Dubois, D., H. Prade, and J.-M. Touscas: 1990, `Inference with imprecise numerical quantifiers'. In: Z. W. Ras and M. Zemankova (eds.): Intelligent Systems. Ellis Horwood, Chapt. 3, pp. 53--72.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

.... 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 (see especially the work by Nilsson [32] Dubois et al. [15], Amarger et al. 1] and Frisch 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53-- 72. Ellis Horwood, 1990.

.... of formal languages that have been explored in model theoretic 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 ....

....flow of reading, some technical details and proofs have been moved to Appendices A E. 2 Model Theoretic Probabilistic Logic In this section, we recall the main concepts from model theoretic probabilistic logic (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] We first define logical constraints and probabilistic formulas, which are interpreted by probability distributions over a set of possible worlds. We then define probabilistic knowledge bases and the notions of satisfiability and logical ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

.... 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

....towards integrating logic oriented and probability based 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

.... 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

.... 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 ....

Dubois, D., H. Prade, and J.-M. Touscas: 1990, `Inference with imprecise numerical quantifiers'. In: Z. W. Ras and M. Zemankova (eds.): Intelligent Systems. Ellis Horwood, Chapt. 3, pp. 53--72.

.... 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 (see especially the work by Nilsson [32] Dubois et al. [15], Amarger et al. 1] and Frisch 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53-- 72. Ellis Horwood, 1990.

.... one has to solve the following problem: given a probabilistic premise, e.g. P (A j B) r in (1) and a target probability (P ( A j B) what are the best upper and lower bounds one can infer for the target probability as a function of the parameters (r) in the premise Dubois, Prade and Toucas [4], for instance, solve the problem for the premises P (A j B) r 1 , P (B j A) r 2 , P (C j B) r 3 , P (B j C) r 4 , and the target probability P (A j C) Lukasiewicz [14] solves the same problem with two generalizations: instead of point valued probabilities in the premise he treats the ....

D. Dubois, H. Prade, and J.-M. Toucas. Inference with imprecise numerical quantifiers. In Z. Ras and M. Zemankova, editors, Intelligent Systems: State of the Art and Future Directions, pages 52--72. Ellis Horwood, 1990.

.... logic, whose roots go back to Boole s book of 1854 The Laws of Thought [6] There is a wide spectrum of formal languages that have been explored in probabilistic logic, which ranges from constraints for unconditional and conditional events (see especially the work by Nilsson [31] Dubois et al. [14], Amarger et al. 1] and 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

.... 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

....only possible within a probabilistic framework, where the degree of certainty associated with a conditional is interpreted as a conditional probability. In fact, probability theory provides a sound and convenient machinery to be used for knowledge representation and automated reasoning (cf. e.g. [5], 4] 14] 18] 23] But its clear semantics and strict rules require a lot of knowledge to be available for an adequate modelling of problems. In contrast to this, usually only relatively few relationships between relevant variables are known, due to incomplete information. Or maybe, an ....

....representation is intended, incorporating only fundamental relationships. In both cases, the knowledge explicitly stated is not sufficient to determine uniquely a probability distribution. One way to cope with this indetermination is to calculate upper and lower bounds for probabilities (cf. 23] [5]) This method, however, brings about two problems: Sometimes the inferred bounds are quite bad, and in any case, one has to handle intervals instead of single values. An alternative way that provides best expectation values for the unknown probabilities and guarantees a logically sound reasoning ....

[Article contains additional citation context not shown here]

D. Dubois, H. Prade, and J.-M. Toucas. Inference with imprecise numerical quantifieres. In Z.W. Ras and M. Zemankova, editors, Intelligent Systems - state of the art and future directions, pages 52--72. Ellis Horwood Ltd., Chichester, England, 1990.

.... 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 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

.... Coming back to our introductory example, we realize that G1 G3 and E1 E2 represent interval restrictions for conditional probabilities, also called conditional constraints (Lukasiewicz 1999b) The literature contains extensive work on reasoning about conditional constraints (Dubois Prade 1988; Dubois et al. 1990; 1993; Amarger et al. 1991; Jaumard et al. 1991; Thone et al. 1992; Frisch Haddawy 1994; Heinsohn 1994; Luo et al. 1996; Lukasiewicz 1999a; 1999b) and their generalizations, for example, to probabilistic logic programs (Lukasiewicz 1998) Now, the main idea of this paper is to use techniques ....

