EasyChair Preprint № 3558

Modeling Fuzzy Temporal Predicate Logic and Fuzzy Temporal Agent

Venkata Subba Reddy Poli

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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, t0t1 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) ĈĎ=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 pΠ =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. [AV B](R(x) or R(z)) 54 These fuzzy quantifiers may be eliminated as R5: : [A]R(x) 55 µvery richXpast(x) = µrichXpast(x) ² Concentration if [A]R(x) then [B]R(y) 56 57 58 59 60 61 62 63 64 65 [tall]Hight(Rama) [[Ao (A→B)]R(y) The fuzzy predicate is given by Patient has cold Hight(Rama, tall) 1 If Patient has cold then Patient has headache where “tall” is fuzzy set, “Hight” is relation and “Rama” 2 The inference is given as using the above fuzzy fact is individual. 3 and fuzzy rule The fuzzy propositions may contain quantifiers like “for 4 [cold ] symptom(Patient) every”, “there exits” “very ”, “More or Less” etc. These fuzzy 5 if [cold ] symptom(Patient) than quantifiers may be eliminated as 6 [headache]symptom(Patient) µvery (x) = µA(x) ² 7 The fuzzy reasoning is given as using Fuzzy for instance, 8 Knowledge Base young= 0.8/x + 0.7/x +0.7/x +0.6/x +0.6/x5 9 1 2 3 4 [cold ] ^[ cold → headache ] symptom (Patient, very young= 0.64/x1 +0.49/x2+0.49/x3+0.36/x4+0.36/x5 10 0.5 headache µ (x) = µ (x) 11 more or less A more or less young 12 V. FUZZY PREDICATE LOGIC =0.89/x +0.83/x +0.83/x +0.77/x +0.77/x 13 1 2 3 4 5 The Predicate is Relation with arguments. The V x (if x is A then x is B ) 14 arguments may be variables or constants. min (R(x,A)→R(y,B)) 15 x Consider the fuzzy proposition 16 The Predicate Logic is a combination of predicates with “for every young person runs fast” may be given as 17 logical operators „not‟, „and‟, „or‟, implication‟, for every ‟ 18 „there exists‟ etc. V x (age(x, young)run(x,fast) 19 Marcus was a man minx (age(x, young)run(x,fast) 20 Man(Marcus) For instance, 21 All man are people young= 0.8/x1 + 0.7/x2+0.7/x3+0.6/x4+0.6/x5 22 V Man(x)→person(x) fast= 0.2/x1 + 0.3/x2+0.4/x3+0.5/x4+0.4/x5 23 Every one loyal to every one youngfast = 0.4/x1 + 0.7/x2+0.8/x3+0.9/x4+.8/x5 24 V x ] y loyal(x,y) minx {youngfast} = 25 Basic resolution minx {0.4/x1 + 0.7/x2+0.8/x3+0.9/x4+0.8/x5 }=0.4/x1 26 p ] x (if x is A then x is B ) 27 pq= (¬p v q) maxx (R(x,A)→R(y,B)) 28 ______Consider the fuzzy proposition 29 q “there exists young person who runs fast” may be given as 30 For instance ] x (age(x, young)run(x,fast) 31 winter maxx (age(x, young)run(x,fast) 32 winter  cold For instance, 33 ______young= 0.8/x1 + 0.7/x2+0.7/x3+0.6/x4+0.6/x5 34 cold fast= 0.2/x1 + 0.3/x2+0.4/x3+0.5/x4+0.4/x5 35 p v q youngfast = 0.4/x1 + 0.7/x2+0.8/x3+0.9/x4+.8/x5 36 ¬p v r maxx {youngfast} = 37 ______maxx {0.4/x1 + 0.7/x2+0.8/x3+0.9/x4+0.8/x5 }=0.9/x4 38 q v r The fuzzy reasoning is drawing conclusions from the 39 for instance fuzzy propositions. Some of the Fuzzy Reasoning rules are 40 winter v summer given as 41 ¬summer v cold R1: [A]R(x,A) 42 ______R(x, B) and R(y,B) 43 Winter v cold 44 The fuzzy predicate may be defined as relation with R(y, AΛB ) 45 arguments and the arguments may be constants or variables R2: R(x, A) 46 R(x,A) R(x,B) or R(y,B) 47 For instance, 48 x is tall R(y, AV B) 49 Height(x,tall) R3: