Application of a Symbolic Probability Theory to Qualitative Reasoning under Uncertain Information

Daniel PACHOLCZYK LERIA - Faculté des Sciences

Université d’Angers, 2 Boulevard Lavoisier 49045 ANGERS CEDEX Tél. 19 33 41 73 54 68 Fax. 19 33 41 73 54 54 pacho@univ-angers.fr

Gilles HUNAULT LERIA - Faculté des Sciences

Université d’Angers, 2 Boulevard Lavoisier 49045 ANGERS CEDEX Tél. 19 33 41 73 54 68 Fax. 19 33 41 73 54 54

gilles.hunault@univ-angers.fr Jean-Marc PACHOLCZYK LAFORIA - Equipe Théorie des représentations cognitives Institut B. PAscal, CNRS (URA 1095), Université Curie, Paris VI, 4 place Jussieu, 75252 PARIS CEDEX 05

Abstract

This paper deals with the Representation and the Symbolic Management of Uncertain information. After a presentation of some examples, we give the outlines of our Symbolic Probability Theory, ending with a discussion about the size of the the model and its influence on the results. The results given here show that such a Symbolic Probability Theory constitutes a satisfying model of management of Qualitative Reasoning under uncertain information encoded in a qualitative way.

1. INTRODUCTION

We present here some examples of Qualitative Reasoning using a Symbolic Probability Theory. The approach developped here is Qualitative; thus, no Numerical computing will be found. During the last ten years, a new concept, viewed as a Symbolic extension of the classical concept of Probability has been studied in order to represent symbolically the Uncertainty in Bayesian Networks or the Uncertainty of statements of the natural Language. It has already lead to the elaboration of many Symbolic Theories [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. The Symbolic Probability Theory used here has already been proposed by D. Pacholczyk [12, 14, 15, 16]. Its semantic model is built upon the substrate of a many-valued Predicate Logic that has a better expressive power [17] and that allows the Management of Uncertainty and Vagueness [11, 12], both expressed in a qualitative way.

A first scale of degrees of Truth enables to express the graduation of the Vagueness. A second scale of Uncertainty degrees, distinct from the first, is used to express the graduation of the Uncertainty. All the definitions and all the results presented here will be translated with the aid of a certainty function of statements, called Cert. Thus, the statement: ”it is rather-probable that Smith be a very man”, whose formal translation is A ⊧rather Cert(Smith is a very rich man) will be expressed as: Cert(Smith is a very rich man) = rather(Probable).

Section 2 will provide some examples written in Natural Language and their treatment using the formulas and rules from our Symbolic Probability Theory. Section 3 deals with the theoretical framework. Section 4 will discuss about the size of the model and the different possible values in a chosen model, showing that coherent overlapping results can be found for any M.

Remark: Let’s recall that the psychologists consider that the symbolic graduation can not be reasonnably apprehended by man above ten degrees [18, 19, 20].

2. THE EXAMPLES

In many situations, the cognitive agent knows that a statement is either true or false but he is unable to decide which one is the correct. He can, at the most, grade symbolically his Certainty ranging from Impossible to Certain. The graduation v_{α} of his certainty will be expressed linguistically by statements such as ”A is v_{α} probable” or ”the Certainty of A is v_{α}”. That’s why we have set the study of the Uncertainty in the framework of a many-valued first order Logic: the cognitive agent has only a partial knowledge of the Truth. Due to the fact that he doesn’t come close enough to this degree of Certainty in a numerical way but in a symbolic way, which is, by nature, discrete, the evaluation of the uncertainty related to a statement can only be an ”approximation”.

Results are shown with the help of a graduation scale of M symbolic values: B_{M}={v_{α}, α = 1...M}. It is totally ordered by the relation v_{α} ≤ v_{β} ⇔ α ≤ β. For example, with M=7, we can propose as linguistic translation of these symbolic degrees the expressions given by the following set: B_{7}={v_{1}...v_{7}}={impossible, verylittle-probable, little-probable, probable, rather-probable, very-probable, certain}. The following examples deal with real situations, that is, situations that can be found in the real word. If it impossible to give a numerical probability about the considered events, it is always possible to give a certainty state or a certainty degree to them. The examples will be treated in L_{7}. The first example (the magician) illustrates the Principle of Total Certainty while the second example (the fire outbreaks) shows also the the Symbolic Generalised Bayes’ Formula whereas the third (shapes of digits and parity) uses also the Propagation Rule.

