Lukasiewicz, T.: 1999b, `Probabilistic deduction with conditional constraints over basic events'. J. Artif. Intell. Res. 10, 199--241.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....In the recent decades, probabilistic reasoning has become an important research topic in artificial intelligence. In particular, extensive research has been carried out on probabilistic reasoning with interval restrictions for conditional probabilities, also called conditional constraints [40]. One important approach for handling conditional constraints is model theoretic probabilistic logic, which comes especially from philosophy and logic, and whose roots go back to Boole s book of 1854 The Laws of Thought [11] There is a wide spectrum of formal languages that have been explored ....

.... comes especially from philosophy and logic, and whose roots go back to Boole s book of 1854 The Laws of Thought [11] There is a wide spectrum of formal languages that have been explored in model theoretic probabilistic logic, ranging from constraints for unconditional and conditional events [16, 20, 2, 52, 19, 23, 33, 44, 39, 40, 42, 45] to rich languages that specify linear inequalities over events [22] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical consequence, and computing tight logically entailed intervals. Another important approach to probabilistic reasoning ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....Background In this section, we describe the probabilistic background of our approach to probabilistic complex value databases. We assume a semantics in which probabilities are defined over a set of possible worlds (see especially [4, 12, 23, 14] where we adopt some technical notions from [19, 20]. A major goal of this section is to give a model theoretic definition of probabilistic conjunction, disjunction, and difference strategies, which have been introduced by an axiomatic characterization in [17] Given the probability ranges of two events e 1 and e 2 , these strategies compute the ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10:199--241, 1999.

....while the latter are appropriate for describing statistical knowledge. In the present paper, we assume that probabilities are de ned over a set of possible worlds. Probabilistic reasoning in its full generality is a quite tricky task and very di erent from classical reasoning (see especially [18], 15] and [14] It should generally be performed by global linear programming methods, rather than by local inference techniques. For this reason, it is generally also computationally more complex than classical reasoning. In particular, the model and xpoint characterization and the proof ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199-241, 1999.

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Lukasiewicz, T.: 1999b, `Probabilistic deduction with conditional constraints over basic events'. J. Artif. Intell. Res. 10, 199--241.

....In the recent decades, probabilistic reasoning has become an important research topic in artificial intelligence. In particular, extensive research has been carried out on probabilistic reasoning with interval restrictions for conditional probabilities, also called conditional constraints [40]. One important approach for handling conditional constraints is model theoretic probabilistic logic, which comes Alternate address: Institut fur Informationssysteme, Technische Universitat Wien, Favoritenstrae 9 11, 1040 Wien, Austria. E mail: lukasiewicz kr.tuwien.ac.at. especially from ....

.... especially from philosophy and logic, and whose roots go back to Boole s book of 1854 The Laws of Thought [11] There is a wide spectrum of formal languages that have been explored in model theoretic probabilistic logic, ranging from constraints for unconditional and conditional events [16, 20, 2, 53, 19, 23, 33, 45, 39, 40, 42, 46] to rich languages that specify linear inequalities over events [22] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical consequence, and computing tight logically entailed intervals. Another important approach to probabilistic reasoning ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

.... probabilistic logic, whose roots go back to Boole s book of 1854 The Laws of Thought [BOO 54] There is a wide spectrum of formal languages that have been explored in probabilistic logic, which ranges from constraints for unconditional and conditional events [AMA 91, FRI 94, LUK 99a, LUK 99b, LUK 01, NIL 86] to rich languages that specify linear inequalities over events [FAG 90] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical consequence, and computing tight logically entailed intervals. Coherence based and ....

LUKASIEWICZ T., "Probabilistic deduction with conditional constraints over basic events", J. Artif. Intell. Res., vol. 10, 1999, p. 199--241.

.... and Haddawy [18] to rich languages that specify linear inequalities over events (Fagin et al. 17] The reasoning methods in probabilistic logic can be roughly divided into local approaches based on local inference rules and global ones using linear optimization techniques (see especially [31, 30] on the issue of local versus global approaches) As shown by Georgakopoulos et al. 21] deciding satisfiability and logical consequence in probabilistic logic is NP and coNP complete, and thus intractable. Moreover, as recently shown in [28] deciding and computing tight logical consequences ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10:199--241, 1999.

