R. Khardon. Translating between Horn representations and their characteristic models. Journal of Artificial Intelligence Research, 3:349--372, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and TRANS ENUM have a large number of applications in many areas of Computer Science, including Distributed Systems, Databases, Boolean Circuits and Artificial Intelligence. There, they have important applications in Diagnosis, Machine Learning, Data Mining, and Explanation Finding, see e.g. [11, 13, 24, 28, 32, 33, 36] and the references therein. Let us call a decision problem # TRANS HYP hard, if problem TRANS HYP can be reduced to it by a standard polynomial time transformation. Furthermore, # is TRANS HYP complete, if # is TRANS HYP hard and, moreover, # can be polynomially transformed into TRANS HYP; that ....

....i k , and X C i j C i 0 Dep(A) for all . k . Informally, X is a minimal key or prime implicant for attribute C i 0 . Theorem 7. Problem FD EQ for instances (A, D) where D is in MAK form is TRANS HYP complete. Some polynomial cases of FD EQ are given in [27] As shown in [33], FD EQ is related to similar problems involving charcteristic models and Horn CNFs. For these and further results about problems in data mining equivalent to TRANSENUM and TRANS HYP, see [33] 3.3 Model Based Diagnosis Di#erent from the heuristic approach, model based diagnosis [41, 8] takes a ....

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R. Khardon. Translating between Horn representations and their characteristic models. J. Artificial Intelligence Research, 3:349--372, 1995.

.... are like transversal computation known to be computationally equivalent to Dualization (see [13] are of interest in various areas such as database theory (e.g. 34, 43] machine learning and data mining (e.g. 4, 5, 10, 18] game theory (e.g. 22, 38, 39] artificial intelligence (e.g. [17, 24, 25, 40]) mathematical programming (e.g. 3] and distributed systems (e.g. 16, 23] to mention a few. While the output CNF # can be exponential in the size of #, it is currently not known whether # can be computed in output polynomial (or polynomial total) time, i.e. in time polynomial in the ....

R. Khardon. Translating between Horn representations and their characteristic models. Journal of AI Research, 3:349-372, 1995.

....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 larger representation in the other. The intertranslatability of the two approaches, in particular for Horn theories, has been addressed in [24, 25, 26, 27, 29]. A number of techniques for efficient model based representation of various fragments of propositional logic have been devised, cf. 25, 29, 30] However, little attention has been paid so far on the important issue of how in this representation different knowledge bases KB 1 ; KB l can ....

....of propositional theories Sigma i . Here, we assume that a theory is a set of models. We focus on those Sigma i s which are Horn theories; such theories are frequently encountered in the context of knowledge representation, and their study in model based reasoning received the main attention in [13, 24, 25, 26, 27, 20], and was further discussed in [29] In particular, we consider the following main problems in the context of model computation. Given the sets of characteristic models M 1 ; M l representing Horn theories Sigma 1 , Sigma l , ffl compute some arbitrary model of the theory Sigma ....

R. Khardon. Translating between Horn Representations and their Characteristic Models. Journal of Artificial Intelligence Research, 3:349--372, 1995.

....exists is at least as hard as the positive duality problem [4] i.e. given two positive DNFs ; decide whether represents the dual of the function represented by . The positive duality problem and equivalent problems have been tackled by many researchers, but no polynomial algorithm is known [23, 4, 15, 11, 24]. This strongly supports that a polynomial time algorithm for the unique bidual Horn extension problem, i.e. deciding whether a pdBf (T ; F ) implicitly defines a total bidual Horn function is difficult to find. We study transformation problems between different representations for bidual Horn ....

....bidual Horn extension problem, i.e. deciding whether a pdBf (T ; F ) implicitly defines a total bidual Horn function is difficult to find. We study transformation problems between different representations for bidual Horn functions, in particular (Horn) DNF formulas and characteristic set [25, 24, 26] (or bases [10] which are a vector based representation of arbitrary Horn functions that has received much interest in the context of knowledge representation and reasoning (see Section 6 for details) We show that the transformation between a Horn DNF of f and its (unique) characteristic set ....

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R. Khardon, Translating between Horn representations and their characteristic models, Journal of Artificial Intelligence Research, 3 (1995) 349--372. 32

....shows that there exists no polynomial total time algorithm for the dualization problem of general Boolean functions unless P=NP. Therefore, research has been focused on important restricted classes of Boolean functions, and in particular on positive (also called monotone) and Horn CNFs (e.g. [3, 7, 12, 16, 17]) Recall that a CNF is positive (resp. Horn) if each clause contains only positive literals (resp. at most one positive literal) It may happen that a CNF is neither positive This work was supported by the Austrian Science Fund (FWF) Project Z29 INF and by Grants in Aid for Scientific ....