R(x,y,A) 50 Rama is tall with fuzziness 0.7 R(y,z,B) 51 Height(Rama,0.7) ______52 where A is fuzzy set, R is relation and x is individual in the R(x,z, AΛ B) 53 Unversed of discourse X. R4: R(x,A) or R(y,A) 54 For instance R(y,B) or R(z,B) 55 “Rama is tall “ may be modulated as 56 57 58 59 60 61 62 63 64 65 R(x, AV B) VR(z, AV B)) x is A and x is B R5: :R(x,A) [A Λ B]R(x) if R(x, A) then R(y,B) if x is A then y is B 1 R(y,Ao A→B) [A ]R(x) [B]R(y) Implication 2 Patient has cold [A‟]R(x) o [A ]R(x) [B]R(y) Comosition 3 If Patient has cold then Patient has headache Consider the two propositions 4 The inference is given as Using the above fuzzy fact x is more or less late 5 and fuzzy rule If x comes late then y comes very late 6 symptom(Patient, cold ) The inference is given as using the above fuzzy fact 7 if symptom(Patient, cold ) than symptom(Patient, and fuzzy rule 8 headache) [very late ] arrival(x) 9 The fuzzy reasoning is given as using Fuzzy [ more or less late ] arrival(x)  [ livery late ] arrival(y) 10 Knowledge Base The fuzzy reasoning is given as 11 [cold ] ^[ cold → headache ] symptom (Patient, [very late]arrival(x) o [ more or less late ] arrival(x)  [ 12 headache) livery late ] arrival(y) 13 The fuzzy logic and fuzzy reasoning are discussed in the 14 following for the FPL. 15 VII. FUZZY TEMPORAL PREDICATE LOGIC 16

17 VI. FUZZY TEMPORAL AGENT 18 First Order Fuzzy Temporal Predicate Logic is 19 The Fuzzy Temporal Agent is an Automated Fuzzy transformation of fuzzy temporal modulations. 20 Reasoning System with fuzzy temporal facts and fuzzy Consider the fuzzy temporal proposition of type “x was 21 temporal rules. These fuzzy temporal facts and fuzzy A” may be modulated as 22 temporal rule are modulated to represent Knowledge to the [Ã]R(x) 23 system. The Fuzzy Temporal Agent is independent The fuzzy temporal predicate may be defined as relation 24 component which performs fuzzy reasoning is shown in Fig.1 with arguments and the arguments may be constants or 25 variables Fuzzy Temporal R(x,Ã) 26 Fuzzy Knowledge Base For instance, 27 Inference x was rich 28 Fuzzy Fuzzy wealth t(x,richXpast) 29 Tempora Temporal The fuzzy temporal predicate logic is the combination of 30 Goals l clauses predicates 31 fuzzy temporal predicate with logical operators. Let A and B be fuzzy sets. 32 33 Fig.1 Fuzzy Temporal Agent x is A 34 R(x, A) 35 Definition: The fuzzy temporal modulations are the arrival(x,late) 36 possibility distribution. x is not A 37 [A]R(x), R(x, not A) 38 where A is fuzzy set, R is relation and x is individual in the arrival(x,not late) 39 Unversed of discourse X. x is A or y is B 40 For instance R(x, A) V R(y,B) 41 “x was rich” may be modulated as represented as arrival(x, late) V arrival(y, very late) 42 [rich]wealth(x), where “young” is fuzzy set, “wealth” is x is A and y is B 43 relation and “x” is individual. R(x, A) Λ R(y,B) 44 The fuzzy temporal modules are type of knowledge arrival(x, late) Λ arrival(y, very late) 45 representation technique for the fuzzy temporal propositions. if x is A then y is B 46 The fuzzy temporal modulations are combined with logical R(x,A)→R(y,B) 47 operators. arrival(x,late )  arrival(y very late ) 48 Let A and B be fuzzy temporal sets. The fuzzy temporal Reasonig is drawing conclusion. The reasoning with 49 operations are given