Dubois, D.; Prade, H.; and Touscas, J.-M. 1990. Inference with imprecise numerical quantifiers. In Ras, Z. W., and Zemankova, M., eds., Intelligent Systems. Ellis Horwood. chapter 3, 53--72.

.... 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 ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

....framework is still NP complete. In early work, Dubois and Prade (1988) use inference rules to model default reasoning with imprecise numerical and fuzzy quantifiers. For this reason, subsequent research on inference rules especially aims at analyzing patterns of human commonsense reasoning (Dubois et al. 1990, 1993; Amarger et al. 1991; Thone, 1994; Thone et al. 1995) Frisch and Haddawy (1994) discuss the use of inference rules for deduction in probabilistic logic. Recent work on inference rules concentrates on integrating probabilistic knowledge into description logics (Heinsohn, 1994) and on ....

....Inspired by previous work on inference rules, we focus our research on the language of conditional constraints over basic events: Dubois and Prade (1988) study the chaining of two bidirectional conditional constraints over basic events ( quantified syllogism rule ) and some of its special cases. Dubois et al. 1990) additionally discuss probabilistic deductions about conjunctions of basic events. Furthermore, they describe the open problem of probabilistic deduction along a chain of more than two bidirectional conditional constraints over basic events. In later work, Dubois Probabilistic Deduction with ....

Dubois, D., Prade, H., & Touscas, J.-M. (1990). Inference with imprecise numerical quantifiers.

....a more detailed theoretical analysis must follow in future work) In [18] we identified important deduction problems in probabilistic propositional logics in which the presented technique works very well. So, the problem of deduction along a bidirectional chain of basic events (see, for example, [8]) results in a quadratic number of variables in the length of the chain (the classical linear programs have an exponential number of variables in the length of the chain) Moreover, probabilistic deduction in hierarchies of basic events and of disjoint basic events also produces a very low number ....

D. Dubois, H. Prade, and J.-M. Touscas, `Inference with imprecise numerical quantifiers', in Intelligent Systems, eds., Z. W. Ras and M. Zemankova, chapter 3, 53--72, Ellis Horwood, (1990).

....of the probabilistic deduction problems. Moreover, it cannot provide any explanatory informations on how the deduced results are obtained. Mainly to overcome these deficiencies, researchers started to work on local techniques based on inference rules. The local approach (see, for example, 7] [9], 2] 8] 25] 11] 13] and [16] is generally performed within more restricted probabilistic languages. The iterative application of inference rules is very rarely and only within very restricted probabilistic languages globally complete (see [11] for an example of globally complete local ....

.... ) z 1 ; z 2 ] AB jC ) z 1 ; z 2 ] We chose these inference rules, since there is already a quite extensive literature on similar inference rules, which are locally complete for biconnected chains of three pairwise different basic events without any taxonomic knowledge beside (see, for example, [9], 2] 25] 8] and [13] Hence, the selected inference rules seem to be quite important, and they also have well explored counterparts in restricted frameworks, which may serve for comparisons. It remains to compute the deduced tightest bounds in the selected inference rules. Let us first give ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

....framework is still NP complete. In early work, Dubois and Prade (1988) use inference rules to model default reasoning with imprecise numerical and fuzzy quantifiers. For this reason, subsequent research on inference rules especially aims at analyzing patterns of human commonsense reasoning (Dubois et al. 1990, 1993; Amarger et al. 1991; Thone, 1994; Thone et al. 1995) Frisch and Haddawy (1994) discuss the use of inference rules for deduction in probabilistic logic. Recent work on inference rules concentrates on integrating probabilistic knowledge into description logics (Heinsohn, 1994) and on ....

....Inspired by previous work on inference rules, we focus our research on the language of conditional constraints over basic events: Dubois and Prade (1988) study the chaining of two bidirectional conditional constraints over basic events ( quantified syllogism rule ) and some of its special cases. Dubois et al. 1990) additionally discuss probabilistic deductions about conjunctions of basic events. Furthermore, they describe the open problem of probabilistic deduction along a chain of more than two bidirectional conditional constraints over basic events. In later work, Dubois Probabilistic Deduction with ....

Dubois, D., Prade, H., & Touscas, J.-M. (1990). Inference with imprecise numerical quantifiers.

