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A Fuzzy Dynamic Belief Logic A Fuzzy Dynamic Belief Logic Xiaoxin Jing and Xudong Luo∗ Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China Keywords: Belief Revision, Fuzzy Logic, Dynamic Logic. Abstract: This paper introduces a new logic approach to reason about the dynamic belief revision by extending well- known Aucher’s dynamic belief revision approach to fuzzy environments. In our system, propositions take a valuation in linguistic truth term set, and a belief revision is also in a qualitative way. Moreover, we reveal some properties of our system in epistemic style and do a comparison between our fuzzy belief system with famous AGM postulats. 1 INTRODUCTION Zhang, 2012). In fact, our belief is often fuzzy. For example, “I think the temperature will be high tomor- The problem of belief change is an active topic in row”. Here “high” is a fuzzy conception because there logic (Aucher, 2006; van Ditmarsch, 2005; Shapiro is not any crisp division between “high” and “low”. et al., 2011) and artificial intelligence (Sardina and And suppose that we want to book an air ticket, we Padgham, 2011; Casali,Godob and Sierra, 2011). Its thought ticket A is good at first and then when we focus is to understand how an agent should change know that the flight would be delayed, we need to re- his belief in the light of new information. Re- vise our previous beliefs about the ticket. On the one searchers have developed various models for belief hand, “the ticket is good” is a fuzzy proposition be- revision. Epistemic plausibility model is one of main cause there are no crisp standards for a good ticket. approaches to model the dynamics of the belief. It It will depend on many factors such as the price, the adds the plausibility ordering in the epistemic model departure time and service, and so on. On the other for each agent, i.e., a pre-order w ≤ v that says agent hand, we might not be able to decide whether or not i considers world w at least as plausible as v. to change our previous belief completely. Rather, we In order to reason about this structure, the epis- can only say the ticket is better or worse than before. temic language is extended with a conditional belief How can we handle this kind of fuzzy belief revision? operator. van Benthem (2007) developed a dynamic Fuzzy theory can provide a powerful tool to han- belief revision frame based on the dynamic epistemic dle this situation, which is studied in various areas, logic and then adds the conditional belief operator in such as mathematics, logic and computer science. it. Baltag (2006) uses the epistemic plausibility model Fuzzy logic began with the 1965 proposal of fuzzy for conditional belief in a multi-agent epistemic envi- set theory by Zadeh (1965) and it is a form of many- ronment, and introduces plausibility pre-order on ac- valued logic (Zadeh, 1975). It deals with reason- tions, notated as action plausibility models to display ing that is approximate rather than accurate. In tra- the dynamic setting, which is somehow the extension ditional logic, a proposition is usually true or false, of well-known Aucher’s method in (Aucher, 2006). while fuzzy logic proposition can have a truth value Another typical approach for belief revision is epis- in-between 0 and 1. In this sense, fuzzy logic is a bet- temic probability model (Baltag and Smets, 2006), ter way for us to handle uncertain reasoning for fuzzy which is closed to the epistemic plausibility model belief revision. with probability measures in the place of plausibility The structure of the paper is listed as follows. Sec- ordering. In these models, various arithmetical for- tion 2 briefs the fuzzy theory, which we will use for mulas are used to compute the probability of belief of developing our logic. Section 3 defines our fuzzy sys- the output-states from the probabilities of the input- tem based on the dynamic belief logic. Section 4 states and the probabilities of actions. studies some properties of our logic. Section 5 gives In the logic above, the beliefs are all crisp. How- a comparison between our fuzzy logic system with ever in real-life, it is not always the case (Wu and AGM postulats. Section 6 discusses the related work to confirm that our work has advanced the state of art. ∗Corresponding author. Jing X. and Luo X.. 289 A Fuzzy Dynamic Belief Logic. DOI: 10.5220/0004257302890294 In Proceedings of the 5th International Conference on Agents and Artificial Intelligence (ICAART-2013), pages 289-294 ISBN: 978-989-8565-39-6 