A Fuzzy Dynamic Belief

Xiaoxin Jing and Xudong Luo∗ Institute of Logic and Cognition, Sun Yat-sen University, Guangzhou, China

Keywords: Belief Revision, , 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 . 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) = sup{µ1(x1) ∧ ... ∧ µn(xn) | f (~x) = y}, (2) where sup denote the supremum operation on a set, ∧ means the conjunction of the items. Let the 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 = {x} be a space of points with a generic element, ED(A,B) = ∑{(µA(x) − µB(x)) | x ∈ [0,1]} (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, τ ∈ LTTS, being the with the values of µ(x) at x representing the member- closest to τ00 should satisfy ship degree of x in A. In fuzzy logic, truth value comes ∀τ0 ∈ LTTS,ED(τ,τ00) ≤ ED(τ0,τ00). (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). τ = (τ00) (6) Defininition 1 (Linguistic Truth). A proposition can take truth on linguistic truth rather than classical proposition truth {0,1}, i.e., 3 FUZZY DYNAMIC BELIEF LTTS = {absolutely-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 alse}.(1) fined on the linguistic truth set and combines with the For convenience, we denote belief dynamic system of (van Ditmarsch, 2005).

LTTSt ={absolutely-true,very-true,true, f airly-true}, 3.1 Language LTTS f ={absolutely- f alse,very- f alse, f alse, f airly- f alse}. 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 ∪ {undecided}. φ := p | ¬φ | φ ∧ ψ | Kjφ | B jφ | [∗φ]ψ Semantics of the terms in this term set are defined where p is an atomic formula as usual, ¬φ and φ ∧ ψ 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, Kjφ means agent j knows proposition φ,B jφ according to the corresponding operation on numeri- means agent j believes proposition φ, and [∗φ]ψ ex- cal variables by using the extension principle as fol- presses that proposition ψ holds after revising an- lows: gent’s belief with formula φ.

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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,∀x ∈ [0,1] µ f alse(x) = 1 − x,∀x ∈ [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 ∈ (0,1) µ (x) = undecided 0 otherwise

3.2 Semantics 41(x,y) = max{0,x + y − 1}, 42(x,y) = x × y, The semantics of our system is completely different 4 = min{x,y}; 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) = min{1,x + y}, 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) = max{x,y}. in three steps. First, we will give the semantics of the simple formulas, (i.e., the no-epistemic formulas), As we can see in the literatures of dynamic then the epistemic formulas with epistemic operator epistemic logic (van Ditmarsch, 2005; Sardina and K, B, and finally we will define the semantics of the Padgham, 2011), the semantic of the epistemic formu- belief revision. las are interpreted in the style of . There- First in the simplest situation, the semantics of the fore, before we take the second step to define the truth non-epistemic formulas (i.e., the ones in the classi- value of the epistemic formulas, we need to define the cal predicate logic). We can obtain the value of the belief-epistemic logic model in advance. formulas from estimates for the valuation of its sub- Defininition 6. A belief-epistemic model is a 4-tuple formulas. For example, if P is a Boolean combina- of (W,R,V,κ j), where tion of P and P , Val(P ) = τ ∈ LTTS, Val(P ) = 1 2 1 1 2 1. W is the set of the possible worlds; τ2 ∈ LTTS, then Val(P) = τ ∈ LTTS, and we can ob- tain τ by doing some operation on τ1 and τ2. Specifi- 2. R is the accessibility relation between the possible cally, we apply the extension principle ⊗ and the lin- worlds; guistic approximation technique on the valuation 3. V is an function from the propositions to LTTS of the sub-formulas, and at last obtain the result we (i.e., we associate each proposition with a set of expected. possible worlds, meaning the valuation of the for- Defininition 5. Let Val(P) = τ ∈ LTTS. Let µτ(x) de- mulas in these possible worlds); and notes the membership degree of a proposition on the 4. κ is a function from the possible worlds in W to linguistic value LTTS when given a value x between natural numbers. [0,1]. Then Given an actual state, with which we associate a 1. Val(¬P) = τ¬P, and µτ¬P (x) = µτP (1 − x),∀x ∈ factual description of the world, an agent may be un- [0,1]; certain about which of a set of different states is actu-

