeed, many of the proponents of this approach to reasoning have been cognitive psychologists [12, 13, 22] In the AI community this approach can be seen as an example of Levesque s notion of vivid reasoning and has already been studied in [14]. The deduction problem KB j= ff can be approached using the following model based strategy: Test Set: A set S of possible assignments. Test: If there is an element x 2 S which satisfies KB, but does not satisfy ff, deduce that KB 6j= ff; Otherwise, KB j= ff. Since, by the model theoretic ....

....queries. For a wide class of queries we show that exact reasoning can be done efficiently, even when the reasoner keeps in KB an approximate representation of the world . We show that the theory developed here generalizes the model based approach to reasoning with Horn expressions, suggested in [14], and captures even the notion of reasoning with theory approximations [33] In particular, our results characterize the Horn expressions for which the approach suggested in [14] is useful and explain the phenomena observed there, regarding the relative sizes of the logical formula representation ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39, 1993.

....Q and can therefore be handled by the same model based representation. We note that this simple approach to dealing with context can be extended to handle a restricted case of default reasoning in the sense defined by Reiter (Reiter 1980) This can be done using results on abduction with models (Kautz, Kearns, Selman 1993; Khardon Roth 1994b) and results on the relation between abduction and default reasoning (Reiter 1987; Selman 1990) The extension holds for the simple form called elementary defaults which consist of rules of the form: if it is consistent to assume l then assume l. A Sampling Approach Let ....

Kautz, H.; Kearns, M.; and Selman, B. 1993. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, 34--39.

....is a wide class of queries for which reasoning is very efficient and exact, even when the model based representation KB provides only an approximate representation of the world . Moreover, we show that the theory developed here generalizes the model based approach to reasoning with Horn theories [KKS93] and captures even the notion of reasoning with Hornapproximations [SK91] Our result characterizes the Horn theories for which the approach suggested in [KKS93] is useful and the phenomena observed there, regarding the relative sizes of the formula based representation and model based ....

....of the world . Moreover, we show that the theory developed here generalizes the model based approach to reasoning with Horn theories [KKS93] and captures even the notion of reasoning with Hornapproximations [SK91] Our result characterizes the Horn theories for which the approach suggested in [KKS93] is useful and the phenomena observed there, regarding the relative sizes of the formula based representation and model based representation of KB is explained and put in a wider context. This paper is available from the Center for Research in Computing Technology, Division of Applied Sciences, ....

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H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39, 1993.

....with its model based representation yields an abductive explanation in time that is polynomial in its size. We give application of the theory to other problems in reasoning, and in particular show that the theory developed here generalizes the model based approach to reasoning with Horn theories (Kautz, Kearns, and Selman, 1993), and captures even the notion of reasoning with Horn approximations (Selman and Kautz, 1991) Finally, we discuss some robustness issues of model based representations and show how they support an incremental approach to reasoning. Learning to Reason: Deduction In Chapter 5 we exhibit the ....

....the notion of reasoning from examples on a qualitative basis. In the AI community this approach can be seen as an example of Levesque s notion of vivid reasoning mentioned above, is somewhat related to Minsky s frames theory (Minsky, 1975) and has already been studied in a narrower context in (Kautz, Kearns, and Selman, 1993). The problem KB j= ff can be approached using the following model based strategy: Algorithm MBR: Test Set: A set S of possible assignments. Test: If there is an element x 2 S which satisfies KB, but does not satisfy ff, deduce that KB 6j= ff; Otherwise, KB j= ff. Figure 4.1: MBR: Model Based ....

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Kautz, H., M. Kearns, and B. Selman. 1993. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39.

....Boolean Formulas (QBF) Yet another related line of research deals with reasoning using models. The model checking approach to reasoning was advocated in (Halpern and Vardi, 1991) Among the related works are those of Khardon and Roth (Khardon and Roth, 1994) and Kautz, Kearns and Selman (Kautz et al. 1993). The work of Cadoli and Schaerf (Cadoli and Schaerf, 1992) on approximating inference, is also relevant. 6 conclusions We started with a simple algorithm DP UD for finding a model for prioritized propositional circumscriptive theories and proved that it works for the tasks of finding a minimal ....

Kautz, H. A., Kearns, M. J., and Selman, B. (1993). Reasoning with characteristic models. In Proceedings AAAI 93, pages 34--39.

....is that deciding whether KB ff holds is intractable in already plain settings; e.g. in the propositional context, it is a well known co NP complete problem. More recently, model based reasoning has been proposed as an alternative form of representing and accessing a logical knowledge base, cf. [13, 24, 25, 26, 29, 30]. It can be seen as an approach towards Levesque s notion of vivid reasoning [31] which for asks for a more straight representations of a knowledge base, from which common sense reasoning is easier and more suitable than from the traditional one. In model based reasoning, KB is represented by a ....

....than by a set of formulas. Reasoning from KB becomes then as easy as to test, given a query ff, whether ff is true in all models of S. For suitable ff, this can be decided efficiently. Moreover, it has also been shown that abduction from a KB represented by its characteristic models is polynomial [24, 25, 29], while this problem is intractable under formula representation [38, 15] This time speed up comes at the price of space; indeed, the formula based and the model based approach are orthogonal, in the sense that while a KB may have small representation in one formalism, it has an exponentially ....