....probabilistic reasoning formalism. Heinsohn [9] presents a probabilistic extension of the description logic , which allows to represent generic probabilistic knowledge about concepts and roles, and which is essentially based on probabilistic reasoning in probabilistic logics, similar to [24,1,6,21]. The work [9] however, does not allow for assertional knowledge about concept and role instances. Also Jaeger [18] gives a probabilistic extension of the description logic , which allows for generic (resp. assertional) probabilistic knowledge about concepts and roles (resp. concept ....

....and atleast restrictions are restricted to simple abstract roles w.r.t. 16] 4 P OqPRUTWVEXpZ In this section, we present the description logic P 0(2 ,3 4 , which is a probabilistic extension of . We first define the syntax of P 5 0( 3 4 , where we use conditional constraints [21] to express probabilistic knowledge in addition to the terminological axioms of . We then define the semantics of P 0(2 ,3 4 , which is based on lexicographic entailment from probabilistic default reasoning; see especially [22,23] for background, intuitions, and further examples. We finally ....

[Article contains additional citation context not shown here]

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....31 E Appendix: Proofs for Section 9 31 1 Introduction During the recent decades, reasoning about probabilities has started to play an important role in artificial intelligence. In particular, reasoning about interval restrictions for conditional probabilities, also called conditional constraints [47], has been a subject of extensive research efforts in the literature. Example 1.1 (Eagles) Suppose that we have the knowledge eagles are birds , birds have legs , and birds fly with a probability of at least 0.95 . What do we conclude about the property of having legs of eagles and their ....

.... probabilistic logic, ranging from constraints for unconditional and conditional events to rich languages that specify linear inequalities over events (see especially the work by Nilsson [52] Fagin et al. 24] Dubois and Prade et al. 18, 22, 2, 21] Frisch and Haddawy [25] and the author [46, 47, 49]; see also the survey on sentential probability logic by Hailperin [38] The main decision and optimization problems in model theoretic probabilistic reasoning are deciding satisfiability, deciding logical consequence, and computing tight logically entailed intervals. Example 1.2 (Eagles ....

[Article contains additional citation context not shown here]

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

.... Proofs for Section 3 13 B Appendix: Proofs for Section 4 13 1 Introduction During the recent decades, there has been a significant amount of research in AI that focuses on probabilistic reasoning with interval restrictions for conditional probabilities, also called conditional constraints [42]. For example, suppose that we have the knowledge ostriches are birds , birds have legs , birds fly with a probability of at least 0.95 , and ostriches fly with a probability of at most 0.05 . What do we then conclude about the property of having legs of birds (resp. ostriches) and their ....

.... for handling conditional constraints is model theoretic probabilistic logic, which can be traced back to Boole [10] There is a wide spectrum of formal languages that have been explored in model theoretic probabilistic logic, ranging from constraints for unconditional and conditional events [15, 19, 2, 18, 23, 33, 47, 41, 42, 44, 48] to rich languages that specify linear inequalities over events [21] The main algorithmic tasks related to model theoretic probabilistic logic are deciding satisfiability, deciding logical consequence, and computing tight logically entailed intervals. In model theoretic probabilistic logic, we ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....representation and reasoning formalisms. Probabilistic propositional logics and their various dialects have been thoroughly studied in the literature (see especially the work by Nilsson [56] Fagin et al. 15] Dubois and Prade et al. 12, 11] Frisch and Haddawy [17] and the second author [41, 42]) Their extensions to probabilistic first order logics can be classified into first order logics in which probabilities are defined over the domain and those in which probabilities are given over a set of possible worlds (see especially the work by Bacchus et al. 2] and Halpern [23] The first ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....probabilistic reasoning formalism. Heinsohn [8] presents a probabilistic extension of the description logic 3 465 , which allows to represent generic probabilistic knowledge about concepts and roles, and which is essentially based on probabilistic reasoning in probabilistic logics, similar to [23, 1, 6, 20]. The work [8] however, does not allow for assertional (i.e. Abox) knowledge about concept and role instances. Also Jaeger [17] gives a probabilistic extension of the description logic 3 4 5 , which allows for generic (resp. assertional) probabilistic knowledge about concepts and roles ....