2. Val(P1 ∧ P2) = (⊗(τP1 ,τP2 ),4); ally the case. This is the set of plausible states. Any 3. Val(P1 ∨ P2) = (⊗(τP ,τP ),5); and of those plausible states may have its own associated 1 2 set of plausible states, relative to that state. In dy- 4. Val(P → P ) 1 2 namic epistemic logic, term ‘plausible’ means ‘acces-  absolutely-true if Val(P ) ≤ Val(P ), = 1 2 sible’. In classical modal logic, the accessibility rela- (1 − µ(x) + µ(y)) if Val(P ) > Val(P ). 1 2 tion is crisp, there are exactly two cases: R(w,w0) and According to (Bonissone and Decker, 1986), 4 ¬R(w,w0). R(w,w0) means that starting with world w, 0 0 denotes the operator given by one of 41, 42 and 43 world w is accessible using R, while ¬R(w,w ) means defined as follows: that starting with world w, world w0 is not accessible using R. Now we want to get more expressive power

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and so we replace the crisp relation by a “soft” acces- Here we get W = W 0 and V = V 0 because the possible sibility relation of R : W ×W → LTTS, where LTTS worlds in these two models are the same and the facts is given by (1). in possible worlds do not change, but for an agent the Given two plausible states, the agent may think preference for the possible worlds (the result of func- that one is more likely to be the actual state than the tion κ) has been changed by revision with [∗φ]. In this other. In other words, the agent has a preference case, the definition becomes: among states. The difference between plausible and Val([∗φ]ψ,(W,R,V,κ),w) = Val(ψ,(W,R0,V,κ0),w0). implausible states is: preferences among plausible states are comparable, some plausible states are pre- There are different methods for belief revision. ferred over other plausible states, while the implausi- Here we proposed a revision method based on ble states are all equally implausible. Preferences are Aucher’s (Aucher, 2006), which is called successful assumed to be partially ordered and we write < for minimal belief revision. an agents preference relation on the set of plausible Defininition 9 (Revision Method). If we revise a states given an actual world. Here we take κ to repre- model M with a formula φ, then sent a preference degree of the possible worlds for an  κ(w)−Min{κ(v)|Val(φ,M,v)∈LTTSt } agent; the less κ the more an agent prefers that pos-   if Val(φ,M,w) ∈ LTTSt , sible world. Then, we can give the semantics of the κ∗(w)= (w)+ −Min{ (v)|Val(¬ ,M,v)∈LTTS } epistemic formulas as follows: κ 1 κ φ t  otherwise. Defininition 7. Given model M and any possible world w in M, Defininition 10 (Validity). 0 0 1. Val(Kjφ,M,w) = inf{Val(φ,M,w ) | w,w ∈ W, 1. Σ |=τ φ iff for any model M and world w in M, if R(w,w0) = absolutely-true}; ∀B ∈ Σ,Val(B) ≥ τ, then Val(φ) ≥ τ. 0 0 00 2. Val(B jφ,M,w) = sup{Val(φ,M,w ) | w,w ,w ∈ 2. Σ |= φ iff ∀τ ∈ LTTS, Σ |=τ φ. W,R(w,w0),R(w,w00) 6= absolutely- f alse, and for all w0,w00 ∈ W,κ(w0) ≤ κ(w00)}. In the above definition, item 1 expresses that the 4 PROPERTIES valuation of Kjφ in possible world w under model M is the infimum of the valuations of φ in worlds w0, where According to the validity condition we defined above, R(w,w0) is absolutely-true. Item 2 expresses the val- we can prove some for our fuzzy belief logic uation of B jφ in possible world w in model M is the system. 0 supremum of the valuations of φ in world w , where 1. Kjφ |= φ. 0 00 R(w,w ) and R(w,w ) are not absolutely- f alse, and This theorem is coincide with the intuitions of hu- 0 the preference degree of w is not less than that of any man beings: if we know a proposition , then is true 00 φ φ other possible world w . in our world.

Defininition 8. Given belief-epistemic logic model Theorem 2. Kjφ |= B jφ. M = (W,R,V,κ) and a belief-epistemic logic model The theorem means if agent j knows proposition M0 = (W 0,R0,V 0,κ0), which is an arbitrary belief- φ, then he must believe it. epistemic model for a set of atoms P, a set of agents N, if Theorem 3. If the frame is transitive then Kjφ |= KjKjφ. Val([∗φ]ψ,M,w) = Val(ψ,M0,w0), where (M0,w0) : (M,w) k [∗φ] k (M0,w0), then M0 is the new model by revising the original model M with [∗φ], 5 COMPARISON WITH THE and after the revision we transfer from the possible AGM POSTULATES world w to a new possible world w0. Formula [∗φ]ψ reads as “after revision with φ, ψ The AGM postulates (named after the names of their holds”. The semantics that we propose is typical for proponents, Alchourrn, Gardenfors,¨ and Makinson) a dynamic modal operator: a state transformer [∗φ] are properties that an operator that performs revision induces a between belief-epistemic should satisfy in order for that operator to be consid- states. That is, if we revise belief-epistemic state w ered rational. The belief set of an agent is denoted in model M = (W,R,V,κ) with formula [∗φ], then by a theory set of K, which is deductively closed set we can get a new belief-epistemic state, denoted as for formulas in the logical language, φ is the revision w0, in a new model, denoted as M0 = (W 0,R0,V 0,κ0). formula. The postulates are listed as follows:

