Fuzzy reasoning spiking neural P system for fault diagnosis

Fuzzy reasoning spiking neural P system for fault diagnosis Hong Peng a,c, , Jun Wang b , Mario J. Pérez-Jiménez c , Hao Wang a , Jie Shao a , Tao Wang b a School of Mathematics and Computer Engineering, Xihua University, Chengdu 610039, China b School of Electrical and Information Engineering, Xihua University, Chengdu 610039, China c Research Group of Natural Computing, Department of Computer Science and Artificial Intelligence, University of Seville, Sevilla 41012, Spain Keywords Fault diagnosis P systems Spiking neural P systems Fuzzy knowledge representation Fuzzy reasoning abstract Spiking neural P systems (SN P systems) have been well established as a novel class of distributed parallel computing models. Some features that SN P systems possess are attractive to fault diagnosis. However, handling fuzzy diagnosis knowledge and reasoning is required for many fault diagnosis applications. The lack of ability is a major problem of existing SN P systems when applying them to the fault diagnosis domain. Thus, we extend SN P systems by introducing some new ingredients (such as three types of neurons, fuzzy logic and new firing mechanism) and propose the fuzzy reasoning spiking neural P systems (FRSN P systems). The FRSN P systems are particularly suitable to model fuzzy production rules in a fuzzy diagnosis knowledge base and their reasoning process. Moreover, a parallel fuzzy reasoning algorithm based on FRSN P systems is developed according to neuron’s dynamic firing mechanism. Besides, a practical example of transformer fault diagnosis is used to demonstrate the feasibility and effectiveness of the proposed FRSN P systems in fault diagnosis problem. 1. Introduction The fault diagnosis of electrical machines have moved in recent years from traditional techniques to artificial intelligence techniques [2,12,24]. In order to guarantee the secure and stable operation of electrical equipment, it is very important to diagnose its faults rapidly and accurately. More recently, a large number of methods or models using artificial intelligence and knowledge engineering have been addressed, such as expert system, Bayesian networks, artificial neural networks, and genetic algorithm [1,5,10,11,21,29]. Unfortunately, however, most of the existing methods are capable of dealing with simple diagnosis reasoning. Consequently, their reasoning processes are often difficult when complex diagnosis are shown. Membrane computing is a class of distributed parallel computing models inspired by the structure and functioning of living cells, as well as from the way the cells are organized in tissues or higher order structures, which was introduced by Gh. Pa ˘un in 2000 [15]. In the membrane computing domain, this kind computing systems (devices) are commonly called P systems. The main ingredients of a P system are (i) the membrane structure (a rooted tree), delimiting compartments where (ii) multisets of objects evolve according to (iii) (reaction) rules of a bio-chemical inspiration [17]. According to their structures, these models can be divided into three categories: cell-like P systems, tissue-like P systems and neural-like P systems. Spiking neural P systems (SN P systems) firstly introduced by Ionescu et al. [8], as main forms of neural-like P systems recently investigated, are incorporated into membrane computing from the way that biological neurons communicate through electrical impulses of identical form (spikes). An SN P system can be viewed as a set of neurons placed in the nodes of a directed graph whose arcs represent the synaptic connections among the neurons. The flow of information is inherently realized by the exciting of pulse potentials, which are encoded by the so-called spikes. The spikes are objects of a unique type and are placed inside the neurons and can be sent from presynaptic to postsynaptic neurons according to specific firing/spiking rules and by means of forgetting rules which are associated with neuron. By applying a firing/spiking rule some spikes are consumed and new spikes are produced. The spikes produced are sent to all neurons linked by a synapse to the neuron where the rule is used. By applying a forgetting rule, spikes are removed from neurons. Currently, SN P systems and a number of variants have been proposed [3,7,8,14,16,20,25,26,28]. However, only a few results applied to practical problems have been addressed [9,17,27]. This paper focuses on the application of SN P systems