Probabilistic Approach to Measuring Inconsistency .1 Measures of Inconsistency

Although it is the overruling opinion that conflicting knowledge bases are undesirable, they easily appear in practice when experts share their knowledge in order to build knowledge of the group. Hunter and Konieczny [39] give the example of a group of clinicians advising on a patient. Although every set of formulasTiwhich represents the opinion ofi-th clinician is consistent, it is possible that their union[Tiis inconsistent.

As inconsistent information is often the only available information, we have to deal with it. Many logical formalisms are proposed for reasoning under inconsis-tency, like paraconsistent logics, default reasoning, possibility theory, belief revi-sion and formal argumentation [5–7,19, 26, 27,38, 40, 41, 53, 54, 56,81, 95, 97–99]. Unlike classical logic, they enable drawing nontrivial conclusions from inconsistent knowledge bases, where two different inconsistent sets can lead to different sets of conclusions.

Development of those techniques points out to the need for analyzing and comparing inconsistent sets [39]. It is definitely not true that all inconsistent sets are equally bad. For example, in the previously mentioned example, it makes the difference whether each two clinicians are in conflict or only several of them taken together are in conflict. Similarly, some inconsistent information are less significant than the other (two conflicting data about weight of a patient in a database are often irrelevant for making a diagnosis from the other information from database).

There are various applications of measures of conflicts. If we have to choose between several inconsistent databases, measures of inconsistency may be used for answering the question which one is the most tolerable. Similarly, in diagnosis, if assumptions are in conflict with observed symptoms, a measure of contradiction may be useful for detecting the wrong assumption. All that indicates that approaches to measure the degree of inconsistency need to be context sensitive (and different approaches lead to measures incompatible with one another [33]).

In one approach, the measure of inconsistency depends on the proportion of the language that is affected by the inconsistency in a theory [31, 51]. The second approach considers the number of formulas needed to produce a contradiction. This idea implies that the set of formulasϕ1,. . .,ϕnis not equivalent to the singleton {ϕ1^ ^ϕn} (this property is valid in the so-called non-adjunctive logics [52]).

The idea to consider the distribution of contradiction among the formulas approach turns to be closely related to probabilistic measure on formulas. In [11,50], this idea was compared with semantic notion of existence of a probability measure which assigns a high probability to each formula of the theory.

Besides existence of probability-based measures of inconsistency of proposi-tional theories, some researchers investigate degrees of inconsistency of sets of probabilistic formulas. In [79,102], the level of inconsistency of a set of probabilis-tic formulas is proportional to distance from the function on formulas (given by the conditions of theory) to the closest probabilistic measure on formulas. This approach has a potential in analyzing CADIAG-2 expert system with applications in medicine.

2.4.2 Probabilistic Approach for Measuring Propositional Theories

According to Sorensen, the significance of any inconsistency is correlated to the minimal number of formulas that are needed for its derivation. The first formalization of this idea was presented in [50] by Knight. He proposed a real numberη∈[0,1] as a degree of consistency of a given set of formulasT, based on the existence of the appropriate probability measure on formulas, and called it η-consistency:

• A knowledge baseTisη-consistent iff there exists a probability measureμon formulas such thatμ(ϕ)ηfor allϕ∈T.

• An η-consistent knowledge base T is maximally η-consistent iff it is not ζ-consistent for allζ<η.

In particular, consistency degree of any consistent theory (knowledge base) is equal to 1. On the other hand, the worst theories (0-consistent) are those containing a contradictory formula. It is proved in [50] that every theory T is maximally η-consistent for some rational numberη.

Example. For a patient with a sniffle, we may consider the following two hypotheses as possible causes:

• A: The patient has a flu.

• B: The patient is allergic to pollen.

In addition, let us assume thatAandBhave not occurred simultaneously in our patient. Denote the last assumption byC. Now our knowledge baseT ¼{A,B,C} is inconsistent, sinceC is equivalent to ØA_ØB. It can be shown that T is ²₃ -consistent.Tis also minimally inconsistent in the sense that any proper subset of Tis consistent.