2.1 The magician

Each night, in a cabaret, a magician performs a card trick. He chooses a person in the room and let him draw a card out of a pack of 32. He then forecasts systematically the event B: ”You drew an ace”. Of course, he has some accomplices in the room, who are certain to draw an ace. The trick would be perfect if, every night, anonymous person didn’t present as volunteers. Let’s suppose that it is little-probable that ”the person that draws the card is an accomplice” and rather-probable that ”the person that draws the card is not an accomplice”. Knowing that, for a normal person, it is very-little-probable that ”the card is an ace”, our magician would like to know, before performing his show, what certainty he can expect on the fact that ”an ace is drawn”. Let A be the event ”the person is an accomplice”. Since we have Cert(B|A) = certain and Cert(B|¬A) = very-little-probable, by the Total Certainty Formula, we get Cert(B) = probable, so it is probable that ”the person draws an ace”. The risks of contest being too big, he abandons his trick.

2.2 Outbreak of fire and prevention

Lets’ now illustrate the theory with an example for a DDSIS (a French acronym for the Direction of a Department of Fire and Help Services). The Director of this Department wants to know the certainty he may have about the risks of a forest fire in his geographic area. He considers then the SDACR (acronym for a Departmental Scheme of Analysis and Coverage of Risks) classification of areas: ”high risks area” (A1), ”low and middle risks area” (A2) or ”extremely weak risks area” (A3). It is then probable that an area is an A1, little-probable that it is an A2 and very-little-probable that it is an A3. He then esteems that it is very-probable that ”there will be risks of a forest fire” (B), knowing the area is an A1, rather-probable that there will be risks of forest fire knowing the area is an A2, and that it is little-probable that there will be risks of forest fire knowing the are is an A3. With the Total Certainty Formula, our director knows then that it is rather-probable that there will be a risk of a forest fire. The DDSIS would also study the need for buying a new HFPT, that is, a heavy FPT (an acronym for Ton Pump Van that is hardware of big capacities for massive fight againt the fire, including helicopters, canad-airs...). He thus considers (C): ”a new HFPT is needed”. Considering the fleet of heavy and light FPT, he esteems that it is impossible that a HFPT is needed in an A3 or in an A2 but that it is rather-probable that it is needed in an A1. After another thought, using the Total Certainty Formula, the DDSIS esteems that it is beforehand also rather-probable that a HFPT is needed.

The DDSIS would now refine this result knowing that its area is an high risks area. From the Bayes’ Formula its director is able to conclude that it is at least very-little-probable and at the most probable that a new HFPT is needed. From these considerations, he decides to postpone the buying to another date.

2.3 Shapes of digits and parity

Let’s now meet a young scholar who has read a digit and wonders about its parity. He would like to know the Certainty he may have that the read digit is even depending only on the shape of it. He then considers three events: ”the read digit owns a line” (D_{1}={1,2,4,5,7}), ”the read digit is made only of curves without being a circle” (D_{2}={3,6,8,9}) and ”the digit is a circle” (D_{3}={0}). He then esteems that it is probable that the digit owns a line, little-probable that the digits is made only of curves without being a circle” and very-little-probable that ”the digit is a circle”. He is then interested in the fact that ”the digit is even” (B={0,2,4,6,8}). Evaluating that it is littleprobable that the digit is even given that it owns a line, probable that it is even given that it is made only of curves, and certain that it is even given that it is a circle, he concludes, using the Total Certainty Formula that it is probable that the digit is even. He wants now to find the degree of certainty that, after having read a digit (and not a letter or a special character), this latter owns a curve. So he considers the three following events: (B) ”the digit is even” (B={0,2,4,6,8}), (C) ”the digit owns a curve ” (C={0,2,3,5,6,8,9}), (D)”the symbol is a digit” (D = {0,1,2,3,4,5,6,7,8,9}). To our scholar, it is certain that the symbol is a digit, veryprobable that the digit owns a curve when the symbol is a digit and at last that it is rather-probable that the digit is even when it owns a curve. So, using the Uncertainty Propagation Rule our scholar knows that is between probable and rather-probable that the digit is even. Confident on the distribution of even numbers among symbols, he decides to go on learning.