....restrictions in are restricted to simple abstract roles w.r.t. 15] 4 P In this section, we present the probabilistic description logic P 1 2( which is a probabilistic extension . We first define the syntax of P 1 2( where we have conditional constraints [20] to express probabilistic knowledge in addition to the terminological axioms of the semantics of P ) which is based on the notion of lexicographic entailment from probabilistic default reasoning; see especially [21, 22] for background, intuitions, and further examples. We finally ....

[Article contains additional citation context not shown here]

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

.... constraints is model theoretic probabilistic logic, whose roots go back to Boole s book of 1854 The Laws of Thought [7] There is a wide spectrum of formal languages that have been explored in probabilistic logic, which ranges from constraints for unconditional and conditional events [2, 15, 27, 28, 29, 32] to rich languages that specify linear inequalities over events [14] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical consequence, and computing tight logically entailed intervals. Coherence based and model theoretic probabilistic ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

.... constraints is model theoretic probabilistic logic, whose roots go back to Boole s book of 1854 The Laws of Thought [7] There is a wide spectrum of formal languages that have been explored in probabilistic logic, which ranges from constraints for unconditional and conditional events [2, 15, 25, 26, 27, 29] to rich languages that specify linear inequalities over events [14] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical consequence, and computing tight logically entailed intervals. Coherence based and model theoretic probabilistic ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....that the problems of reference class reasoning have already been solved in the area of reasoning with interval restrictions lukasiewicz.tex; 1 09 2001; 20:56; p. 2 Probabilistic Default Reasoning with Conditional Constraints 3 for conditional probabilities, also called conditional constraints [62]. There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model theoretic notion of logical entailment [20, 25, 3, 45, 76, 24, 29, 43, 65, 62, 59] which can be traced back to Boole [14] and those that are based on entailment ....

.... Default Reasoning with Conditional Constraints 3 for conditional probabilities, also called conditional constraints [62] There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model theoretic notion of logical entailment [20, 25, 3, 45, 76, 24, 29, 43, 65, 62, 59] (which can be traced back to Boole [14] and those that are based on entailment under de Finetti s notion of coherence and its generalizations (see especially the work by Coletti [18] and Gilio [36] However, we will see that neither logical entailment nor the weaker notion of entailment under ....

[Article contains additional citation context not shown here]

Lukasiewicz, T.: 1999d, `Probabilistic deduction with conditional constraints over basic events'. J. Artif. Intell. Res. 10, 199--241.

....Background In this section, we describe the probabilistic background of our approach to probabilistic complex value databases. We assume a semantics in which probabilities are defined over a set of possible worlds (see especially [4,14,27,16] where we adopt some technical notions from [22,23]. A major goal of this section is to give a model theoretic definition of probabilistic conjunction, disjunction, and difference strategies, which have been introduced by an axiomatic characterization in [20] Given the probability ranges of two events 54 and ,9 , these strategies compute ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10:199--241, 1999.

.... and Haddawy [18] to rich languages that specify linear inequalities over events (Fagin et al. 17] The reasoning methods in probabilistic logic can be roughly divided into local approaches based on local inference rules and global ones using linear optimization techniques (see especially [31, 30] on the issue of local versus global approaches) As shown by Georgakopoulos et al. 21] deciding satisfiability and logical consequence in probabilistic logic is NP and coNP complete, and thus intractable. Moreover, as recently shown in [28] deciding and computing tight logical consequences ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10:199--241, 1999.

.... constraints is model theoretic probabilistic logic, whose roots go back to Boole s book of 1854 The Laws of Thought [8] There is a wide spectrum of formal languages that have been explored in probabilistic logic, which ranges from constraints for unconditional and conditional events [2, 13, 19, 20, 22, 23] to rich languages that specify linear inequalities over events [12] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical entailment, and computing tight logically entailed intervals. Coherence based and model theoretic probabilistic ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

.... Frisch and Haddawy [17] to rich languages that specify linear inequalities over events (Fagin et al. 16] The reasoning methods in probabilistic logic can be roughly divided into local approaches based on local inference rules and global ones using linear optimization techniques (see especially [30, 29] on the issue of local versus global approaches) As shown by Georgakopoulos et al. 20] deciding satisfiability and logical consequence in probabilistic logic is NP and co NP complete, and thus intractable. Moreover, as recently shown in [27] deciding and computing tight logical consequences ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Journal of Artificial Intelligence Research, 10:199--241, 1999.

