## Belief Functions and Default Reasoning (2000)

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Citations: | 35 - 4 self |

### BibTeX

@MISC{Benferhat00belieffunctions,

author = {S. Benferhat and A. Saffiotti and P. Smets},

title = {Belief Functions and Default Reasoning},

year = {2000}

}

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### Abstract

We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilon-belief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show that these structures can be used to give a uniform semantics to several popular non-monotonic systems, including Kraus, Lehmann and Magidor's system P, Pearl's system Z, Brewka's preferred sub-theories, Geffner's conditional entailment, Pinkas' penalty logic, possibilistic logic and the lexicographic approach. In the second part, we use epsilon-belief assignments to build a new system, called LCD, and show that this system correctly addresses the well-known problems of specificity, irrelevance, blocking of inheritance, ambiguity, and redundancy.

### Citations

7493 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ...s paper (1980), many proposals for reasoning with default information have appeared in the literature. Some of them are based on the use of uncertainty models such as probability theory (Adams, 1975, =-=Pearl, 1988-=-), or possibility theory (Dubois & Prade, 1988, Benferhat et al., 1992). Up to now, however, no single system has been reported that fulfils all of the desiderata above. In this paper, we show how we ... |

2456 |
A Mathematical Theory of Evidence
- Shafer
- 1976
(Show Context)
Citation Context ...p to now, however, no single system has been reported that fulfils all of the desiderata above. In this paper, we show how we can use belief functions, originally developed for modelling uncertainty (=-=Shafer, 1976-=-, Smets, 1988, Smets and Kennes, 1994), to build a non-monotonic system that gives a satisfactory answer to all of the above issues. There have already been a few works on representing default informa... |

1484 | A logic for default reasoning - Reiter - 1980 |

816 |
Statistical reasoning with imprecise probabilities
- Walley
- 1991
(Show Context)
Citation Context ...rmalized belief functions. e-Bel-Fct July 27, 1999 8 It is important to emphasize that our approach is based on the transferable belief model (Smets, 1997a), and not on upper and lower probabilities (=-=Walley, 1991-=-). Accordingly, the term bel(A) should be read as the belief given by the agent to the fact that the actual world belongs to A, and not as the lower envelop of a family of probability functions repres... |

560 | Nonmonotonic reasoning, preferential models and cumulative logics
- Kraus, Magidor
- 1990
(Show Context)
Citation Context ...and Dubois and Prade (1995) have shown that if we allow arbitrary (not necessarily extreme) values, then the corresponding consequence relation does not satisfy the rationality postulates defined by (=-=Kraus et al., 1990-=-). Lehmann and Magidor (1992) have shown that e-consequence is equivalent to system P 3 . This means that Adams' construction gives a (probability-based) semantics for P, unfortunately, this also mean... |

395 | The transferable belief model
- Smets, Kennes
- 1994
(Show Context)
Citation Context ...e system has been reported that fulfils all of the desiderata above. In this paper, we show how we can use belief functions, originally developed for modelling uncertainty (Shafer, 1976, Smets, 1988, =-=Smets and Kennes, 1994-=-), to build a non-monotonic system that gives a satisfactory answer to all of the above issues. There have already been a few works on representing default information with belief functions, e.g., (Hs... |

364 |
Fuzzy sets as a basis for a theory of possibility
- Zadeh
- 1978
(Show Context)
Citation Context ...lled a consonant belief function, in xhich case, for all A,BW, bel(AB) = min(bel(A), bel(B)), and pl(AB) = max(pl(A), pl(B)). In this case, bel is a necessity measure and pl is a possibility measure (=-=Zadeh, 1978-=-, Dubois and Prade, 1988). If m has at most one focal element AsW, As, i.e., m(A) = s, m(W) = 1--s, m(elsewhere) = 0, ss[0,1], then its related belief function is called a simple support function. Bel... |

241 | What does a conditional knowledge base entail
- Lehmann
- 1992
(Show Context)
Citation Context ... to 0 or to 1." The study of such expressions is the object of non-standard analysis: we address the reader to (Robinson, 1965), (Keisler, 1976) and (Nelson, 1977) for some foundational work, and=-= to (Lehmann and Magidor, 1992-=-), (Weyderd, 1995) and (Wilson, 1996) for applications to probability theory. For the limited goals of this paper, however, we do not need a full theory of non-standard analysis. In what follows, we o... |