3. THE MODEL

Two operators are defined in B_{7}: a symbolic sum noted S and a symbolic complement n, both analogous to the usual operators of classical Probabilities. The choice of S is not unique and we take the T-conorm associated to Lukasiewicz’s implication. S(v_{α}, v_{β}) is defined by v_{α} + v_{β} − 1 if v_{α} + v_{β} − 1 < M and M otherwise; n is defined by n(v_{α}) = M + 1 − v_{α}. The justification of the algebraic construction of our symbolic model can be found in [11, 14].

3.1 The axiomatics of Certainty

We now define a function of Certainty denoted Cert that will apply to Boolean statements. So, ”A is v_{α} probable” is translated into Cert(A) = v_{α}. This predicate satisfies postulates that use the previous S and n. Their formal definitions and the intuitive human motivations for them can be found in [11, 14]. These Postulates are:

[P1] If A⊧_{M}ϕ ≡ ψ, then Cert(ϕ)=Cert(ψ)

[P2] If A⊧_{M}ϕ, then Cert(ϕ)=v_{M}

[P3] If A⊧_{1}ϕ, then Cert(ϕ)=v_{1}

[P4] Cert(¬ϕ)=n(Cert(ϕ))

[P5] If A⊧_{1}ϕ ∩ ψ then Cert(ϕ ∪ ψ)=S(Cert(ϕ),Cert(ψ)).

3.2 Certainty of a logical implication and Generalised Modus Ponens Rule

Let’s consider two formulae A and B verifying Cert(A)=v_{α}, Cert(B)=v_{β} and Cert(A∩B)=v_{λ}. Then v_{δ} = v_{α} → v_{λ} = v_{M} if α ≤ λ else v_{M−α+λ} which is Lukasiewicz’s implication ([14], p.423). So this leads to Cert(A⊃B)=Cert(A) → Cert(A∩B). This property is called the Generalised Modus Ponens Rule: If Cert(A)=v_{α} and Cert(A⊃B)= v_{δ}, then Cert(B)=v_{β} with v_{β} ∈ [n(S(n(v_{α}),n(v_{δ}))),v_{δ}].

3.3 The notion of Conditional Certainty

Inspired with the definition of conditional probability, we defined in [12, 14, 16] a symbolic division operator, noted C (Conditioning Criterion), or, in an equivalent way, an symbolic multiplication operator noted I (Independence Criterion). The postulates of I are:

[I1] (∀v_{α})(∀v_{β})[I(v_{α},v_{β})=I(v_{β},v_{α})]

[I2] (∀v_{α})[I(v_{1},v_{α})=v_{1}]

[I3] (∀v_{α})[I(v_{α},v_{M})=v_{α}]

[I4] (∀v_{α})(∀v_{β})(∀v_{γ})[v_{β} ≤ γ ⇒ I(v_{α},v_{β})≤ I(v_{α},v_{γ})]

[I5] C(v_{α},v_{λ})={v_{β})|I(v_{α} ,v_{β})=v_{λ}}

[I6] (∀v_{α})(∀v_{β})(∀v_{γ}) [I(I(v_{α} ,v_{β}),v_{γ})=I(v_{α},I(v_{β},v_{γ}))]

[I7] (∃v_{α} ∈[v_{2} ,v_{a}])[I( v_{α},v_{α})=v_{α}]. The operator I thus defined by this axiomatics is not unique. Among the different tables that exist we have chosen the one given by v_{β} ∈ C(v_{α},v_{λ}) ⇔ v_{λ}=I(v_{α},v_{β}). Then the operator C corresponding to I is defined in an unique way (Cf. Table 1). Please note that, in a symbolic way, the ”division” does not give an element but an interval of Certainty. This is due to the fact that the approximation used for the discretisation does not allow us to give a unique value. In other terms, C(v_{α}, v_{β},) is the degree of Certainty of the ”division” of Cert(A ∩ B) by Cert(B). We then set, in a similar way to the classical Probability: the Conditional Certainty of B given A noted Cert(A|B) is such that Cert(A|B) ∈ C(Cert(A),Cert(A∩B)).

