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Fuzzy Temporal Predicate Logic for Incomplete Information EasyChair Preprint № 3558 Modeling Fuzzy Temporal Predicate Logic and Fuzzy Temporal Agent Venkata Subba Reddy Poli EasyChair preprints are intended for rapid dissemination of research results and are integrated with the rest of EasyChair. June 5, 2020 Manuscript Click here to download Manuscript: IJFSifuzzy14ftpl.pdf ModelingFuzzy Fuzzy Temporal Temporal Predicate Logic Logic and Fuzzy for TemporalIncomplete Agent 1 2 Information 3 4 5 P.Venkata Subba Reddy 6 Computer Science and Engineering 7 Sri Venkateswara University College of Engineering 8 Tirupati-517502, India 9 [email protected] 10 11 12 13 Abstract—The fuzzy temporal logic is fuzzy logic to deal time 14 constraints of incomplete information. The Fuzzy Temporal II. FUZZY LOGIC 15 logic is to solve independent time constraints problems of Zadeh [11] introduced fuzzy set as model to deal with 16 incomplete information in AI. The Knowledge Representation imprecise, inconsistent and inexact information. The fuzzy set (KR) is the main component to solve the problems in Artificial 17 A of X is defined by its membership function µ (x) and take 18 Intelligence (AI). In this paper, fuzzy predicate temporal logic is A the values in the unit interval [0, 1] 19 studied for incomplete information with time constraints. Fuzzy µ (x) →[0, 1], where x єX is in some Universe of 20 Temporal Agent is discussed for Fuzzy Automated Reasoning. A discourse. 21 The Logic Programming Prolog is discussed for Fuzzy Temporal Predicate Logic For example, 22 Consider the fuzzy proposition “x is tall” and the fuzzy set 23 Keywords— fuzzy logic; fuzzy reasoning; fuzzy agent; fuzzy „tall” is defined as 24 modules; temporal logic fuzzy temporal predicate logic, logic µ (x)→[0, 1], x Є X 25 tall programming tall= 0.5/x1 +0.6/x2 + 0.7/x3 +0.8/x4 +0.9/x5 26 . Let A, B and C be the fuzzy sets, and the 27 I. INTRODUCTION 28 operations on fuzzy sets are given below 29 The problem solving in the system may be viewed as a AVB=max(µA(x) , µB(x)} Disjunction 30 collection of intelligent agents [3]. The Intelligent agent is AΛB=min(µA(x) , µB(x)} Conjunction 31 the agent which deals with independent component. There are A′=1- µA(x) Negation 32 intelligent agents for automated reasoning systems like A→B=min {1, (1- µA(x) +µB(x)} Implication 33 TMS[6] to deal with incomplete information, which are non AXB=min { µA(x) , µB(y)}/(x,y) Relation 34 fuzzy based. AoR=min x{ µA(x) , µR(x,y)}/y Composition 35 The fuzzy agent is fuzzy based automated system to The fuzzy propositions may contain quantifiers like 36 deals with incomplete information which consists of fuzzy “very ”, “More or Less” etc. These fuzzy quantifiers may be 37 modulations The fuzzy agent is fuzzy based automated eliminated as[10] 38 system to deals with incomplete information The µvery (x) = µA(x) ² Concentration 39 Knowledge Representation (KR) is key component of µmore or less(x) = µA(x) 0.5 Diffusion 40 Intelligence Agent to solve the problems. There are KR III. FUZZY TEMORAL LOGIC 41 methods like predicate logic for complete information [6]. 42 The information available to the system may be incomplete . the temporal logic is a logic with time constraints and 43 information. The fuzzy logic [10] deals incomplete Time variables “t1-t0” like “before”, “meet”, “after”, where 44 information with belief rather than probable [5]. The fuzzy starting time t0 and ending time t1. 45 modulations[8] and fuzzy predicate logic[9] are discussed for Fuzzy temporal logic has to deal with incomplete 46 incomplete information. information of time constraints[1]. 47 The fuzzy propositions may contain time constrains. For A temporal variable is “t1-t0”, where t0 is starting time 48 instance “ The train “x” will come shortly”. This situation is and t1 ending time. 49 falls under fuzzy temporal. There is a need for KR for fuzzy For instance “past”=t 1-t0, t0<t1 50 temporal logic. “Present”= t1 approximately =t0 51 The Incomplete information time constraints are “feature”=t 0-t1, t0>t1 52 represented with fuzzy temporal modulations and later A temporal set is set of temporal variables with interval 53 represented by Fuzzy Temporal Predicate Logic (FTPL). The ”t1-t0 54 Fuzzy Temporal Agent is discussed for Automation with .The Temporal logic is interpreted in simple methodl. 