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1. K ∗ φ is a theory type 2005). In most of the literature on fuzzy modal logic, 2. φ ∈ K ∗ φ success they deal the fuzzy environment with the many-value logic, while in this paper a new method is proposed 3. K ∗ φ ⊆ K + φ upper bound to depict the fuzziness, i.e. the linguistic variable 4. if ¬φ ∈ K, then K + φ ⊆ K ∗ φ lower bound terms (LTTS). On one hand, LTTS can give the 5. K ∗ φ = K⊥ iff φ is inconsistent triviality propositions’ truth value in a continuous degree, like Lukasiewicz many-value logic. On the other hand, 6. if φ is equivalent to ψ then K ∗ φ = K ∗ ψ exten- LTTS is more closer to our daily language and it may sionality play an more important part in application. 7. K ∗ (φ ∧ ψ) ⊆ (K ∗ φ) + ψ iteration upper bound In the study of fuzzy belief logic (Zhang and Liu, 8. if¬ψ ∈ K ∗ φ, then (K ∗ φ) + ψ ⊆ K ∗ (φ ∧ ψ) iter- 2012), researchers formalize reasoning about fuzzy ation lower bound belief and fuzzy common belief, reduce the belief degrees to truth degrees, and finally prove the com- In this section we will check whether or not the pleteness of the fuzzy belief and fuzzy common be- AGM postulates are fulfilled after an update in our lief logic. However, they have just studied the fuzzy logic system. As a preliminary step, we need to define version of belief and knowledge in the state situation. belief sets, and decide the type of the revision formu- While in daily life, actually if our belief or knowl- las. In the reminder, let (M,s) be a belief state, where edge have changed, there must be a new actions or an M is defined in Definition 6; let Σ = {φ | M,s |= Bφ} event happened. Like Public Announcement Logic be a belief set, and ψ be a revision formula; and we re- (van Benthem, 2002), we may know φ after someone strict the revision formulas on propositional formulas. has declared proposition φ or some other proposition Then formally, we have: ψ related to φ. We can say that the actions or events Defininition 11. actually cause to the belief or knowledge changing. K = {χ | Val(Bχ,M,s) 6= absolutely- f alse}; So, it is really important to consider the dynamics K ∗ φ = {χ | Val([∗φ]Bχ,M,s) 6= absolutely- f alse}; when we reason about the beliefs. However, the ex- K + φ = K ∪ {φ}. isting work did this little. Now we are going to check whether or not the 8 In addition, the dynamic fuzzy logic can handle postulates can be verified. Formally, we have the fol- the fuzzy environments (Hughes,Esterline and Kimi- lowing theorem: aghalam, 2012). However, belief revision has not be handled like what we did in this paper. On the other Theorem 4. In our fuzzy belief logic system, AGM hand, although dynamic epistemic logic (van Ben- postulates them, 2002; Ditmarsch,Hoek and Kooi, 2007) and • K1,K5,K6 are verified; dynamic belief logic (van Ditmarsch, 2005; van Ben- • K3,K4,K7,K8 are undecided; and them, 2007) are studied vastly, there are few literature to expand the dynamic epistemic logic and dynamic • K2 is weakly satisfied, i.e., if we revise our previ- ous belief with a proposition of φ and Val(φ) = belief logic into the fuzzy realm. However in this pa- per, we give a new approach to reason about dynamic {τ | τ ∈ LTTSt } in the revision method pro- posed in Definition 9, then we will believe propo- belief revision in fuzzy environments. 0 0 sition φ to an extent of τ such that τ ∈ LTTSt and τ0 = Val(φ,M,v), where κ(v) ≤ κ(v0) and 0 Val(φ,M,v),Val(φ,M,v ) ∈ LTTSt . 7 CONCLUSIONS

