R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. Acta Informatica, 36(4):267--286, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(0101) 2 C ( Sigma 2 ) while (0000) 2 C ( Sigma 2 ) it holds that C ( Sigma 2 ) Sigma 1 . We remark that the characteristic set of a Horn theory without negative clauses has been studied in the context of relational databases, where it is known as the generating set [3] see [28] for a discussion. Observe that b Sigma is not uniquely defined; we use this as a conversion of a set of models into an equivalent formula, which is needed in some contexts. 6 Throughout this paper, we suppose that sets of vectors S f0; 1g are represented in the standard way, i.e. each ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with Examples: Propositional Formulae and Database Dependencies. Acta Informatica, to appear.

....generating explanations and prime implicates using SOL resolution. He proposed a strategy which processes, starting from the empty set, clauses from a theory incrementally. However, due to possible large intermediate results, this method is not total polynomial time in general. Khardon et al. [13] show how computing all keys of a relational database schema, which is constrained by a Boolean formula , can be polynomially transformed into computing all explanations of a query q from , where is Horn. Thus, our algorithm EXPLANATIONS can be used for efficiently generating all keys of a ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. Acta Informatica, 36(4):267--286, 1999.

....ones. A sound and complete axiomatization of RUD : is an interesting and important result. A Completeness Proof To simplify the notations, the completeness proof exploits an equivalence between RUD : and positive propositional calculus. Similar equivalences have appeared in the literature [2, 13, 16, 17]. Definition 9 Let B be a set of Boolean variables. If p is a Boolean variable, then p and :p are literals. For convenience, p can be denoted p. Greek letters ff and fi are used to denote literals. ff equals ff. A set T of literals is called a valuation iff every Boolean variable occurs exactly ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. To appear, 1999.

....dependency from each lower level to the level above it. In many real situations, however, the dimensional normal form may be too strict. The authors also study a problem related to attributes that are not applicable for all instances, for example, all products do not have a colour. Khardon et al. [KMR99] have studied Boolean dependencies in a context of example based reasoning. Boolean dependencies are a generalisation to the functional dependencies [Cod72] While the functional dependency has a form X Y, where X and Y are sets of attributes, the Boolean dependencies allow us to use Boolean ....

Khardon, R., Mannila, H., Roth, D.: Reasoning with examples: propositional formulae and database dependencies, Acta Informatica, 36(4): 267-286, 1999.

.... 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, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. Acta Informatica, To appear. 23

....every month belongs to a single year. We also say that a month rolls up to its year. On the other hand, WEEK and MONTH are not comparable by because months do not divide evenly into weeks, nor vice versa. The level PRICE BRACKET denotes a set of consecutive price intervals, for example, 1 10] [11 20], 21 30] and so on. We have PRICE PRICE BRACKET, and a price rolls up to its containing price bracket. Roll up dependencies (RUDs) extend functional dependencies (FDs) by allowing attributes to be compared for equality at a specified level. For example, we may find that the tax rate does ....

....ones. A sound and complete axiomatization of RUD : is an interesting and important result. A Completeness Proof To simplify the notations, the completeness proof exploits an equivalence between RUD : and positive propositional calculus. Similar equivalences have appeared in the literature [2, 11, 13, 14]. Definition 9. Let B be a set of Boolean variables. If p is a Boolean variable, then p and :p are literals. For convenience, p can be denoted p. Greek letters ff and fi are used to denote literals. ff equals ff. A set T of literals is called a valuation iff every Boolean variable occurs ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. To appear, 1999.

....commonly used in the logical database design to express integrity constraints, and thus to express domain knowledge. The problems of inferring functional dependencies from relations have been studied in [36, 30] Thorough theoretical studies of functional dependencies in relational databases (see [11, 14, 42, 15, 28]) have established a close connection between them and Horn clauses. Horn clauses were introduced in formal logic (see [37, 22] and gained prominence in logic programming (see [12] and artificial intelligence (see [8, 10, 25] In artificial intelligence, the implementation of a knowledge base ....

....dependencies D, a theory Sigma is called an Armstrong relation for D if the set of all the functional dependencies that hold in Sigma coincides with the closure b D. The concept of Armstrong relations is very important in the theory of relational databases, and has been well studied (see [1, 2, 28, 35]) It is known that for any set of functional dependencies there exists an Armstrong relation. However, such relation may not be Boolean. If we restrict the set of relations to theories in f0; 1g n , there are sets of functional dependencies D for which there is no Armstrong relation. For ....

R. Khardon, H. Mannila and D. Roth. Reasoning with Examples: Propositional Formulae and Database Dependencies , Technical Report TR-15-95, Aiken Computation Laboratory, Harvard University, Cambridge, MA, 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, Heikki Mannila, and Dan Roth. Reasoning with examples: Propositional formulae and database dependencies. Technical Report, TR-15-95, Harvard University, 1995.