Knight proved the following characterization of minimally inconsistent knowl-edge bases:

• If T is a minimally inconsistent set of formulas, then T is also maximally 1_{j jT}¹ -consistent.

For arbitrary inconsistent sets, the following generalization holds:

• IfT⁰is the smallest minimally inconsistent subset ofT, thenTis^{j j1T}_{j j}⁰_T -consistent.

Note that maximality ofηis not stated in the previous result. Precise relationship between syntactic notion of cardinality of the smallest minimal inconsistent subset of a knowledge base and semantic approach based on probability measures is studied in [11]. For that purpose, the following notion is introduced:

• A theoryTisn-consistent iff each subsetT⁰ofTwithnelements is consistent.

The notion ofn-consistency is a syntactical notion and it is not comparable with the Knight’sη-consistency. The corresponding semantical notion is the notion ofn-probability:

• A theory T is n-probable if there exists a probability measure μ such that μ ϕð Þ>1¹_n for all ϕ∈T.

The notion ofn-probability is similar to a special case of Knight’sη-consistency, for η¼1¹_n. It turns out that then-consistency is a weaker property than then -probabil-ity, i.e., everyn-probable theory isn-consistent, but there is ann-consistent theory that is not n-probable (for arbitrary n). The semantical notion that turns out to be the probabilistic analogue ofn-consistency is local variant of then-consistency is given by

• T is locallyn-probable if each subset ofTwithn+ 1 elements isn-probable.

A theoryTis locallyn-probable if and only if it isn-consistent.

In [11] we also analyze inconsistent knowledge split into two parts: one part consists of consistent set of the “facts” {ϕ1,. . .,ϕk}, representing the knowledge of kexperts, and the second part consisting of some statements believed to be probable by at least one expert, on the basis of his knowledge.

Conditional probabilities are used to introduce the notion ofn-probability modulo

“facts.” The additional condition is that the expert’s beliefs are highly compatible.

A theory T is said to be n-probable modulo {ϕ1,. . .,ϕk}, if there exists a probability measureμwith the following properties:

1. μðϕ1^ ^ϕkÞ>0: 2. μ ϕi ϕj

>1¹_nfor all i, j∈f1;. . .;kg:

3. For allψ ∈T, there existsi∈{1, . . .,k} such thatμ ψ ϕ_i

>1¹_n. Then the following result holds:

• If theoryTisn-probable modulo {ϕ1,. . .,ϕk}, then Tis (n k+ 1)-probable, hence (n k+ 1)-consistent.

A syntactic notion ofn-consistency modulo a set of formulas is also introduced:

T isn-consistent moduloSiff for anyψ1,. . .,ψn∈T, the theory {ψ1,. . .,ψn}[S is consistent. For the finiteS, the following result is proved:

• IfTisn-probable modulo {ϕ1,. . .,ϕk}, thenTis (n k+ 1)-consistent modulo {ϕ1,. . .,ϕk}.

2.4.3 Measuring Inconsistency in Probabilistic Knowledge Bases

In [79, 102], an approach to measuring degree of inconsistency of probabilistic knowledge bases is developed. Probabilistic knowledge base is any finite set of conditional statements, with probability values attached. The measures of

inconsistency are based on distance from the knowledge base to the set of probability measures.

In [102], a probabilistic knowledge base is a set of probabilistic constraints of the form (ϕ|ψ)[d], where the pair of formulas (ϕ|ψ) is a conditional statement and d∈[0,1]. A knowledge base is said to be satisfiable, if there is a probability measure \mu on formulas, such thatμ(ϕ|ψ)¼dfor every element (ϕ|ψ)[d] of the base. In order to avoid case differentiation when μ(ψ)¼0, it is assumed that μ(ϕ|ψ)¼diffμ(ϕ ^ψ)¼dμ(ψ). Inc*(T), is obtained by minimization of the above sum, whenμranges the set of probability measures. The following properties ofInc* are shown in [102]:

• Consistency: IfTis consistent, thenInc*(T)¼0.