as fuzzy temporal predicate logic is given as 50 x is A x is late 51 [A]R(x) if x comes late then y comes very late 52 x is ¬A The fuzzy temporal fact and rule may be modulated as 53 [¬A]R(x) Negation arrival(x, late ) 54 x is A or y is B arrival(x, more or less late )  arrival(y, very late ) 55 [A ]R(x)V [B]R(y) Disjunction arrival(x, late )o arrival(x, more or less late )  arrival(y, 56 57 58 59 60 61 62 63 64 65 very late ) Yes arrival(y, lateo(more or less late  very late )) The prolog will give a fuzziness is 0.64 for “y is late”. let fuzziness late =0.8 1 arrival(y, 0.8o(0.89  0.64 )) IX. CONCLUSION 2 arrival(y, min{0.8, min(1, 1-(0.89 + 0.64 )) The AI Problems may contain temporal constraints with 3 arrival(y, min{0.8,0.53) incomplete information. The Knowledge Representation is the 4 arrival(y, 0.53) key factor for problem representation In AI. The traditional 5 i.e., y is late with fuzziness 0.53 predicate logic is difficult to study with the fuzzy temporal 6 constraints. The fuzzy temporal logic is studied for reasoning 7 VIII. PROLOG FOR FUZZY TEMPORAL PREDICATE in AI problems with incomplete information time constraints. 8 LOGIC The fuzzy temporal agent is independent intelligent problem 9 The Prolog is mainly used for Predicate Logic. The solving system with time constraints and incomplete 10 Prolog is a Logic Programming language [2]. It contains information for Automation. The fuzzy temporal modulations 11 mainly predicates and Clauses. are discussed for fuzzy temporal predicate logic to represent 12 A predicate is a relation with name of the relation and incomplete information of time constraints . The Prolog is 13 arguments. The arguments may be containing variables or discussed for fuzzy temporal agent for automated reasoning. 14 constants. 15 16 For instance, ACKNOWLEDGMENT 17 arrival(x, late ) A clause is combination of and/ or more predicates for the The author thank the Program Chair iFUZZY 2014 and 18 referees for their valuable suggestions. 19 rules 20 arrival(y, very late ) :-arrival(x, more or less late ) 21 Consider the logic programming for fuzzy temporal logic 22 x is late References if x comes late then y comes very late 23 24 arrival(x, late ) 25 arrival(x, more or less late )  arrival(y, very late ) [1] Allen, J.F. Natural Language Understanding, Benjamin 26 arrival(x, late )o arrival(x, more or less late )  arrival(y, Cummings,1994 very late ) [2] Bratko, Ivan, Prolog programming for artificial intelligence. 27 Harlow, England ; New York: Addison Wesley, 2001. 28 arrival(y, lateo(more or less late  very late )) The fuzzy reasoning may be given as [3] A.Chavez,A.Moukas and P.Maes, ”Challenger: A multiagent 29 system for Distributed Resource allocation” Proce.Of. Int 30 arrival(x, late ) Conference on Autonomous agents, Morina del rey, 31 arrival(y, very late ) :-arrival(x, more or less late ) Colofornial,1997. 32 ______[4] N. Rescher, Many-Valued Logic, McGrow-Hill, New York, 1969. 33 arrival(y, lateo(more or less late  very late )) [5] Shafer, G, A Mathematical Theory of Evidence , Priceton, NJ, 34 The Logic Programming may be written in SWI-Prolog as University Press, 1976. 35 fuzzy(A,B,M) :- A > B, M is 1-A+B. [6] Elaine Rich, Kevin Knight: “Artificial Intelligence”, 3rd Edition, 36 fuzzy(A,B,M) :- M=1. Tata McGraw Hill, 2009. 37 fuzzy1(C,M,F):-C