....the global precision of probabilistic deductions. However, all global techniques developed so far run in exponential time in the size of the probabilistic deduction problems. Mainly this lack of efficiency of global techniques encouraged many researchers to work on local methods (see e.g. 8] [9], 1] 11] 28] 16] and [19] Probabilistic deductions by local techniques are generally performed from more restricted languages. These restrictions are necessary if attempting to gain global precision and efficiency. The local approach generally does not guarantee the global precision of ....

....the viewpoint of linear programming. We present a new, efficient linear programming approach to probabilistic deduction from probabilistic knowledge bases over conjunctive events. We show that it solves the classical problem of probabilistic deduction along a chain of basic events (see especially [9], 1] 28] and [20] in a very efficient and elegant way. As in [20] we are also concerned about probabilistic deduction under taxonomic knowledge in this paper. We elaborate how taxonomic knowledge can be exploited for an increased efficiency in both our new and the classical approach to ....

[Article contains additional citation context not shown here]

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53-- 72. Ellis Horwood, 1990.

....of the probabilistic deduction problems. Furthermore, it cannot provide any explanatory informations on how the deduced results are obtained. Mainly to overcome these deficiencies, researchers started to work on local techniques based on inference rules. The local approach (see, for example, 7] [9], 2] 8] 28] 11] 15] and [18] is generally performed within more restricted probabilistic languages. The iterative application of inference rules is very rarely and only within very restricted languages globally complete (see [11] for an example of globally complete local probabilistic ....

D. Dubois, H. Prade, and J.-M. Touscas. Inference with imprecise numerical quantifiers. In Z. W. Ras and M. Zemankova, editors, Intelligent Systems, chapter 3, pages 53--72. Ellis Horwood, 1990.

....framework is still NP complete. In early work, Dubois and Prade (1988) use inference rules to model default reasoning with imprecise numerical and fuzzy quantifiers. For this reason, subsequent research on inference rules especially aims at analyzing patterns of human commonsense reasoning (Dubois et al. 1990, 1993; Amarger et al. 1991; Thone, 1994; Thone et al. 1995) Frisch and Haddawy (1994) discuss the use of inference rules for deduction in probabilistic logic. Recent work on inference rules concentrates on integrating probabilistic knowledge into description logics (Heinsohn, 1994) and on ....

....Inspired by previous work on inference rules, we focus our research on the language of conditional constraints over basic events: Dubois and Prade (1988) study the chaining of two bidirectional conditional constraints over basic events ( quantified syllogism rule ) and some of its special cases. Dubois et al. 1990) additionally discuss probabilistic deductions about conjunctions of basic events. Furthermore, they describe the open problem of probabilistic deduction along a chain of more than two bidirectional conditional constraints over basic events. In later work, Dubois Probabilistic Deduction with ....

Dubois, D., Prade, H., & Touscas, J.-M. (1990). Inference with imprecise numerical quantifiers.

.... and no prior probability information is required in order to start the inference process in the approach described in this paper (contrary to Quinlan (1983) s INFERNO system or Baldwin (1990) s support logic programming) Other works have been published, that handle probability bounds (see (Dubois et al. 1990) for a survey) However, these works always assume knowledge about unconditional probabilities (i.e. P(A) P(A X) in our framework) and are often oriented towards the computation of unconditional probabilities P(B) This is not true here. The reasoning systems of Bacchus (1990) aim at embedding ....

....to present computational methods that can handle imprecisely known conditional probabilities. This work pursues an earlier investigation. In Dubois and Prade (1988) see also L aSomb (1990) a first local pattern of reasoning, corresponding to the transitive chaining syllogism was studied. In (Dubois et al. 1990) two other local patterns enable us to estimate conditional probabilities involving conjunctions of events or contexts in their expression. A more complete set of propagation rules is presented in (Amarger et al. 1991) After presenting the problem is section 2, section 3 recalls how our problem ....

[Article contains additional citation context not shown here]

D. Dubois, H. Prade, and J-M. Toucas (1990) Inference with imprecise numerical quantifiers. In : Intelligent Systems: State of the Art and Future Directions (Z.

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Dubois, D., H. Prade, and J.-M. Touscas: 1990, `Inference with imprecise numerical quantifiers'. In: Z. W. Ras and M. Zemankova (eds.): Intelligent Systems. Ellis Horwood, Chapt. 3, pp. 53--72.

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