....22] However, it is still open whether the positive dualization problem has a polynomial total time algorithm or not. For Horn functions, our state of knowledge is less advanced; we have a quasi polynomial total time algorithm for generating a DNF which contains all prime implicants of a Horn CNF [17], but it is not known whether the Horn dualization problem has a polynomial (or even only quasi polynomial) total time algorithm. Note, however, that this problem is at least as hard as the positive dualization problem, since by changing the polarities of all literals in a positive CNF, we obtain ....

R. Khardon. Translating between Horn representations and their characteristic models. Journal of Artificial Intelligence Research, 3:349--372, 1995.

....by a CNF hypothesis. Indeed, every DNF formula can be transformed into an equivalent CNF formula (and vice versa) but the number of literals is not constant as shown in Example 2.5. This has also been observed in [Mooney, 1995] and [Blockeel, 1998] Related results can be found in, e.g. [Khardon, 1995; Khardon and Roth, 1996] where the computational complexity of the representations and the tranformation between DNF and CNF expressions is discussed. Example 2.5 In Example 2.3 (page 17) the DNF hypothesis consists of 4 literals, whereas the CNF description contains only 3 literals 4. Note the ....

R. Khardon. Translating between horn representations and their characteristic models. Journal of Artificial Intelligence Research, 3:349372, 1995.

.... like transversal computation known to be computationally equivalent to problem DUALIZATION (see [15] are of interest in various areas such as database theory (e.g. 38, 49] machine learning and data mining (e.g. 6, 7, 12, 22] game theory (e.g. 26, 42, 43] artificial intelligence (e.g. [21, 28, 29, 44]) mathematical programming (e.g. 5] and distributed systems (e.g. 18, 27] to mention a few. While the output CNF can be exponential in the size of , it is currently not known whether can be computed in output polynomial (or polynomial total) time, i.e. in time polynomial in the ....

R. Khardon. Translating between Horn representations and their characteristic models. Journal of Artificial Intelligence Research, 3:349-372, 1995.

.... can be a succinct representation of , and eliciting all models from it, as needed by the algorithm in [15] is not feasible in polynomial time in the input size of C ( Algorithms for computing the Horn envelope for certain classes of formula representations of a theory are contained in [4, 5, 15, 16, 21, 22]. In particular, 22, 4, 5] explicitly consider formula representations, while [15, 16] cover implicitly formula representations of theories in terms of disjunctions of minterms (i.e. terms such that every variable occurs either positively or negatively in them) The papers [21, 22] suggested a ....

.... in [15] is not feasible in polynomial time in the input size of C ( Algorithms for computing the Horn envelope for certain classes of formula representations of a theory are contained in [4, 5, 15, 16, 21, 22] In particular, 22, 4, 5] explicitly consider formula representations, while [15, 16] cover implicitly formula representations of theories in terms of disjunctions of minterms (i.e. terms such that every variable occurs either positively or negatively in them) The papers [21, 22] suggested a general framework for knowledge compilation, in which the concept of Horn envelope is ....

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R. Khardon. Translating between Horn Representations and their Characteristic Models. Journal of Articial Intelligence Research, 3:349-372, 1995.

....Horn theory R which includes , i.e. logically implies R. While there is more than one Horn core in general, the Horn envelope is always unique. Semantical and computational issues on Horn cores and the Horn envelope have been studied extensively, and a number of results have been obtained, cf. [21, 16, 2, 17, 18, 3, 4, 10, 1]. The main results of the present paper can be summarized as follows. We present characterizations of the Horn cores and the Horn envelope of a Horn di erence 1 n 2 , which will form a basis of the algorithms discussed in this paper. We either present a polynomial time algorithm or prove ....

.... 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, ....

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R. Khardon. Translating between Horn Representations and their Characteristic Models. Journal of Articial Intelligence Research, 3:349-372, 1995.

.... 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 set P and negative example set N is the set of concepts in C consistent with P and N . A version space V has ....

R. Khardon. Translating between horn representations and their characteristic models. Journal of AI Research, 3:349--372, 1995.