236 |
Reasoning About Change: Time and Causation from the Standpoint of Artificial Intelligence
- Shoham
- 1988
(Show Context)
Citation Context ...r approach to defining non-monotonic systems using (epsilon-) belief functions. 3.3 A preferential view Non-monotonic consequence relations are often characterized in terms of preferential semantics (=-=Shoham, 1988-=-, Kraus et al., 1990). In a nutshell, a preferential model is a structure W = (S, f,swhere S is a finite set of states, f maps each state to a world, andsThessometimes called a preference relation. Gi... |

236 | Nonstandard Analysis - Robinson - 1966 |

235 | Possibilistic logic
- Dubois, Lang, et al.
- 1994
(Show Context)
Citation Context ...orithm coincides with the one of system Z tells us that the LC consequence precisely coincides with Pearl's system, and those systems equivalent to system Z like the one based on possibilistic logic (=-=Dubois et al., 1994-=-, Benferhat et al., 1992). Theorem 2. For a given D, a + lcsb if, and only if, a + Zsb. Proof. Immediate consequence of Lemma 8. y The stratification produced by Pearl's algorithm or by our LC algorit... |

219 |
Probability and the logic of conditionals
- Adams
- 1966
(Show Context)
Citation Context ...since Reiter's paper (1980), many proposals for reasoning with default information have appeared in the literature. Some of them are based on the use of uncertainty models such as probability theory (=-=Adams, 1975-=-, Pearl, 1988), or possibility theory (Dubois & Prade, 1988, Benferhat et al., 1992). Up to now, however, no single system has been reported that fulfils all of the desiderata above. In this paper, we... |

213 | Preferred Subtheories: An Extended Logical Framework for Default Reasoning - Brewka - 1989 |

189 | Theoretical foundations for non-monotonic reasoning in expert systems - Gabbay - 1985 |

173 | System Z : a natural ordering of defaults with tractable applications for default reasoning - Pearl - 1990 |

166 | Inconsistency management and prioritized syntax-based entailment
- Benferhat, Cayrol, et al.
- 1993
(Show Context)
Citation Context ...quence relations + lcd and + Z are incomparable. Another solution to get non-monotonic inferences that go beyond those of system P is to use the so-called lexicographic approach (Dubois et al., 1991, =-=Benferhat et al., 1993-=-, Lehmann, 1993). One well know drawback of this approach is that it is syntax-sensitive: repetitions of the same default in ∆ may change the result. We show that LCD is not sensitive to duplications.... |

133 | On interacting defaults - Reiter, Criscuolo |

131 |
The combination of evidence in the transferable belief model
- Smets
- 1990
(Show Context)
Citation Context ...ence A: bel(X|A) = bel(XA c ) -- bel(A c ) 1 -- bel(A c ) , pl(X|A) = pl(XA) pl(A) . A formal justification of both Dempster's rule of combination and Dempster's rule of conditioning can be found in (=-=Smets, 1990, 1997b) a-=-nd (Klawonn and Smets, 1991). Distinctness is defined in (Smets, 1992). 2.4. Infinitesimals and the induced negligibility relation In this work, we consider "extreme" belief functions that t... |

107 | Combining knowledge bases consisting of first-order theories - Baral, Kraus, et al. - 1992 |

102 |
A mathematical theory of hints: An approach to DempsterShafer theory of evidence
- Kohlas, Monney
- 1995
(Show Context)
Citation Context ...functions. In order to make this paper self-contained, we recall here the basic definitions. For a more complete exposition, we refer the reader to (Shafer, 1976, Smets, 1988, Smets and Kennes, 1994, =-=Kohlas and Monney, 1995-=-, Smets, 1998). Let W be a finite set of worlds, one of them being the actual world. A (normal 4 ) basic belief assignment on W is a function m: 2 Ws[0,1] that satisfies, m() = 0, AWsm(A) = 1. The ter... |