paradigm to the fault diagnosis problem. In addition to distributed and parallel computing advantage, the following features can be summarized from inherent mechanism of SN P systems: (i) high understandability (due to their directed graph structure), (ii) dynamically (neuron’s firing and spiking mechanisms make them suitable to model dynamic behaviors of a system), (iii) synchronization (that makes them suitable to describe concurrent events or activities), (iv) non-linearly (that makes them suitable to process non-linear situation), (v) nondeterministically. As we know, fault diagnosis problem can refer to the handling of the uncertainty and incompleteness, the representation of diagnosis knowledge and reasoning [10,24]. These features and advantage of SN P systems seem to be suitable to model diagnosis knowledge and reasoning to fault diagnosis. However, major drawback of existing SN P systems and their variants is not capable of handling incomplete and uncertain information. In this paper we propose a class of extended SN P systems, called fuzzy reasoning spiking neural P systems (FRSN P systems). Some new ingredients are introduced to extend original SN P systems, including three types of neurons (proposition neurons, ‘‘AND’’-type and ‘‘OR’’-type rule neurons), fuzzy truth value, fuzzy logic and new firing mechanism. The FRSN P systems can well model and visualize fuzzy production rules in a diagnosis knowledge base due to their graphical nature. Combination of neuron’s new firing mechanism and fuzzy logic ensures to automatically accomplish dynamic fuzzy reasoning. Furthermore, a fuzzy reasoning algorithm based on matrix operations is developed, which can give full play to maximally parallel computing advantage of the FRSN P systems. The present paper is organized as follows. Section 2reviews some concepts of SN P systems, and then describes the proposed FRSN P systems. Section 3describes a modeling method of fuzzy production rules based on the FRSN P systems. A fuzzy reasoning algorithm based on the proposed FRSN P systems is presented in Section 4. A fault diagnosis example is provided in Section 5, and conclusions are discussed in Section 6. 2. FRSN P systems 2.1. SN P systems Here, we briefly review the basic concepts of SN P systems in the computing form (i.e., able to take some inputs and provide some outputs) (see [8,17]). Definition 1. A SN P system of degree mP1, is a construct of the form P¼ðA; r 1 ;...; r m ;syn;I;OÞ where (1) A={a} is the singleton alphabet (the object ais called spike); (2) r 1 ,..., r m are neurons, of the form r i =(n i ,r i ) with i2{1, ...,m}, where (i) n i P0 is the initial number of spikes contained by neuron r i ; (ii) r i is a finite set of rules of the form E=a c !a p ;d where Eis a regular expression over a,cP1 and p,dP0, with cPp;ifp= 0, then Eis the empty string, d= 0, and a s belongs to the language generated by expression Efor no rule E/a c ?a p ;d(cPpP1) from r i . (3) syn #{1,2,...,m}{1,2,...,m} with i–jfor all (i,j)2syn,16i,j6m(synapses between neurons). (4) Iand Oare input neuron set and output neuron set, respectively. In the above definition, the rule E/a c ?a p ;dwith pP1 is called firing/spiking rule; the rule E/a c ?a p ;dwith p=d=0is written in the form a c ?kand is called forgetting rule. The firing mechanism of neurons can be explained as follows: if a neuron r i contains kspikes, a k belongs to the language generated by expression E(see [23] for further details about language theory), and kPc, the firing/spiking rule E/a c ?a p ;d2r i (with pP1) in neuron r i is enabled and can be applied. This means that cspikes are consumed, kcspikes remain in the neuron, the neuron is fired, and then it produces pspikes after dtime units. If d= 0 the spikes are emitted immediately. In the case dP1, if the rule is used at step t, the neuron is ‘‘closed’’ or ‘‘blocked’’ at steps t,t+1,...,t+d1, and it cannot receive new spikes from other neurons. At step t+d, the neuron emitts spikes and becomes again open, hence it can receive spikes. The pspikes emitted by the neuron r i are replicated and they go to all neurons r j such that (i,j)2syn (each such neuron r j of them receives pspikes). A forgetting rule a c ?kis applicable to a neuron wheteher the neuron contains exactly cspikes and then all cspikes are removed. SN P systems are synchronized because a global clock is assumed, marking the time for the whole system. Besides, SN P systems are non-deterministic because two rules E 1 =a c 1 !a p 1 ;d 1 and E 