Remark: we have not defined the conditional Certainty but rather a symbolic interval of conditional Certainty, interval to which belongs the conditional Certainty of B given A. This relative imprecision translates the fact that man apprehends better the ”symbolic Multiplication” than the ”symbolic Division”.

Table 1: Symbolic Division C in B_{7}

v_{1}  v_{2}  v_{3}  v_{4}  v_{5}  v_{6}  v_{7}
v_{1}  [v_{1},v_{7}]  ∅  ∅  ∅  ∅  ∅  ∅
v_{2}  {v_{1}}  [v_{2},v_{7}]  ∅  ∅  ∅  ∅  ∅
v_{3}  {v_{1}}  [v_{2},v_{6}]  {v_{7}}  ∅  ∅  ∅  ∅
v_{4}  {v_{1}}  [v_{2},v_{5}]  {v_{6}}  {v_{7}}  ∅  ∅  ∅
v_{5}  {v_{1}}  [v_{2},v_{4}]  {v_{5}}  {v_{6}}  {v_{7}}  ∅  ∅
v_{6}  {v_{1}}  [v_{2},v_{3}]  {v_{4}}  {v_{5}}  {v_{6}}  {v_{7}}  ∅
v_{7}  {v_{1}}  {v_{2}}  {v_{3}}  {v_{4}}  {v_{5}}  {v_{6}}  {v_{7}}

3.4 Properties and Combination of Uncertainties

We will now present some results allowing us to handle the uncertainty using our system. More complete results can be found in ([14] §3.2, §4.6).

- Link between the different operators

{A⊧_{M}¬A∪B,Cert(A)=v_{α}, Cert(B)=v_{β}}⇒ v_{α} ≤ v_{β}.. It is the symbolic generalization of the classical result: if A ⊂B then pr(A) ≤ pr(B). If {Cert(A)=v_{α},Cert(B)=v_{β} then {Cert(A∩B)=v_{γ}, Cert(A∪B)=v_{δ}} with v_{γ} ∈[T(v_{α},v_{β}), v_{α} ∧ v_{β}] and v_{δ} ∈ [v_{α} ∨ v_{β}, S(v_{α},v_{β})]. Lukasiewicz’s implication also gives δ = α + β − γ, generalizing the classical property : P r(A ∪ B) = P r(A) + P r(B) − P r(A ∩ B).

- C-conditional Detachment Rule

Let Cert(A)=v_{α} and Cert(B|A)=v_{μ}. If v_{λ}=I(v_{α},v_{μ}), then Cert(B) = v_{β} with v_{β} ∈ [ v_{λ}, v_{α} → v_{λ}].

- Compound Certainties Formula

If Cert(A)=v_{α} and Cert(B|A)=v_{μ} then Cert(A∩B) =I(v_{α},v_{μ}).

- Uncertainty Propagation Rule

Let A,B,C be events such that Cert(A)=v_{α}, Cert(B|A)=v_{μ_{1}} and Cert(C|B)=v_{μ_{2}}. Then Cert(C) =v_{γ}, with v_{γ} ∈ [I(v_{δ}, v_{μ_{2}}), v_{δ} → I(v_{α} → v_{δ}, v_{μ_{2}})] where v_{δ} = I(v_{α}, v_{μ_{1}}.

- Total Certainty Formula

Consider a partition A_{1}...A_{n} of the universe of discourse. Consider also an event B. If we know on the one hand the certainty of each A_{i} and on the other hand for each A_{i} the certainty of B given A_{i}, then we can compute the certainty of B. Let A_{1},..A_{n} be such that Cert(A_{1}∪...A_{n})=v_{M} and such that for all i and j of {1...n} with i ≠ j, Cert(A_{i} ∩ A_{j}) = v_{1} and such that for every i of {1...n}, Cert(A_{i})=v_{α_{i}} and Cert(B|Ai)=v_{μ_{i}}. Then, setting v_{γ_{1}}=I(v_{α_{1}},v_{μ_{1}}) and for all k=2,...n, v_{γ_{k}}=S(v_{γ_{k−1}},I(v_{α_{k}},v_{μ_{k}})), we get Cert(B) = v_{γ_{n}}.