.... constraints is model theoretic probabilistic logic, whose roots go back to Boole s book of 1854 The Laws of Thought [7] There is a wide spectrum of formal languages that have been explored in probabilistic logic, which ranges from constraints for unconditional and conditional events [2, 15, 24, 25, 27, 28] to rich languages that specify linear inequalities over events [14] The main problems related to model theoretic probabilistic logic are checking satisfiability, deciding logical entailment, and computing tight logically entailed intervals. Coherence based and model theoretic probabilistic ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....reasoning In this paper, we address this important question. As a first idea, we may suspect that the problems of reference class reasoning have already been solved in the area of reasoning with interval restrictions for conditional probabilities, also called conditional constraints [59]. There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model theoretic notion of logical entailment [16, 21, 2, 40, 39, 20, 25, 38, 60, 59, 56] which can be traced back to Boole [11] and those that are based on entailment ....

.... in the area of reasoning with interval restrictions for conditional probabilities, also called conditional constraints [59] There is extensive work on reasoning about conditional constraints, which can be roughly divided into approaches that use the model theoretic notion of logical entailment [16, 21, 2, 40, 39, 20, 25, 38, 60, 59, 56] (which can be traced back to Boole [11] and those that are based on entailment under de Finetti s notion of coherence (see especially the work by Coletti [14] and Gilio [31] However, we will see that neither logical entailment nor the weaker notion of entailment under coherence show the ....

[Article contains additional citation context not shown here]

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....Background In this section, we describe the probabilistic background of our approach to probabilistic complex value databases. We assume a semantics in which probabilities are de ned over a set of possible worlds (see especially [3, 9, 17, 10] Note that we adopt some technical notions from [13, 14]. The main aim of this section is to give a model theoretic de nition of probabilistic conjunction, disjunction, and di erence strategies, which have been introduced by an axiomatic characterization in [12] Given the probability ranges of two events e 1 and e 2 , these strategies compute the ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Journal of Articial Intelligence Research, 10:199-241, 1999.

.... knowledge, while the latter are appropriate for representing subjective knowledge (also called degrees of belief) One important element of many formal languages for representing probabilistic knowledge are interval restrictions for conditional probabilities, also called conditional constraints [47]. In particular, the literature contains extensive work on probabilistic reasoning about propositional conditional constraints (cf. especially the work by Dubois and Prade s group [17, 1, 16] Frisch and Haddawy [20] and the author [43, 47] As an important formalism for reasoning with classical ....

.... probabilities, also called conditional constraints [47] In particular, the literature contains extensive work on probabilistic reasoning about propositional conditional constraints (cf. especially the work by Dubois and Prade s group [17, 1, 16] Frisch and Haddawy [20] and the author [43, 47]) As an important formalism for reasoning with classical knowledge, logic programming [39, 3] started in the early 1970 s [36] based on earlier work in automated theorem proving, and began to flourish especially with the spreading of PROLOG. Logic programming is now a well established knowledge ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....and conditional entailment, respectively. We show that the new notions of z entailment, lexicographic entailment, and conditional entailment for conditional constraints properly extend the classical notion of logical entailment for conditional constraints. Note that all proofs are given in (Lukasiewicz 2000). Preliminaries We now introduce some necessary technical background. We assume a finite nonempty set of basic propositions (or atoms) We use and to denote the propositional constants false and true, respectively. The set of classical formulas is the closure of [f ; g under the Boolean ....

Lukasiewicz, T. 1999b. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res. 10:199--241.

....interval Bayesian, credal) tree iff the associated undirected graph D is a directed tree (that is, a directed acyclic graph in which every node has exactly one incoming arrow, except for the root that does not have any) 2. 2 CONDITIONALS We will use the language of conditionals (see especially [13, 21, 24, 25]) to represent Bayesian networks, interval Bayesian networks, and credal networks. Let U = fX 1 ; X n g with n 1 be a set of discrete random variables, where each variable X i 2 U has a finite and nonempty domain DX i = fx i;1 ; x i;d i g. The set of basic events BU INFSYS RR ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....interval Bayesian, credal) tree iff the associated undirected graph D is a directed tree (that is, a directed acyclic graph in which every node has exactly one incoming arrow, except for the root that does not have any) 2. 2 CONDITIONALS We will use the language of conditionals (see especially [13, 21, 25, 26]) to represent Bayesian networks, interval Bayesian networks, and credal networks. Let U = fX 1 ; Xn g with n 1 be a set of discrete random variables, where each variable X i 2 U has a finite and nonempty domain DX i = fx i;1 ; x i;d i g. The set of basic events BU contains all ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199-- 241, 1999.