2 =a c 2 !a p 2 ;d 2 can have L(E 1 )\L(E 2 )–;. Therefore, it is possible that two or more rules of the system can be enabled in a neuron. In this case, one of them is non-deterministically chosen to be used. Moreover, in each time unit, if a neuron can use a rule, the rule must be used. Each neuron deals with its spikes in the sequential manner, only using one rule in each time unit, but the rules are used in parallel for all neurons of the system. An instantaneous description or a configuration at any instant of a SN P system is described by both the number of spikes in each neuron and the state of the neuron, more precisely, by the number of steps to count down until it becomes open (this number is zero if the neuron is already open). The initial configuration is described by the number of spikes initially placed in each neuron, n 1 ,n 2 ,...,n m , with all neurons being open. A configuration is a halting configuration if all neurons are open and no rule of the system is applicable to it. Using the rules described above, one can define transitions among configurations. We say that configuration C 1 yields configuration C 2 in one transition step, denoted by C 1 ) P C 2 , if we can pass from C 1 to C 2 by applying the rules from the system following the previous remarks. Acomputation of P is a (finite or infinite) sequence of configurations such that: 1. the first term of the sequence is the initial configuration of the system; 2. each non-initial configuration of the sequence is obtained from the previous configuration by a transtion step; and 3. if the sequence is finite (called halting computation) then the last term of the sequence is a halting configuration. A computation in a system as above starts in the initial configuration. In order to compute a function f:N k ?Nwe introduce knatural numbers q 1 ,...,q k in the system by ‘‘reading’’ from the environment a binary sequence z¼10 q 1 1 10 q 2 1 1...10 q k 1 1 This means that the input neuron of P receives a spike in each step corresponding to a digit 1 from the string zand no spike otherwise. Note that we input exactly k+ 1 spikes, i.e., after the last spike we assume that no further spike is coming to the input neuron. With any computation (halting or not) we can associate a spike train, sequence of symbols 0 and 1 describing the behavior of the output neuron: if the output neuron spikes, then we write 1; otherwise we write 0. Besides, we also associate other forms of computation results according to different computing purposes, such as, the distance between two consecutive steps when there are spikes which exit the system. 2.2. FRSN P systems The goal of this paper is to develop a new modeling method for fault diagnosis inspired by SN P systems. As stated above, fault diagnosis can involve such a problem, that is, representation of fuzzy diagnosis knowledge and fuzzy reasoning. However, current forms of existing SN P systems and their variants cannot deal with the problem. In order to make them able to handle fuzzy diagnosis knowledge and fuzzy reasoning, we will extend the definition of SN P systems and propose a new class of extended SN P systems, called fuzzy reasoning spiking neural Psystems (FRSN P systems, in short). Definition 2. A FRSN P system of degree mP1, is a construct of the form P¼ðA; r 1 ;...; r m ;syn;I;OÞ where (1) A={a} is the singleton alphabet (the object ais called spike); (2) r 1 ,..., r m are neurons, of the form r i =( a i , s i ,r i ) with i2{1, ...,m} where (i) a i 2[0,1] and it is called the (potential) value of spike contained in neuron r i (also called pulse value); (ii) s i 2[0,1] is the truth value associated with neuron r i ; (iii) r i is a firing/spiking rule contained in neuron r i , of the form E/a a ?a b , where a ,b2[0,1]. (3) syn #{1,2,...,m}{1,2,...,m} with i–jfor all (i,j)2syn,16i,j6m(synapses between neurons); (4) Iand Oare input neuron set and output neuron set, respectively. Now, we explain how the FRSN P systems are extended from SN P systems. First of all, content of neuron is denoted by a fuzzy truth value in [0,1] instead of the number of spikes in SN P systems, which can be interpreted as the (potential) value of spike from the view point of biological neuron. For a neuron r i ,if a i > 0, we say the neuron contains a spike with (potential) value a i ; otherwise, the neuron contains no spike. Secondly, each neuron in FRSN P systems will associate with either a fuzzy proposition or a fuzzy production rule, and s i 2[0,1] will be used to express