- Symbolic Generalised Bayes’ Formula With the same partition of A_{i} as above, we can evaluate the certainty of all A_{i} given B: Cert(A_{i}|B)= v_{η_{i}} with v_{η_{i}} ∈ C(v_{γ_{n}},I(v_{α_{i}},v_{μ_{i}})).

Remarks: First, lets’ remark that for every odd M, a table for I and C can be constructed. There are of course more than one solution, but one is particularly easy to describe: the one related to Lukasiewics’s implication. For I (resp. C), it can be described (or defined) as follows :

I(v_{a},v_{b}) is: min(v_{a},v_{b}) for v_{a}, v_{b} ∈ {v_{1}, v_{M}}, v_{2} for a + b ≤ M + 1, v_{a+b−M} otherwise. C(v_{a},v_{b}) is: [v_{1},v_{M}] for v_{a}=v_{1} and v_{b}=v_{1}, { v_{1}} for v_{a} > v_{1} and v_{b}=v_{1}, [v_{2},v_{a} → v_{b}] for v_{a} > v_{1} and v_{b}=v_{2}, {v_{a} → v_{b}} if v_{a} ≥ v_{b}, ∅ otherwise.

It is important to note that there is no objective way to decide which size is the best for a given problem, and which operator I should be taken. It is only a matter of choice. Taking M=7 gives more precise results that M=5 but it requires also to choose more carefully I. Let’s suppose that we have chosen to work with 7 degrees of certainty, each degree being related to a linguistic expression. If we try to solve a problem for which the elementary elements have a probability of one tenth, we will have to add some degrees. But there are many ways to do it. The first possibility is to recreate a scale of degrees having the right number of degrees (that is 10 for our case). But then, there will be no correspondence between the scale of the problem and the linguistic scale. So the result will not be expressible in a linguistic way. On the contrary, if we replace each degree of L_{7} by a fixed number of degrees (2 for our example), then, each specific degree of the problem will be a specialisation of a degree of L_{7} For instance probable will give the two qualifiers rather-probable and very-probable.

We will then be able to match each specific degree of the problem with a linguistic expression. The result will then always be given precisely with the new graduation though it may not be very expressive. But it will be expressed in natural language. The problem of the graduality has already been studied in [10] but this is not the method used here. Let’s see how L_{5}, L_{7}, L_{9}, L_{11} fit to represent our examples. We may choose any linguistic qualifiers though the derivation of qualifiers from one M to another may not be explicit. The correspondence between a scale and another one is not obvious. For instance, we could use an hierarchic scale of degrees by refining the leftmost term, but we prefer the last one of Figure 1 which seems more reliable for our examples. The comparison can be done as follows: for each scale (that is for each choice of M), we compute the certainty of the same event. The results are then given in the following Table. One has to be careful: [v_{3}, v_{4}] for M=7 and [v_{3}, v_{4}] for M=9 does not mean the same thing. Also, some choices have been made: v_{6} for M=7 may be taken as v_{7} or v_{8} for M=9. But the reader could check as we did that even with the other possible choices, the results still hold.

Table 2: Comparison of results for different M

Problem magician (1) magician (2)  fire
Rule  Total  Bayes’  Uncertainty
Certainty  Formula  Propagation
M = 5  v_{3}  [v_{2},v_{4}]  [v_{3},v_{4}]
M = 7  v_{4}  { v_{6} }  [v_{4},v_{5}]
M = 9  v_{5}  { v_{8} }  [v_{4},v_{6}]
M = 11  v_{6}  { v_{10} }  [v_{4},v_{7}]

With all these examples, the results show coherent: the intervals overlap or belong to the others, but we have never disjoint intervals or isolated values. But, it has to be noted that for a same value of M, the results are more or less satisfying for different problems. Whereas there seems to be a better value of M, that is a value giving a smaller interval which in turn, gives a stricter approximation, it is not possible to fix and use a unique value of M for all problems, neither to increase the number to give a more precise value of the intervals for all cases. This is shown by the Figure 1 where the rectangles indicate the interval of solution. The scales with M=5 or M=7 are probably too small, that is, having too few values but, globally, the most adequate size seems to be M=9. In fact, a cognitive agent, for a predicative domain, does not tend to use the qualifier under the mean value. Being positivist, he will choose the event that is the more probable. This means that he may have to express his certainty not on an event but on its complementary. It follows that the expressive power of a scale is related not to the whole set of terms but to the number of terms above the mean value (which is, by itself, non informative).