55 Prolog. A Temporal logic is logic combinations of temporal sets. 56 57 58 59 60 61 62 63 64 65 0.5 Let (I, t) and (J.t) are temporal sets. µmore or less richXpast(x) = µrichXpast(x) Diffusion For instance “ x was rich” The operations on fuzzy temporal are similar to fuzzy sets Was rich=richXpast. type-2 are given as 1 (Rich, past) ĈVĎ=max{µĈ(x,t) , µĎ(x,t)} Disjunction(Ĉ overlap Ď) 2 not(I), t)=not (I,t) ĈΛĎ=min{µĈ(x,t) , µĎ(x,t)} Conjunction(Ĉ before Ď) 3 (not(rich), past)=not (rich,past) ĈĎ=min{1, 1-µ (x,t) +µ (x,t)}Implication(Ĉ proceeds Ď) 4 Ĉ Ď (I,t) and (J,t) = (I,t) Λ(J,t) conjunction ĈxĎ=min{µĈ(x,t) ,µĎ(x,t)} Relation 5 “x was rich and poor” 6 (rich,past) and (poor,past) = (rich, past) Λ(poor, past) IV. FUZZY MODULATIONS 7 (I,t) or (J,t) =(I,t) V (J,t) disjunction The Automated Fuzzy Reasoning System is 8 “x was rich or poor” problem solving system using fuzzy reasoning with fuzzy 9 (rich,past) or (poor,past) = (rich, past) V (poor, past) facts and rules. These fuzzy facts and rules are modulated to 10 If (I,t) then (J,t)= (I,t) (J,t) implication represent the Knowledge available to the system. The fuzzy 11 “if x was rich thenx was poor” agent is independent component which performs fuzzy 12 If (rich,past) then (poor,past) = (rich, past) (poor, past) reasoning. 13 Definition: Let p be the fuzzy temporal proposition of the The fuzzy modulations for Knowledge representation are 14 form like „ x was A”. The fuzzy temporal set à may be type of modules for fuzzy propositions “x is A”. “x is A” is 15 defined in terms of possibility Π as 16 defined as pΠ =A, where “R” is relation and xЄX is universe of 17 R(x) [A]R(x), 18 discourse. where A is fuzzy set, R is relation and x is individual in the 19 For instance , Unversed of discourse X. 20 The fuzzy proposition may contain time variables like . For instance 21 “ x was rich” “Rama is tall “represented as 22 was rich=Πwealth(x)= rich X past [tall]Hight(Rama), where “tall” is fuzzy set, “Hight” is 23 Definition: The fuzzy temporal set à is characterized relation and “Rama” is individual. 24 by membership function µÃ:XxT [0,1], xЄX and TЄA The fuzzy modules[13] are knowledge representation 25 Suppose X is a finite set. The fuzzy temporal set à of X technique of the fuzzy propositions. The Fuzzy Modulations 26 may be represented by are combined with logical operators. 27 Ã=⌠⌠µÃ(x,t)/x/t= ∑∑ µÃ(x,t) = (µÃ(x1,t1)/x1 + µÃ(x2,t1)/x2 Let A and B be fuzzy sets. 28 +…+ µÃ(xn,t1)/xn)/t1 x is ¬A 29 + (µÃ(x1,t2)/x1 + µÃ(x2,t2)/x2 +…+ µÃ(xn,t2)/xn)/t2 +…+ [¬A]R(x) 30 (µÃ(x1,tm)/x1 + µÃ(x2,tm)/x2 +…+ µÃ(xn,t1)/xn)/tm x is A or x is B 31 à ′=1-µÃ(x,t) [A V B]R(x) 32 Ã= { (0.1/x1+ 0.2/x2+0.3/x3+0.35/x4+0.4/x5)/t1 x is A and x is B 33 +(0.4/x1+0.45/x2+0.5/x3+0.55/x4+0.6/x5)/t2 [A Λ B]R(x) 34 +(0.7/x1+0.75/x2+0.8/x3+0.85/x4+0.9/x5)/t3 } if x is A then x is B 35 Train arrival= { (0.2x1+ [A → B]R(x) 36 0.3/x2+0.5/x3+0.7/x4+0.9/x5)/before Some of the Fuzzy Reasoning rules are given as 37 +(0.6/x1+0.65/x2+0.7/x3+0.8/x4+0.9/x5)/normal R1: [A]R(x) 38 +(0.7/x1+0.75/x2+0.8/x3+0. 85/x4+0.9/x5)/after } [B](R(x) or R(y)) 39 For instance “Train came in normal time” 40 “Train will come after 10 minutes” [AΛB]R(y) 41 “Train left 10 munities before” R2: [A]R(x) 2 42 “usually the train x is late” µÃ(x) [B](R(x) or R(y) 0.5 43 “the train x comes more or less in time” µÃ(x) 44 “x was rich” [AVB ]R(y) 45 Π was rich (x)=rich X past 46 where rich X past = min { rich, past} R3: [A](R(x,y) 47 rich= 0.5/x1 +0.5.5/x2 + 0.7/x3 +0.7.5/x4 +0.8/x5 [B](R(y,z) 48 past= 0.4/t1 +0.6/t2 + 0.7/t3 +0.8/t4 +0.85/t5 ______________ 49 was rich =rich X past =min { 0.5/x1 +0.5.5/x2 + 0.7/x3 [AΛ B](R(x,z) 50 +0.7.5/x4 +0.8/x5 , 0.4/t1 +0.6/t2 + 0.7/t3 +0.8/t4 +0.85/t5 } R4: [A](R(x) or R(y)) 51 = 0.4/t1 +0.55/t2 + 0.7/t3 +0.7.5/t4 +0.8/t5 [B](R(y) or R(z)) 52 The fuzzy temporal propositions like “x was A” 53 may contain quantifiers like “very ”, “More or Less” etc.
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