Formal Models for Artificial Intelligence Methods: Mathematical Models for Reasoning Under Uncertainty

Appendix I Formal Models for Artificial Intelligence Methods: Mathematical Models for Reasoning Under Uncertainty

In the first section fundamental notions of measure theory [ 251 ] are introduced in order to define probability space [ 118 , 119 ], which is a basic definition of probability theory. The Bayesian model [ 119 ], which is a basis for constructing probabilistic

reasoning systems described in Chap. 12 , 24 is introduced in the second section. The third section contains basic notions [ 67 , 271 ] of the Dempster-Shafer theory.

I.1 Foundations of Measure Theory and Probability Theory

Let us begin with fundamental definitions of measure theory, i.e. σ-algebra, measur- able space, measure, and measure space.

Definition I.1 conditions.

• ∅ ∈ M. • If A 1 , A 2 , A 3 ,...

∈ M, then ∞

i =1 A i ∈ M.

Definition I.2 space .

If a set A belongs to a σ-algebra M, then we say that A is M-measurable, or simply measurable , if it is clear what the underlying σ-algebra is.

Definition I.3

24 This model is also used for defining statistical pattern recognition algorithms, discussed in Chap. 10 . © Springer International Publishing Switzerland 2016

293 M. Flasi´nski, Introduction to Artificial Intelligence, DOI 10.1007/978-3-319-40022-8

294 Appendix I: Formal Models for Artificial Intelligence Methods … M

Definition I.4

μ : M −→ R ∪ {∞}

is called a measure iff it satisfies the following conditions. • For each set A ∈ M: μ(A) ≥ 0.

• μ(∅) = 0. • If A 1 , A 2 , A 3 ,... ∈ M are pairwise disjoint, then μ( ∞ i

=1 A i ) = i =1 μ(A i ) . Definition I.5

is called a measure space. Definition I.6

of events that we want to analyze. Each such a set consists of elementary events. The function (measure) P is called a probability measure, and a number P(A), A ∈ F is

called a probability of an event A.

I.2 Bayesian Probability Theory

After defining notions of probability space, space of elementary events and proba- bility of an event, we can introduce the foundations of Bayesian probability theory.

Definition I.7 conditional probability of an event A ∈ F, assuming the event B has occurred, is given by a formula

Now we introduce the total probability theorem.

1 , B 2 ,..., B n ∈F fulfill the following conditions.

• P(B i )>

0, for each i = 1, 2, . . . , n. •B i ∩B j •B 1 ∪B 2 ∪···∪B n

25 We say that σ( M ) M .

Appendix I: Formal Models for Artificial Intelligence Methods … 295 Then for each event A ∈ F the following formula holds

P(A) =

P(A |B i ) · P(B i ).

i =1

The following theorem, called Bayes’ rule, results from the total probability theorem and a definition of conditional probability.

Theorem I.2 Let the assumptions of Theorem I.1 be fulfilled. Then the following formula holds

for each k = 1, 2, . . . , n. At the end of this section we define independence of events and conditional

independence of events. Definition I.8

dent iff the following condition holds

P(A ∩ B) = P(A) · P(B).

If P(B) > 0, then this condition is equivalent to

P(A |B) = P(A).

A notion of independence can be extended to any finite collection of events. Definition I.9

1 ,..., A n ∈ F are (mutually) independent iff for any sub-collection of k events A i 1 ,..., A i k the follow- ing condition holds

P(A i 1 ∩···∩A i k ) = P(A i 1 ) · · · · · P(A i k ). Definition I.10

Events A, B are conditionally independent given an event C iff the following condition holds

P(A ∩ B|C) = P(A|C) · P(B|C).

This condition is equivalent to

P(A |B ∩ C) = P(A|C).

Similarly to a notion of independence, a conditional independence can be extended to any finite collection of events.

296 Appendix I: Formal Models for Artificial Intelligence Methods …

I.3 Basic Notions of Dempster-Shafer Theory

In this section we introduce the basic notions of the theory of belief functions used in Sect. 12.2 .

Definition I.11 discourse (a frame of discernment). A function m : 2 −→ [0, 1] is called a basic belief assignment (a mass assignment function) iff it fulfills the following conditions.

• m(∅) = 0. •

A m(A) = 1. Definition I.12

A function Bel : 2 −→ [0, 1] is called a belief function iff

Bel(A) =

m(B)

B :B⊆A

Definition I.13

A function Pl : 2 −→ [0, 1] is called a plausibility function iff

Pl(A) =

m(B)

function in the following way:

Pl(A) = 1 − Bel( ¯A),

where ¯A is a complement of the set A.

Appendix J Formal Models for Artificial Intelligence Methods: Foundations of Fuzzy Set and Rough Set Theories

The basic definitions of fuzzy set theory [ 321 ] and rough set theory [ 216 ], introduced

in Chap. 13 for defining imprecise (vague) notions, are contained in here.

J.1 Selected Notions of Fuzzy Set Theory Basic notions of fuzzy set theory such as fuzzy set, linguistic variable, valuation in

fuzzy logic are introduced. Then selected definitions of the Mamdani fuzzy reasoning [ 191 ] (fuzzy rule, fuzzification operator, Mamdani minimum formula of reasoning, center of gravity of membership function) are presented.

Definition J.1 Let U be a nonempty space, called a universe of discourse. A set A in the space U, A ⊆ U, is called a fuzzy set iff

A = {(x, μ A ( x)) : x ∈ U},

where

μ A : U −→ [0, 1]

is a membership function, which is defined in the following way ⎧

⎪ ⎩ s, s ∈ (0, 1), x belongs toA with a grade of membership s.