Copyright c 2013 SCITEPRESS (Science and Technology Publications, Lda.) ICAART2013-InternationalConferenceonAgentsandArtificialIntelligence Defininition 2 (Extension Principle). Suppose f is a function with n arguments x1;:::;xn,denoted by~x. Let µi(xi) be the membership function of argument xi (1 ≤ i ≤ n ). Then µ(y) = supfµ1(x1) ^ ::: ^ µn(xn) j f (~x) = yg; (2) where sup denote the supremum operation on a set, ^ means the conjunction of the items. Let the fuzzy set corresponding to µ be B, and let the fuzzy set corre- sponding to µi be Ai. For convenience, we denote the operation of the extension principle as ⊗. i.e., B = ⊗(A1;:::;An; f ): (3) If an operation on some linguistic terms is not closed in the predefined linguistic term set, a linguis- Figure 1: Linguistic truth. tic approximation technique in necessary in order to find a term in the term set, whose meaning (member- Section 7 concludes the paper with future work. ship function) is the closest to that of the result of the operation. Defininition 3 (Linguistic Approximation). The most 2 PRELIMINARIES straight forward approach, the BEST FIT, uses the Euclidean Distance (ED) as follows: This section recaps fuzzy set and logic theory. Let q 2 X = fxg be a space of points with a generic element, ED(A;B) = ∑f(µA(x) − µB(x)) j x 2 [0;1]g (4) denoted as x, of X. A fuzzy set A in X is characterised between fuzzy sets A and B defined on [0;1], to eval- by a membership function of µ(x), which associates uate which one in the term set is the closest to the set with each point in X a real number in interval [0;1], being approximated. Namely, t 2 LTTS, being the with the values of µ(x) at x representing the member- closest to t00 should satisfy ship degree of x in A. In fuzzy logic, truth value comes 8t0 2 LTTS;ED(t;t00) ≤ ED(t0;t00): (5) in continuous degrees. There are different forms for truth value. By a linguistic variable we mean a vari- For convenience, we denote the above operation of able whose values are words or sentences in a natural linguistic approximation as , i.e. language (Zadeh, 1975; Luo et al., 2002). t = (t00) (6) Defininition 1 (Linguistic Truth). A proposition can take truth on linguistic truth rather than classical proposition truth f0;1g, i.e., 3 FUZZY DYNAMIC BELIEF LTTS = fabsolutely-true;very-true;true; LOGIC f airly-true;undecided; f airly- f alse; This section will present our logic, which will be de- f alse;very- f alse;absolutely- f alseg:(1) fined on the linguistic truth set and combines with the For convenience, we denote belief dynamic system of (van Ditmarsch, 2005). LTTSt =fabsolutely-true;very-true;true; f airly-trueg; 3.1 Language LTTS f =fabsolutely- f alse;very- f alse; f alse; f airly- f alseg: We define the language of our fuzzy dynamic belief logic as follows: Thus, we have Defininition 4 (Language of belief-epistemic logic). LTTS = LTTSt [ LTTS f [ fundecidedg: f := p j :f j f ^ y j Kjf j B jf j [∗f]y Semantics of the terms in this term set are defined where p is an atomic formula as usual, :f and f ^ y as shown in Table 1 (Luo et al., 2002). are the Boolean combination of the propositional for- The operation on linguistic variables are defined mulas, Kjf means agent j knows proposition f,B jf according to the corresponding operation on numeri- means agent j believes proposition f, and [∗f]y ex- cal variables by using the extension principle as fol- presses that proposition y holds after revising an- lows: gent’s belief with formula f. 290 AFuzzyDynamicBeliefLogic Table 1: Linguistic truth. 1 if x = 1 1 if x = 0 µ (x) = µ (x) = absolutely−true 0 otherwise absolutely− f alse 0 otherwise 2 2 µvery−true(x) = µtrue(x) µvery− f alse(x) = µ f alse(x) µtrue(x) = x;8x 2 [0;1] µ f alse(x) = 1 − x;8x 2 [0;1] 1=2 1=2 µ f airly−true(x) = µtrue(x) µ f airly− f alse(x) = µ f alse(x) 1 if x 2 (0;1) µ (x) = undecided 0 otherwise 3.2 Semantics 41(x;y) = maxf0;x + y − 1g; 42(x;y) = x × y; The semantics of our system is completely different 4 = minfx;yg; from the one in the classical two-valued logic. The 3 truth value of the proposition is taken on the lan- and 5 denotes the operator given by one of 51 ,52 guage truth term set, LTTS, given by (1). The func- and 53 defined as follows: tions of different linguistic truths’ membership de- grees have been listed in Table 1 and their figure have 51(x;y) = minf1;x + yg; been showed in Fig.1 correspondingly. 52(x;y) = x + y − x × y; Then we will give the semantics of the formulas 53(x;y) = maxfx;yg: in three steps.
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