Figure 1: Scales of qualifiers and interval of solutions (fire outbreaks example) M=3 impossible M=5 impossible M=7 impossible M=9 impossible M=11 impossible  probable probable probable probable probable  certain certain certain certain certain less-than-probable  more-than-probable very-little-probable  little-probable  rather-probable  very-probable frankly-improbable really-improbable little-probable  rather-probable really-probable frankly-probable really-improbable little-improbable  rather-probable really-probable frankly-very-improbable really-very-improbable  really-very-probable frankly-very-probable

So, in a scale with 7 terms, only 3 are pertinent, which is too restrictive. This fact is corroborated in psychology [20, 21] where 5,7... values are said to be pertinent and linguistically fit to give precise evaluations. On practical problems, a scale of 11, 13 or 15 can then be used. Fixing the size of the scale is not enough: for the same M, we have different operators and we have to find the most adapted operators for our problem.

Remark: Let’s compare now our approach to that of Aleliunas, Darwiche, Darwiche & Ginsberg and of Spohn. Our concept of Certainty lead to properties that are similar to those of Coherent State of Belief (axioms A0-A4) postulated by Darwiche & Ginsberg [8]. More, the T-conorm S that we introduced by axiom P5 stands for the summation operator of their theorem 1. Their symbolic theory is founded only on a Propositional Logic whereas ours, based upon a many-valued Predicate Logic, owns a greater expressive power. In particular, we can, at the same time, represent and manage the Imprecision and the Uncertainty. Note also that, in their theory, no link is established between their summation operator and the implication, an inferential Process of deductive type is not proposed. On the other hand, our system can, with this link, spread the Uncertainty as we will see in the next paragraph. A link with the structure of Aleliunas’ Probability Algebra can be made [5]. The operator I (respectively n) taking the place of * (resp. i), the axioms 1-10 proposed in [5] are verified. Thus, the axiomatics of I (resp. C) leads to a particular structure of Probability Algebra, finite and totally ordered.

We have added to the theory a notion of Independence closely linked to the Conditional Certainty. A link being made with the implication, we add to the Symbolic extension of Bayes’ Theorem the symbolic rule of Generalised Modus Ponens as second means of propagation for the Uncertainty. Our axiomatics of the Concept of Certainty does not give (in the prepositional case) an Aleliunas’ Probabilistic Logic [3, 4]. Indeed, it is easy to verify that the axioms 7-9 are not satisfied by our concept. Spohn has also proposed a theory allowing to treat uncertain information [22]. He approaches the concept of Certainty (degree of strength of the Belief in his theory) in an ordinal way. He especially introduced the concept of ordinal degree of conditional Certainty. Substituting successively + * and / of the classical probabilities by his operators min + and -, one then finds again the results of his theory which thus presents as an ordinal, non probabilistic model of inductive reasoning. So, the theory used here presents a similarity with that of Spohn. In both cases, we are looking for a non probabilistic (in the classical sense) model of the concepts of certainty and of conditional certainty, even if the objectives are different. In both cases, one can propose generalisations of classical probabilistic results. But let’s point out several important differences. Spohn’s approach being ordinal, the operators used are that defined in IN. It is clear that they differ basically from the operators S, I and n that we have built in the scale of certainty. The scale of Graduation in his theory is infinitely denumerable whereas ours, by definition, is finite.

4. CONCLUSION

In this paper, we have first studied three different examples to illustrate the fact that, in situations where the numerical approach is ill adapted, our model works simply and gives results that conform to the human intuition. Then, we gave the axiomatics of the operators of a Symbolic Probability Theory that manage the Uncertainty of the statements of the Natural Language evaluated in a Qualitative way, exposing the different rules that may be used. The problem of the choice of the size of the model has then been discussed, showing that for any value of M, there is an overlapping of the solution of size M with the solution for another M. Clearly, our theory brings new tools to Linguistics for the explicit treatment of uncertain statements of the Natural Language, in particuliar the Conditionals of Language.

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