100 | Representing default rules in possibilistic logic
- Benferhat, Dubois, et al.
- 1992
(Show Context)
Citation Context ...nformation have appeared in the literature. Some of them are based on the use of uncertainty models such as probability theory (Adams, 1975; Pearl, 1988), or possibility theory (Dubois & Prade, 1988; =-=Benferhat et al., 1992-=-). Up to now, however, no single system has been reported that fulfils all of the desiderata above. In 1 Note that this is different from the situation in which the same argument supports both a concl... |

95 | Default reasoning: causal and conditional theories - Geffner - 1992 |

88 | Qualitative probabilities for default reasoning, belief revision, and causal modeling - Goldszmidt - 1996 |

84 | Another perspective on default reasoning
- Lehmann
- 1995
(Show Context)
Citation Context ...nd + Z are incomparable. Another solution to get non-monotonic inferences that go beyond those of system P is to use the so-called lexicographic approach (Dubois et al., 1991, Benferhat et al., 1993, =-=Lehmann, 1993-=-). One well know drawback of this approach is that it is syntax-sensitive: repetitions of the same default in ∆ may change the result. We show that LCD is not sensitive to duplications. Example 7. Con... |

59 |
Belief functions
- Smets
- 1988
(Show Context)
Citation Context ...ver, no single system has been reported that fulfils all of the desiderata above. In this paper, we show how we can use belief functions, originally developed for modelling uncertainty (Shafer, 1976, =-=Smets, 1988-=-, Smets and Kennes, 1994), to build a non-monotonic system that gives a satisfactory answer to all of the above issues. There have already been a few works on representing default information with bel... |

58 | 1997] “Internal Set Theory, a new approach to nonstandard analysis
- Nelson
(Show Context)
Citation Context ...recise meaning to the expression "infinitely close to 0 or to 1." The study of such expressions is the object of non-standard analysis: we address the reader to (Robinson, 1965), (Keisler, 1=-=976) and (Nelson, 1977-=-) for some foundational work, and to (Lehmann and Magidor, 1992), (Weyderd, 1995) and (Wilson, 1996) for applications to probability theory. For the limited goals of this paper, however, we do not nee... |

46 | Using crude probability estimates to guide diagnosis - Kleer - 1990 |

38 | Propositional nonmonotonic reasoning and inconsistency in symmetric neural networks - Pinkas - 1991 |

34 | Generalized qualitative probability: Savage revisited - Lehmann - 1996 |

34 | Implicit Ordering of Defaults in Inheritance Systems
- Touretzky
- 1982
(Show Context)
Citation Context ...tionality postulates proposed by Kraus, Lehmann and Magidor (1990) (see section 2.1). . Specificity: results obtained from more specific classes should override those obtained from more generic ones (=-=Touretzky, 1984). For example, from the defau-=-lt base D = {"Birds fly", "Penguins do not fly", "Penguins are birds"}, one should deduce that Tweety, who is both a penguin and a bird, does not fly, since penguins are ... |

33 |
Foundations of Infinitesimal Calculus
- Keisler
- 1976
(Show Context)
Citation Context ...s, we need to give precise meaning to the expression "infinitely close to 0 or to 1." The study of such expressions is the object of non-standard analysis: we address the reader to (Robinson=-=, 1965), (Keisler, 1976-=-) and (Nelson, 1977) for some foundational work, and to (Lehmann and Magidor, 1992), (Weyderd, 1995) and (Wilson, 1996) for applications to probability theory. For the limited goals of this paper, how... |

30 | Average-case Analysis of a Search Algorithm for Estimating Prior and Posterior Probabilities in Bayesian Networks with Extreme Probabilities - Poole - 1993 |

30 | Possibilistic and standard probabilistic semantics of conditional knowledge bases - Benferhat, Dubois, et al. - 1999 |

28 |
The dynamic of belief in the transferable belief model and specialization-generalization matrices
- Klawonn, Smets
- 1992
(Show Context)
Citation Context ... c ) -- bel(A c ) 1 -- bel(A c ) , pl(X|A) = pl(XA) pl(A) . A formal justification of both Dempster's rule of combination and Dempster's rule of conditioning can be found in (Smets, 1990, 1997b) and (=-=Klawonn and Smets, 1991). Distinc-=-tness is defined in (Smets, 1992). 2.4. Infinitesimals and the induced negligibility relation In this work, we consider "extreme" belief functions that take values which are infinitely close... |