.... was proposed by Geffner [18, 19] and an infinitesimal belief function approach was suggested by Benferhat et al. 7] Coming back to our introductory example, we realize that G1 G3 and E1 E2 express interval restrictions for conditional probabilities, also called conditional constraints [37]. The literature contains extensive work on reasoning about conditional constraints (see especially [12, 2, 16, 34, 37] and their generalizations (for example, to probabilistic logic programs [33] Now, the main idea of this paper is to use techniques for default reasoning from conditional ....

.... Benferhat et al. 7] Coming back to our introductory example, we realize that G1 G3 and E1 E2 express interval restrictions for conditional probabilities, also called conditional constraints [37] The literature contains extensive work on reasoning about conditional constraints (see especially [12, 2, 16, 34, 37]) and their generalizations (for example, to probabilistic logic programs [33] Now, the main idea of this paper is to use techniques for default reasoning from conditional knowledge bases in order to perform probabilistic reasoning from statistical knowledge and degrees of beliefs. More ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199--241, 1999.

....semantics of n valued first order logics. Our aim is to bring together the advantages of two different approaches. While the probabilistic formalism provides a well defined semantics, truth functional logics are often more compelling from the computational side (see, for example, 22] and [23] for the subtleties of probabilistic propositional deduction) In this paper, we now present a new probabilistic semantics of n valued firstorder logics that lies between the purely probabilistic semantics and the truthfunctional semantics given by the n valued Lukasiewicz logics Ln . More ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proc. of the 6th International Conference, pages 380--391. Morgan Kaufmann Publishers, 1998.

....This technique reduces the original entropy maximization task to solving a modified and relatively small optimization problem. 1 Introduction Probabilistic propositional logics and their various dialects are thoroughly studied in the literature (see especially [19] and [5] see also [15] and [16]) Their extensions to probabilistic first order logics can be classified into first order logics in which probabilities are defined over the domain and those in which probabilities are given over a set of possible worlds (see especially [2] and [9] The first ones are suitable for describing ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 1999. To appear.

....example, the requested least upper bound for u 1 = u 2 = u and x 1 = x 2 = x is shown in Fig. 5 for u; x 2 [0; 1] r 1 = r 2 = 0:15, v 1 = 0:8, y 1 = 0:8, and s 1 2 f0:05; 0:1g. The requested least upper bound for u 1 u 2 or x 1 x 2 is the maximum value over [u 1 ; u 2 ] Theta [x 1 ; x 2 ] Lukasiewicz s1=0.05 z2 0.99 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u x s1=0.1 z2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 ....

Lukasiewicz, T. (1998c). Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference, pp. 380--391. Morgan Kaufmann.

....while the latter are appropriate for describing statistical knowledge. In the present paper, we assume that probabilities are de ned over a set of possible worlds. Probabilistic reasoning in its full generality is a quite tricky task and very di erent from classical reasoning (see especially [19], 15] and [14] It should generally be performed by global linear programming methods, rather than by local inference techniques. For this reason, it is generally also computationally more complex than classical reasoning. In particular, the model and xpoint characterization and the proof ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. J. Artif. Intell. Res., 10:199-241, 1999.

....b) The problem of computing the tight answer for 9(Hj ) x1 ; x2 ] to a ground probabilistic logic program is NP hard. Proof sketch. a) The NP complete problem of graph 3 colorability [12] can be polynomially reduced to the described problem of probabilistic logic program satisfiability (see also [20]) b) The claim follows immediately from a) since a probabilistic logic program P is satisfiable iff j = fx1=1; x2=1g is the tight answer for 9(h j ) x1 ; x2 ] to P [ f(h j ) 1; 1]g, where h is a new 0 ary predicate symbol that does not occur in P . 2 Note that if we restrict our considerations ....

T. Lukasiewicz, `Probabilistic deduction with conditional constraints over basic events', in Principles of Knowledge Representation and Reasoning: Proc. of the 6th International Conference. Morgan Kaufmann Publishers, (1998).