....Horn clauses have the expressive power sufficient for many applications. A Horn theory is characterized by the condition that the intersection of any two models is again a model. A theory can be viewed as the set of its models, and reasoning with models has been developed in recent AI studies (see [25, 29, 27]) From the database theory point of view, the set of models is in fact a relation. This relation may have functional dependencies, which reveal important structural properties of the theory by describing the intrinsic determinants of values of certain attributes. The knowledge of functional ....

R. Khardon. Translating between Horn representations and their characteristic models, Journal of Artificial Intelligence Research, 3 (1995), 349 - 372.

.... ( Sigma) can be a succinct representation of Sigma, and eliciting all models from it, as needed by the algorithm in [15] is not feasible in polynomial time in the input size of C ( Sigma) Algorithms for computing the Horn envelope restricted to special classes of formulas are contained in [14, 2, 15, 4, 5, 16]. In particular, 14, 2, 4, 5] consider CNF formulas, while [15, 16] cover the case of theories which are represented by a disjunction of minterms (i.e. terms such that all variables occur either positively or negatively in them) All these algorithms require exponential time. We have shown that ....

.... it, as needed by the algorithm in [15] is not feasible in polynomial time in the input size of C ( Sigma) Algorithms for computing the Horn envelope restricted to special classes of formulas are contained in [14, 2, 15, 4, 5, 16] In particular, 14, 2, 4, 5] consider CNF formulas, while [15, 16] cover the case of theories which are represented by a disjunction of minterms (i.e. terms such that all variables occur either positively or negatively in them) All these algorithms require exponential time. We have shown that the Horn envelope of a disjunction of Horn theories Sigma 1 ; ....

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R. Khardon. Translating between Horn Representations and their Characteristic Models. Journal of Artificial Intelligence Research, 3:349--372, 1995.

.... 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 positive example set P and negative example set N is the set of concepts in C consistent with P and N . A version space V has ....

Roni Khardon. Translating between horn representations and their characteristic models. Journal of AI Research, 3:349--372, 1995.

....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 larger representation in the other. The intertranslatability of the two approaches, in particular for Horn theories, has been addressed in [24, 25, 26, 27, 29]. A number of techniques for efficient model based representation of various fragments of propositional logic have been devised, cf. 25, 29, 30] However, little attention has been paid so far on the important issue of how in this representation different knowledge bases KB 1 ; KB n can ....

....= Sigma 1 Delta Delta Delta Sigma l of propositional theories Sigma i . We focus on those Sigma i s which are Horn theories; such theories are frequently encountered in the context of knowledge representation, and their study in model based reasoning received the main attention in [13, 24, 25, 26, 27, 20], and was further discussed in [29] In particular, we consider the following main problems in the context of model computation. Given the sets of characteristic models M 1 ; M l representing Horn theories Sigma 1 , Sigma l , IFIG RR 9803 3 ffl compute some arbitrary model of the ....

R. Khardon. Translating between Horn Representations and their Characteristic Models. Journal of Artificial Intelligence Research, 3:349--372, 1995.

....in several forms in databases. In particular, the problem of translating between a set of functional dependencies and their corresponding Armstrong relation [16, 17] is at least as hard as this problem and equivalent to it in special cases [8] Further discussion of these issues is given by [12, 18]. Notice that in general the output for this problem may be exponentially larger than its input, and thus the question is whether it can be solved in time polynomial in both its input size and output size. We say that an algorithm is output T ( time algorithm for the problem if it runs in time T ....

....width(L; queries. 2 Thus we see that the connection to hypergraph transversals holds not only for the verification problem but also for the generation problem. We note that for the case of functional dependencies with fixed right hand side, and for keys, even simpler algorithms can be used [16, 12]. In this case one can access the database and directly compute Bd Gamma (MTh) according to the appropriate representation as sets, this corresponds to the so called agree sets of the relation) Then a single run of an HTR subroutine suffices. The current result holds even if the access to the ....

R. Khardon. Translating between Horn representations and their characteristic models. Journal of AI Research, 3:349--372, 1995.

....the learnability of such representations is studied in [21] The question of translating between characteristic models and propositional expressions (which is relevant in database theory as well) has also been studied. Some results on the complexity of this and related questions are described in [17]. Most of the work on reasoning assumes that the knowledge base is given in some form, and the question of how this knowledge might be acquired is not considered. While in this paper we also take this point of view, we are interested in studying the entire process of learning a knowledge base ....

R. Khardon. Translating between Horn representations and their characteristic models. Journal of AI Research, 3:349--372, 1995.

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R. Khardon. Translating between Horn representations and their characteristic models. Journal of Artificial Intelligence Research, 3:349--372, 1995.

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