24 | Nonmonotonic inference based on expectations - Gtirdenfors, Makinson - 1994 |

23 |
The normative representation of quantified beliefs by belief functions
- Smets
- 1997
(Show Context)
Citation Context ...n bel(B|A) and bel(W|A), which corresponds to studying normalized belief functions. e-Bel-Fct July 27, 1999 8 It is important to emphasize that our approach is based on the transferable belief model (=-=Smets, 1997-=-a), and not on upper and lower probabilities (Walley, 1991). Accordingly, the term bel(A) should be read as the belief given by the agent to the fact that the actual world belongs to A, and not as the... |

22 |
A general approach for inconsistency handling and merging information in prioritized knowledge bases
- Benferhat, Dubois, et al.
- 1998
(Show Context)
Citation Context ...ned at the semantical level, hence looking for the syntactic counterpart is an important point. One possible way to find syntactic counterparts is to explore the compilation techniques developped in (=-=Benferhat et al., 1998-=-). Their work provides a syntactic inference for reasoning from stratified knowledge basessusing utilitarist semantics (like the one based on penalty logic). The idea is to transformsinto a new knowle... |

17 | A general approach to specificity in default reasoning - Delgrande, Schaub - 1994 |

17 |
with the collaboration of farreny h., martin- clouaire r. and testemale c). possibility theory an approach to computerized processing of uncertainty
- Dubois, Prade
- 1988
(Show Context)
Citation Context ...ant belief function, in xhich case, for all A,BW, bel(AB) = min(bel(A), bel(B)), and pl(AB) = max(pl(A), pl(B)). In this case, bel is a necessity measure and pl is a possibility measure (Zadeh, 1978, =-=Dubois and Prade, 1988-=-). If m has at most one focal element AsW, As, i.e., m(A) = s, m(W) = 1--s, m(elsewhere) = 0, ss[0,1], then its related belief function is called a simple support function. Belief functions can be par... |

15 | System-Z+: A formalism for reasoning with variable-strength defaults - Goldszmidt, Pearl - 1991 |

15 | Maximum entropy and variable strength defaults - Bourne, Parsons - 1999 |

13 | Conditional objects, possibility theory and default rules - Dubois, Prade - 1995 |

12 |
The concept of distinct evidence
- Smets
- 1992
(Show Context)
Citation Context ...(A) . A formal justification of both Dempster's rule of combination and Dempster's rule of conditioning can be found in (Smets, 1990, 1997b) and (Klawonn and Smets, 1991). Distinctness is defined in (=-=Smets, 1992). 2.4. In-=-finitesimals and the induced negligibility relation In this work, we consider "extreme" belief functions that take values which are infinitely close to 0 or to 1, much in the spirit of Adams... |

12 | Inconsistency in possibilistic knowledge bases - to live or not to live with it - Dubois, Lang, et al. - 1991 |

11 |
The alpha-junctions: combination operators applicable to belief functions
- Smets
- 1997
(Show Context)
Citation Context ...n bel(B|A) and bel(W|A), which corresponds to studying normalized belief functions. e-Bel-Fct July 27, 1999 8 It is important to emphasize that our approach is based on the transferable belief model (=-=Smets, 1997-=-a), and not on upper and lower probabilities (Walley, 1991). Accordingly, the term bel(A) should be read as the belief given by the agent to the fact that the actual world belongs to A, and not as the... |

10 | What is a default priority - Boutilier - 1992 |

10 |
Default reasoning and the transferable belief model
- Smets, Hsia
- 1990
(Show Context)
Citation Context ...d a non-monotonic system that gives a satisfactory answer to all of the above issues. There have already been a few works on representing default information with belief functions, e.g., (Hsia, 1991, =-=Smets and Hsia, 1991-=-). These works require the assessment of numerical values, whose origin is often an open question. Finding a representation free from such assessments would somehow avoid the problem of the origin of ... |

10 | Diverse Confidence Levels in a Probabilistic Semantics for Conditional Logics - Snow - 1999 |

9 | Complexity results for default reasoning from conditional knowledge bases - Eiter, Lukasiewicz |

8 | Probability and the logic of conditionals, in Aspects of Inductive Logic - Adams - 1966 |