• Inconsistency: IfTis inconsistent, thenInc*(T)>0.

• Monotonicity:Inc*(T)Inc*(T[{(ϕ|ψ)[d]}).

• Super-additivity: IfT\T⁰ ¼∅, thenInc*(T[T⁰)Inc*(T) +Inc*(T⁰).

• Weak Independence: If no propositional letter fromϕ,ψappears inT, thenInc* (T)¼Inc*(T[{(ϕ|ψ)[d]}).

• Independence: If (ϕ|ψ)[d] does not appear in any minimal inconsistent subset of T, thenInc*(T)¼Inc*(T[{(ϕ|ψ)[d]}).

• Penalty: If (ϕ|ψ)[d] does appear in some minimal inconsistent subset ofT, then Inc*(T)<Inc*(T [{(ϕ|ψ)[d]}).

• Continuity: The function Inc*({(ϕi|ψi)[xi]|i¼1,. . .,n}) is continuous in all variablesxi.

All of the above properties also hold for the normalizationInc0*ofInc*, defined byInc0ð Þ ¼T ^Inc_{j jT}^{ð ÞT}.

In [79], approach from [102] is extended in several ways:

• Interval-valued probabilistic constraints of the form ϕ ψ l;u

½ (where 0lu1 ) are allowed, with the standard probabilistic interpretation (if l¼u, we writeϕ ψ

l

½ as above).

• Inc* is generalized to the family of measuresDC^p(called “p-distances to consis-tency”), forp∈[1, +1] (actually,DC^pΓis defined, whereΓrepresents correctly evaluated conditional statements, which should not be considered when assigning thep-distance of a theory), defined by the so-calledp-norm. For the finite p the corresponding p-norm is defined by kx1;. . .;x_nk_p¼Σⁿ1j jx_i^p1

p, while for p ¼+ 1 the corresponding p-norm is defined by kx1,. . .,xnk1¼ max(|x1|,. . .,|xn|).

• The extended definition of conditional probability from [102], which allowed null probability for the conditioning event, is replaced by standard definition.

• Inconsistency values range over the setft,tþε, 1=ε:t∈g, whereε>0 is an infinitesimal.

Here, operations of nonstandard field of real numbers are not considered, andεis only used to distinguish knowledge bases that are “closer to consistency” from the rest.

, thenpdistance to consistency DC^pis defined as follows:

DC^pð Þ ¼T

Note that 1-norm is used in the definition of Inc*. Let us try to clarify the definition ofDC^pon the following CADIAG-2 example taken from [79]:

Example. Consider the unsatisfiable set of CADIAG-2 rules:

T¼ D36 S157

Letμnbe the sequence of probability measures on the set of formulas generated byD36,D81 andS157, such that:

Then the following equalities also hold:

• μ_nD36 S157

The following properties of the measuresDC^pare presented in [79]:

• DC^pð Þ T DC^pðT[T⁰Þ:

• DC^p(T)¼0 iffTis satisfiable.

• Ifp<q, thenDC^p(T)DC^q(T).

• Ifp 6¼q, then DC^pand DC^q induce distinct orderings, i.e., we can construct TandT⁰such thatDC^p(T)>DC^p(T⁰) andDC^q(T) <DC^q(T⁰).

It is also shown thatDC^pð Þ T j jT^1p, which enabled normalizationDC^pofDC^p (see [79] for details), with the property.

• Ifp<q, thenDC^pð Þ T DC^qð Þ, contrary toT DC^p.

The same author presented similar approach for measuring inconsistency in fuzzy knowledge bases in [80]. Measures presented in this section are also used to develop strategies for repairing inconsistent knowledge bases.

2.5 Evidence and PST Logics