© Springer International Publishing Switzerland 2016 297 M. Flasi´nski, Introduction to Artificial Intelligence, DOI 10.1007/978-3-319-40022-8

298 Appendix J: Formal Models for Artificial Intelligence Methods … Definition J.2

A linguistic variable 26 is a quadruple L = (N, T, U, M), where

N is a name of the variable, T is a set of possible linguistic values for this variable, U is a universe of discourse, M is a semantic function that ascribes a meaning M(t) for every linguistic value

t ∈ T; a meaning is represented with a fuzzy set A ∈ X. Definition J.3 Let there be given propositions P: “x is P” and Q: “x is Q”, where P

and Q are vague notions with fuzzy sets P and Q ascribed to by a semantic function. Let μ P and μ Q

be their membership functions, respectively. A valuation in fuzzy propositional logic is defined with the help of the truth degree function T in the following way.

• T(P) = μ P ( x) . • T(¬P) = 1 − T(P). • T(P ∧ Q) = min{T(P) , T(Q)}. • T(P ∨ Q) = max{T(P) , T(Q)}. • T(P ⇒ Q) = max{T(¬P) , T(Q)}.

Now, we introduce selected notions of the Mamdani model of reasoning 27 Definition J.4

A rule (R k , COND k , ACT k ) is called a fuzzy rule iff: COND k is of the form :

1 is A 1 ∧···∧x n is A n , where x k , i

kk

= 1, . . . , n is a linguistic variable, A i , i = 1, . . . , n is a linguistic value, ACT k is of the form :

y k is B k , where y k is a linguistic variable, B k is a linguistic value. 28 Definition J.5 Let X ⊆ R be a domain of a variable x. A singleton fuzzification

operation of a value x of a variable x is a mapping of a value x to a fuzzy set A ⊆ X with a membership function given by the following formula:

26 There are several definitions of linguistic variable. In its original version [ 322 ] linguistic variable was defined on the basis of context-free grammar. However, because of the complex form of such

a definition, nowadays a linguistic variable is defined with a set of linguistic values explicitly. We assume such a convention in the definition.

27 We assume a definition of a rule as in Appendix F .

28 The definition concerns rules for which a condition is of the form of a conjunction and a consequent is a single element (i.e., of the form of a canonical MISO). In practice, fuzzy rules can be defined

in such a form (with certain assumptions).

Appendix J: Formal Models for Artificial Intelligence Methods … 299

In order to define fuzzy reasoning, we introduce firstly a concept of fuzzy relation. Definition J.6 Let X, Y be nonempty (non-fuzzy) sets. A fuzzy relation R is a fuzzy

set defined on the Cartesian product X × Y such that

R = {((x, y) , μ R ( x, y)) : x ∈ X, y ∈ Y, μ R : X × Y −→ [0, 1]}. Definition J.7 Let X, Y be sets, A ⊆ X, B ⊆ Y be fuzzy sets with membership

functions μ A and μ B . A result of fuzzy reasoning from A to B in the Mamdani model according to a minimum rule 29 is a fuzzy relation on X × Y, denoted A → B, in which a membership function is defined with the following formula

μ A →B ( x, y) = min{μ A ( x), μ B ( y) }. Definition J.8 Let A ⊆ X ⊆ R be a fuzzy set with a membership function μ A .A

defuzzification operation with the help of a center of gravity consists in ascribing a value x to a set A according to the following formula

X x μ A ( x x) dx

X μ A ( x) dx

assuming the existence of both integrals.

J.2 Selected Notions of Rough Set Theory Let U be a nonempty space containing objects considered, called a universe of dis-

course , A be a finite nonempty set of attributes describing objects belonging to U. For every attribute a ∈ A let us define a set of its possible values V a , called the domain

of a. Let a(x) denote a value of an attribute a for an object x ∈ U. Definition J.9 Let B ⊆ A. A relation I B ⊆ U × U is called a B-indiscernibility

relation iff it fulfills the following condition. ( x, y) ∈I B ⇔ ∀a ∈ B : a(x) = a(y), for every x, y ∈ U. We say that objects x, y are B-indiscernible.

29 Such a somewhat complicated formulation is used by the author in order to avoid naming the formula: a fuzzy implication A → B. The Mamdani minimum formula, although very useful in

practice, does not fulfil a definition of fuzzy implication.

300 Appendix J: Formal Models for Artificial Intelligence Methods … Since a relation I B is an equivalence relation, it defines a partition of a universe U

into equivalence classes. Let us introduce the following definition. Definition J.10 Let I B be a B-indiscernibility relation in a universe U, x ∈ U. A

B-elementary set [x] I B is the equivalence class of an object x, i.e.

[x] I B = {y ∈ U : (x, y) ∈ I B }.

Let X ⊆ U. Definition J.11

A B-lower approximation of the set X is a set

BX = {x ∈ U : [x] I B ⊆ X}.

Definition J.12

A B-upper approximation of the set X is a set

BX = {x ∈ U : [x] I B

Definition J.13

A B-boundary region of the set X is a set

B BOUND X = BX\BX.

Definition J.14 The set X is called a B-rough set iff the following condition holds

BX

Definition J.15 The set X is called a B-exact (B-crisp) set iff the following condition holds

BX = BX.

Definition J.16 Let card(Z) denote the cardinality of a nonempty set Z. A coefficient of an accuracy of approximation of the set X with respect to a B-indiscernibility relation is given by the following formula

card(BX)

α B ( X) =

card(BX)