Allocation of arguments and evidence theory. (English) Zbl 0874.68275

Summary: The Dempster-Shafer theory of evidence is developed here in a very general setting. First, its symbolic or algebraic part is discussed as a body of arguments which contains an allocation of support and an allowment of possibility for each hypothesis. It is shown how such bodies of arguments arise in the theory of hints and in assumption-based reasoning in logic. A rule of combination of bodies of arguments is then defined which constitutes the symbolic counterpart of Dempster’s rule. Bodies of evidence are next introduced by assigning probabilities to arguments. This leads to support and plausibility functions on some measurable hypotheses. As expected in Dempster-Shafer theory, they are shown to be set functions, monotone or alternating of infinite order, respectively. It is shown how these support and plausibility functions can be extended to all hypotheses. This constitutes then the numerical part of evidence theory. Finally, combination of evidence based on the combination of bodies of arguments is discussed and a generalized version of Dempster’s rule is derived. The approach to evidence theory proposed is general and is not limited to finite frames.


68T27 Logic in artificial intelligence
Full Text: DOI


[1] Besnard, P.; Kohlas, J., Evidence theory based on general consequence relations, Int. J. Foundations Comput. Sci., 6, 119-135 (1995) · Zbl 0830.68112
[2] Darwiche, A. Y., Argument calculus and networks, (Heckermann, D.; Mamdani, A., Proc. 9th Conf. on Uncertainty in Artificial Intelligence (1993), Kaufmann, Morgan: Kaufmann, Morgan Los Altos, CA)
[3] De Kleer, J., An assumption-based TMS, Art. Int., 28, 127-162 (1989)
[4] Dempster, A., Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist., 38, 325-339 (1967) · Zbl 0168.17501
[5] Dubois, D.; Prade, H., Théorie des possibilités. Applications à la représentation des connaisances en informatique (1985), Masson: Masson Paris · Zbl 0674.68059
[6] Elvang-Gøransson, M.; Krause, P. J.; Fox, J., Acceptance of arguments as ‘logical uncertainty’, (Clarke, M.; Kruse, R.; Moral, S., Symbolic and Quantitative Aproaches to Reasoning and Uncertainty (1993), Springer: Springer Berlin), 85-90
[7] Feller, W., (An Introduction to Probability Theory and Its Applications, Vol. 1 (1968), Wiley: Wiley New York) · Zbl 0155.23101
[8] Halmos, P. R., Lectures on Boolean Algebras (1963), Van Nostrand: Van Nostrand Princeton, NJ · Zbl 0114.01603
[9] Hestir, K.; Nguyen, H. T.; Rogers, G. S., A random set formalism for evidential reasoning, (Goodman, I. R.; etal., Conditional Logic in Expert Systems (1991)), 309-344
[10] Kohlas, J., The reliability of reasoning with unreliable arguments, Ann. Oper. Res., 32, 67-113 (1991) · Zbl 0732.68095
[11] Kohlas, J., Support- and plausibility functions induced by filter-valued mappings, Int. J. Gen. Syst., 21, 343-363 (1993) · Zbl 0780.60006
[12] Kohlas, J., Symbolic evidence, arguments, supports and valuation networks, (Clarke, M.; Kruse, R.; Moral, S., Symbolic and Quantitative Approaches to Reasoning and Uncertainty (1993), Springer: Springer Berlin), 186-198
[13] Kohlas, J., Mathematical foundations of evidence theory, (Coletti, G.; Dubois, D.; Scozzafava, R., Mathematical Models for Handling Partial Knowledge in Artificial Intelligence (1995), Plenum: Plenum New York), 31-64 · Zbl 0859.68103
[14] Kohlas, J.; Monney, P. A., Probabilistic assumption-based reasoning, (Heckermann, D.; Mamdani, A., Proc. 9th Conf. on Uncertainty in Artificial Intelligence (1993), Kaufmann, Morgan: Kaufmann, Morgan Los Altos, CA) · Zbl 0917.68202
[15] Kohlas, J.; Monney, P. A., Representation of evidence by hints, (Yager, R.; etal., Recent Advances in Dempster-Shafer Theory (1994), Wiley: Wiley New York), 473-492
[16] Kohlas, J.; Monney, P. A., A Mathematical Theory of Hints. An Approach to Dempster-Shafer Theory of Evidence, (Lecture Notes in Economics and Mathematical Systems, Vol. 425 (1995), Springer: Springer Berlin) · Zbl 0833.62005
[17] Laskey, K. B.; Lehner, P. E., Belief maintenance: an integrated approach to uncertainty management, (Proc. Amer. Ass. AI (1988)), 210-214
[18] Neveu, J., Bases mathématiques du calcul des probabilités (1964), Mason: Mason Paris · Zbl 0137.11203
[19] Nguyen, H. T., On random sets and belief functions, J. Math. Anal. Appl., 65, 531-542 (1987) · Zbl 0409.60016
[20] Pearl, J., Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference (1988), Morgan Kaufman: Morgan Kaufman San Mateo, CA
[21] Provan, G. M., A logic-based analysis of Dempster-Shafer theory, Int. J. Approx. Res., 4, 451-495 (1990) · Zbl 0706.68087
[22] Shafer, G., A Mathematical Theory of Evidence (1976), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0359.62002
[23] Shafer, G., Allocations of probability, Ann. Probab., 7, 827-839 (1979) · Zbl 0414.60002
[24] Shafer, G., An Axiomatic Study of Computations in Hypertrees, (School of Business WP 232 (1991), The University of Kansas: The University of Kansas Lawrence)
[25] Shenoy, P.; Shafer, G., Axioms for probability and belief function propagation, (Shachter; etal., Uncertainty AI, 4 (1990)), 575-610
[26] Sikorski, R., Boolean Algebras (1960), Springer: Springer Berlin · Zbl 0191.31505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.