This paper is a fuzzy extension of dynamic belief re- 6 RELATED WORK vision with quantitative method depicting belief revi- sion. We give the syntax and semantics of the fuzzy Generally speaking, so far few people have study the dynamic logic and use the Linguistic Truth Term Set fuzzy form of dynamic belief logic although fuzzy as the truth values for the fuzzy propositions. Then modal logic is the foundation of the fuzzy belief logic we expose some properties for the logic. Our fuzzy and fuzzy dynamic logic because they are actually the method to deal with dynamic belief logic is not only extension of modal logic. of great importance for theory study like epistemic Researchers started to study fuzzy modal logic logic and belief logic, but also of great application in in about 1970 (Schotch, 1975; Zadeh, 1965; Zadeh, Artificial Intelligence and Multi-agent Systems. 1975). Moreover, some scholars have developed In the future, we will give the complete axiomatic a complete system of fuzzy modal logic (Mironov, system and prove the soundness and of

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our logic,2 and we will also integrate the fuzzy logic Schotch, P.K. (1975) Fuzzy Modal Logic. In Proceeding for other dynamic belief logic besides Aucher’s the- of the 5th Internation Symposial on Multiple-Valued ory. Logic 176-182. Shapiro, S., Pagnucco, M., Lesperance, Y., Levesque, H.J. (2011) Iterated Belief Change in the Situation Calcu- lus. In Artificial Intelligence 175(1), 165-192. ACKNOWLEDGEMENTS Sardina, S., Padgham, L. (2011)A BDI Agent Programming Language with Failure Handling, Declarative Goals, and Planning. In Autonomous Agents and Multi-Agent This paper is supported by National Natural Science Systems 23(1), 18-70 Foundation of China (No. 61173019), Bairen plan van Ditmarsch, H. (2005) Prolegomena to Dynamic Logic of Sun Yat-sen University, National Fund of Phi- for Belief Revision. In Synthese 147(2), 229-275 losophy and Social Science (No. 11BZX060), Of- van Benthem, J. (2007)Dynamic Logic for Belief Revision. fice of and Social Science of Guangdong In Journal of Applied NonClassical 17(2), 129- Province of China (No. 08YC-02), Fundamental Re- 155 search Funds for the Central Universities in China, van Benthem, J. (2002) ’One is a Lonely Number’: On and major projects of the Ministry of Education (No. the Logic of Communication.In Proceedings of Logic 10JZD0006). We also appreciate reviewers’ valuable Colloquium ’02 96-129 comments, which helped us to improve the paper sig- Wu, X., Zhang, J. (2012) Rough set models based on ran- nificantly. dom fuzzy sets and belief function of fuzzy sets. In International Journal of General Systems 41(2), 123- 141. Zadeh, L.A. (1965) Fuzzy sets. In Information and Control REFERENCES 8(3), 338-353 Zadeh, L.A. (1975)The concept of a linguistic variable and its application to approximate reasoning. In Informa- Aucher, G. (2006) A Combined System for Update Logic tion Sciences 8(3),199-249 and Belief Revision. In Intelligent Agents and Multi- Agent Systems, Intelligent Agents and Multi-Agent Zhang, J., Liu, X. (2012) Fuzzy belief measure in ran- Systems.3371(132), 1-18 dom fuzzy information systems and its application to knowledge reduction, In Neural Computing & Appli- Baltag, A., Smets, S. (2006) Dynamic Belief Revision over cations Multi-Agent Plausibility Models. In Proceedings of the LOFT’06, University of Liverpool.11-24. Bonissone P.P., Decker K.S. (1986) Selecting Uncertainty Calculi and Granularity: An Experiment in Trading- off Precision and Complexity. In Uncertainty in Ar- ticficial Intelligenc 217-247 Casalia, A., Godob, L., Sierra, C. (2011) A Graded BDI Agent Model to Represent and Reason about Prefer- ences.In Artificial Intelligence 175(7-8), 1468-1478 Ditmarsch H., Hoek, W., Kooi, B. (2007) Dynamic Epis- temic Logic, Synthesis Library, 337, Springer. Hughes, J., Esterline, A., Kimiaghalam, B. (2012): Fuzzy Dynamic Logic, in http://phiwumbda.org/ ∼jesse/papers/fuzzy.ps Luo, X., Zhang, C., and Jennings N. R. (2002) A Hybrid Model for Sharing Information Between Fuzzy, Un- certain and Default Reasoning Models in Multi-Agent Systems. In International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10(4), 401- 450. Mironov, A.M. (2005) Fuzzy Modal Logics. In Journal of Mathematical Sciences 128(6), 3461-3483. Rott, H. (2012) Bounded Revision, Two-Dimensional Be- lief Change Between Conservative and Moderate Re- vision. In Journal of Philosophical Logic41(1), 173- 200.

2The proof of the completeness is very complicated, be- yond the scope of this paper. We will discuss the issue in a seperate paper.

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