....knowledge is integrated into a terminological language. We choose taxonomic and probabilistic knowledge bases over conjunctive events as a concrete framework in which our motivating ideas shall be realized. In this framework, the deduction of probabilistic knowledge is NP hard (we show in [19] that it is even NP hard for probabilistic knowledge bases over basic events) while the deduction of taxonomic knowledge can be done in linear time in the size of the taxonomic knowledge base. Hence, each inference rule that exploits taxonomic knowledge can also be applied in linear time in the ....

....complexity point of view, it is reasonable to focus on a more restricted class of probabilistic deduction problems. Surprisingly, even the problem of computing the tight answer for a probabilistic query over basic events to a probabilistic knowledge base over basic events is NP hard, as we show in [19]. While already in the framework of taxonomic formulas over conjunctive events, the problem of deciding whether a taxonomic formula is a logical consequence of a taxonomic knowledge base can be solved in linear time in the size of the taxonomic knowledge base. More precisely, taxonomic formulas ....

[Article contains additional citation context not shown here]

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proc. of the 6th International Conference. Morgan Kaufmann Publishers, 1998.

....semantics of n valued first order logics. Our aim is to bring together the advantages of two different approaches. While the probabilistic formalism provides a well defined semantics, truth functional logics are often more compelling from the computational side (see, for example, 22] and [23] for the subtleties of probabilistic propositional deduction) In this paper, we now present a new probabilistic semantics of n valued firstorder logics that lies between the purely probabilistic semantics and the truthfunctional semantics given by the n valued Lukasiewicz logics Ln . More ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proc. of the 6th International Conference, pages 380--391. Morgan Kaufmann Publishers, 1998.

....example, the requested least upper bound for u 1 = u 2 = u and x 1 = x 2 = x is shown in Fig. 5 for u; x 2 [0; 1] r 1 = r 2 = 0:15, v 1 = 0:8, y 1 = 0:8, and s 1 2 f0:05; 0:1g. The requested least upper bound for u 1 u 2 or x 1 x 2 is the maximum value over [u 1 ; u 2 ] Theta [x 1 ; x 2 ] Lukasiewicz s1=0.05 z2 0.99 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u x s1=0.1 z2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 ....

Lukasiewicz, T. (1998c). Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference, pp. 380--391. Morgan Kaufmann.

.... Box 940, D 58084 Hagen, Germany E mail: gabriele.kern isberner fernuni hagen.de Copyright c fl 1999 by the authors IFIG RR 9903 2 1 Introduction Probabilistic propositional logics and their various dialects are thoroughly studied in the literature (see especially [27] and [9] see also [21] and [22]) Their extensions to probabilistic first order logics can be classified into first order logics in which probabilities are defined over the domain and those in which probabilities are given over a set of possible worlds (see especially [2] 3] and [13] The first ones are suitable for ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. Accepted for publication in J. Artif. Intell. Res., 1999. To appear.

....example, the requested least upper bound for u 1 = u 2 = u and x 1 = x 2 = x is shown in Fig. 5 for u; x 2 [0; 1] r 1 = r 2 = 0:15, v 1 = 0:8, y 1 = 0:8, and s 1 2 f0:05; 0:1g. The requested least upper bound for u 1 u 2 or x 1 x 2 is the maximum value over [u 1 ; u 2 ] Theta [x 1 ; x 2 ] Lukasiewicz s1=0.05 z2 0.99 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 u x s1=0.1 z2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 ....

Lukasiewicz, T. (1998c). Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference, pp. 380--391. Morgan Kaufmann.

....many valued logic programming in this framework is computationally more complex than classical logic programming. More precisely, some deduction problems that are P complete for classical logic programs are shown to be co NP complete for probabilistic many valued logic programs (see also [28] [29], and [30] for other work on the subtleties and the computational complexity of probabilistic deduction) We then focus on many valued logic programming in Pr n as an approximation of probabilistic many valued logic programming. Crucially, many valued logic programming in Pr n has a model ....

T. Lukasiewicz. Probabilistic deduction with conditional constraints over basic events. In Principles of Knowledge Representation and Reasoning: Proceedings of the Sixth International Conference, pages 380--391. Morgan Kaufmann Publishers, 1998.

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