[Article contains additional citation context not shown here]

H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proc. AAAI-93, 1993.

....using the set ae. Proof: i) is immediate from Theorem 6.3, since g = ae H lub . For (ii) notice that if the query ff is a Horn query, then model based reasoning with ae is correct, by Theorem 6.4. For correct reasoning with general queries, it is possible to use the reasoning algorithm from [KKS93]. 9 While the oracle RQ has not been studied elsewhere, it is similar to superset queries that have been considered in the literature [Ang88] 7 Learning to Reason with Horn Queries Model based representations are not unique in supporting Learning to Reason. In this section we exhibit another ....

H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In Proceedings of the National Conference on Artificial Intelligence, pages 34--39, 1993.

....R. Khardon and D. Roth Proof. i) is immediate from Theorem 4, since g = ae H lub . For (ii) notice that if the query ff is a Horn query, then model based reasoning with ae is correct, by Theorem 5. For correct reasoning with general queries, it is possible to use the reasoning algorithm from [Kautz et al. 1993]. 7. LEARNING TO REASON WITH HORN QUERIES Model based representations are not unique in supporting Learning to Reason. In this section we exhibit another instance of Learning to Reason in which (formulabased) reasoning is computationally hard. The corresponding learning problem has not been ....

Kautz, H., Kearns, M., and Selman, B. 1993. Reasoning with characteristic models. In Proc. of the National Conference on Artificial Intelligence (1993), pp. 34--39.

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Kautz, Henry; Kearns, Michael; and Selman, Bart 1993. Reasoning with characteristic models. Technical report, AT&T Bell Laboratories, Murray Hill, NJ.

....representations, and stop when the smaller one is completed. The characteristic models in a closed set can be efficiently found by selecting each model which is not equal to the intersection of any two models in the set. The clausal theory can be found using the algorithms described in [5] and [11]. We will return to the problem of generating efficient representations in Section 6 below. 4 Deduction using Characteristic Models One of the most appealing features of Horn theories is that they allow for fast inference. In the propositional case, queries can be answered in polynomial time [6] ....

H. Kautz, M. Kearns, and B. Selman, Reasoning with characteristic models, Proceedings AAAI-93 , Washington, DC (1993) 34--39.

....checking of Horn formulas (see [ Dowling and Gallier, 1984 ] and because of its connection to logic programming. Unfortunately, not every set can be represented by Horn formulas. CM denotes the class of all sets generated by a set of characteristic models. The idea originated in [ Kautz et al. 1993 ] Since every Horn set is closed under bitwise multiplication, it makes sense to try to represent a Horn set not as the set of all its models but as the set of its characteristic models, so that the original set is obtained as the closure of the set of characteristic models under bitwise ....

....vectors rep 2 The idea is later generalized in [Khardon and Roth, 1994] to capture non Horn sets, and was successfully applied to model based reasoning. resentable as bitwise product of some vectors from M . The set of models of a set of characteristic vectors M is precisely closure(M ) In [ Kautz et al. 1993 ] it has been proved that H and CM are orthogonal, i.e. there are situations when one is better than the other for compact knowledge representation. However, in Section 4 we give an intriguing method for simulating CM by H: It is also proved in [ Kautz et al. 1993 ] that abduction in CM can be ....

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H.A. Kautz, M.J. Kearns, B. Selman. Reasoning with characteristic models. Proceedings of AAAI, 1993.

....ff be a formula in conjunctive normal form. It is possible to determine if S j= ff in time O(jM j Delta jffj 2 ) where jM j is the total length of the representation of M . Finally, using more sophisticated data structures we can bring the complexity down to truely linear time, O(jM j jffj) [Kautz et al. 1993] . 5 Abduction using Characteristic Models Another central reasoning task for intelligent systems is abduction, or inference to the best explanation [Peirce, 1958] In an abduction problem, one tries to explain an observation by selecting a set of assumptions that, together with other background ....

Kautz, Henry; Kearns, Michael; and Selman, Bart 1993. Reasoning with characteristic models. Technical report, AT&T Bell Laboratories, Murray Hill, NJ.

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Kautz, H., Kearns, M.J., Selman, B.S., 1993. Reasoning with Characteristic Models, Proceedings of AAAI-93, 34-39.

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H. Kautz, M. Kearns, and B. Selman. Reasoning With Characteristic Models. In Proc. 11th National Conference on Artificial Intelligence (AAAI-93), pp. 34--39, 1993.

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Kautz, H., Kearns, M.J., Selman, B.S., 1993. Reasoning with Characteristic Models, Proceedings of AAAI-93, 34-39.

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Henry Kautz, Michael Kearns, and Bart Selman. Reasoning with characteristic models. In Proceedings of the Eleventh Conference of the AAAI, pages 34--39, 1993.

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H. Kautz, M. Kearns, and B. Selman. Reasoning with characteristic models. In AAAI-93, pages 34--39. S. M. Kosslyn. Image and Mind. Harvard Press, 1983.

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