....e.g. 0101) 2 C ( Sigma 2 ) while (0000) 2 C ( Sigma 2 ) it holds that C ( Sigma 2 ) Sigma 1 . We remark that the characteristic set of Horn theories without negative clauses has been studied in the context of relational databases, where it is known as the generating set [3] see [28] for a discussion. Throughout this paper, we suppose that sets of vectors S f0; 1g n are represented in the standard way, i.e. each model v 2 f0; 1g n is stored as a sequence v 1 v 2 Delta Delta Delta v n of 0 s and 1 s. However, our algorithms can be adapted for other forms of storage, ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with Examples: Propositional Formulae and Database Dependencies. manuscript, Acta Informatica.

....seeks an efficient translation from a set of (characteristic) models into a Horn expression that explains it. We denote this translation problem by SID (for Structure Identification) Interestingly, the same constructs appear in the theory of relational databases. As shown in a companion paper [KMR95], there is a correspondence between Horn expressions and Functional Dependencies, and a correspondence between characteristic models and an Armstrong relation. The equivalent question of translating between functional dependencies and Armstrong relations has been studied before [BDFS84, MR86, ....

....problem in [KPS93] and we resolve it here. We note that these hardness results are, in some sense, not too bad since the duality problem, DME, has been shown to possess sub exponential n O(logn) time algorithm [FK94] These hardness results can be alternatively derived by combining results from [EG94, BI93, KMR95, KR94]. Next we consider the corresponding decision problem. The problem of Characteristic Models Identification (CMI) is the problem of deciding, given a Horn expression H and a set of models G, whether G = char(H) We show that CCM,SID, and CMI are equivalent under polynomial reductions. Namely, the ....

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R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. In preparation., 1995.

....An example to the notions introduced here is presented at the end of the section. 1 We note that this direction was studied independently in the Relational Data Base community [2, 16] The results on model based reasoning have immediate implications in this domain which are described elsewhere [9]. Definition 3.1 (Order) We denote by the usual partial order on the lattice f0; 1g n , the one induced by the order 0 1. That is, for x; y 2 f0; 1g n , x y if and only if 8i; x i y i . For an assignment b 2 f0; 1g n we define x b y if and only if x Phi b y Phi b (Here Phi is the ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. Technical Report TR-15-95, Aiken Computation Lab., Harvard University, February 1995.

.... deduction algorithm using this set, and abduction can be performed in polynomial time, while using formulas it is NP Hard (Kautz et al. 1995) Furthermore, an algorithm for default reasoning using characteristic models has been developed, for cases where formula based algorithms are not known (Khardon Roth, 1995). Hence, the question arises, whether one can efficiently translate a Horn formula into its set of characteristic models c fl1995 AI Access Foundation and Morgan Kaufmann Publishers. All rights reserved. Khardon and then use this set for the reasoning task. We denote this translation problem by ....

....seeks an efficient translation from a set of characteristic models into a Horn expression that explains it. We denote this translation problem by SID (for Structure Identification) Interestingly, the same constructs appear in the theory of relational databases. As shown in a companion paper (Khardon, Mannila, Roth, 1995), there is a correspondence between Horn expressions and Functional Dependencies, and a correspondence between characteristic models and an Armstrong relation. The equivalent question of translating between functional dependencies and Armstrong relations has been studied before (Beeri, Dowd, ....

[Article contains additional citation context not shown here]

Khardon, R., Mannila, H., & Roth, D. (1995). Reasoning with examples: Propositional formulae and database dependencies. Tech. rep. TR-15-95, Aiken Computation Lab., Harvard University.

....independently in the Relational Data Base community (where they are called generators ) 2, 25] for the special case of definite Horn expressions. The results in this paper have immediate implications in this domain (e.g. bounding the size of Armstrong relations) which are described elsewhere [18]. In addition, we consider the problem of performing abduction using a model based approach and show that for any propositional knowledge base, using a model based representation yields an abductive explanation in time that is polynomial in the size of the model based representation. Some of our ....

.... In this section we consider in detail the case of Horn formulas and show that in this case our notion of characteristic models coincides with the notion introduced in [14] Characteristic models for Horn expressions also coincide with the notion of generators in relational database theory [2, 18]. We then discuss the issue of using a fixed model based representation for answering unrestricted queries. We show that this extension, discussed in [14] relies on a special property of Horn formulas and does not generalize to other propositional languages. An example that explains this ....

R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. Technical Report TR-15-95, Aiken Computation Lab., Harvard University, Feb. 1995.

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R. Khardon, H. Mannila, and D. Roth. Reasoning with examples: Propositional formulae and database dependencies. Acta Informatica, 36(4):267--286, 1999.

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R. Khardon, H. Mannila and D. Roth. \Reasoning with examples: propositional formulae and database dependencies". Acta Informatica, 36, 267-286, 1999.

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