the truth value of the fuzzy proposition or confidence factor (CF) of the fuzzy production rule. Thirdly, each neuron contains only one spiking (firing) rule, of the form E/ a a ?a b , where E=a n is called the firing condition and nis the number of input synapses from other neurons to the neuron. The firing condition E=a n indicates that if the neuron receives nspikes the spiking rule can be applied; otherwise the rule cannot be enabled until nspikes are received. When the number of spikes received by a neuron is less than n, value of the spikes received will be updated according to logical ‘‘AND’’ or ‘‘OR’’ operations. Fourthly, the firing mechanism of neurons in FRSN P systems can be described as follows. For the neuron r i , if its firing rule E/a a ?a b can be applied, this means that its pulse value a > 0 is consumed (removed), the neuron fires, and then it produces a spike with value b. Once the spike with value bis excited from neuron r i , all neurons r j with (i,j)2syn will immediately receive the spike. Further, three types of neurons are defined (see Definitions 3–5 below). The three types of neurons use different ways to handle both a and b, and bis relative with both a and s i . Finally, time delay is ignored in FRSN P system, thus all neurons are always open. As stated above, neurons in FRSN P system are classified into three classes: proposition neuron, ‘‘AND’’-type rule neuron and ‘‘OR’’-type rule neuron. These types of neurons are defined as follows. Definition 3. The proposition neurons are a class of neurons, which are associated with propositions in a fuzzy knowledge base. A proposition neuron can be denoted by r =( a , s ,r), where a is its pulse value, s is the truth value of the proposition associated with it, and ris its spiking rule of the form E/a a ?a a . If a proposition neuron r is an input neuron in P , then we have a = s ; otherwise, a equals to logical ‘‘OR’’ operation of all pulse values received from other neurons. After the neuron updates its content, the truth value of the corresponding proposition will be equal to its pulse value, i.e., s = a . When the neuron fires and applies its firing rule, it will produce a spike with pulse value a .Fig. 1 shows a proposition neuron. Definition 4. The ‘‘AND’’-type rule neurons are a class of neurons, which are associated with fuzzy production rules with ‘‘AND’’-type antecedent part, where confidence factor (CF) of each the rule is denoted by s . Fig. 2 shows a ‘‘AND’’-type rule neuron, which is labeled by the symbol ‘‘AND’’. When a ‘‘AND’’-type rule neuron receives n pulse values from other neurons, a 1 , a 2 ,..., a n , it uses logic operator ‘‘AND’’ to combine its all inputs, i.e., a = min ( a 1 , a 2 ,..., a n ), where a i 2[0, 1], 1 6i6n. If its firing condition Eis satisfied, it fires and applies its spiking rule E/a a ?a b to produce a spike with value b= a  s , hence b= min( a 1 , a 2 ,..., a n ) s ; otherwise, it only updates its content by using pulse values of the received pikes. Definition 5. The ‘‘OR’’-type rule neurons are a class of neurons, which are associated with fuzzy production rules with ‘‘OR’’-type antecedent part, where confidence factor (CF) of each the rule is denoted by s . Fig. 3 shows a ‘‘OR’’-type rule neuron, which is labeled by the symbol ‘‘OR’’. When a ‘‘OR’’-type rule neuron receives n pulse values from other neurons, a 1 , a 2 ,..., a n , it uses the logic operator ‘‘OR’’ to combine its all inputs, i.e., a = max ( a 1 , a 2 ,..., a n ), where a i 2[0, 1], 1 6i6n. Similarly, when Eis satisfied, it fires and applies its spiking rule E/a a ?a b to produce a spike with value b= a  s , hence b= max( a 1 , a 2 ,..., a n ) s ; otherwise, it only updates its content by using pulse values of the received pikes. 3. Modeling fuzzy production rules using FRSN P systems 3.1. Fuzzy production rules In fault diagnosis application, diagnosis knowledge extracted from real-world data are usually expressed by fuzzy production rules. To date, fuzzy production rule is a widely used tool for fuzzy knowledge representation (see [4,6,13,19,18,22,30]). Usually, a simple fuzzy production rule is of the form R i :IF p j THEN p k ðCF ¼ s i Þ ð1Þ where R i indicates ith fuzzy production rule of a fuzzy diagnosis knowledge base and s i 2[0,1] is its confidence factor. p j and p k are two propositions and their truth values are real numbers in [0,1]. In addition to above simple fuzzy production rule, composite fuzzy production rules are widely used in fuzzy diagnosis knowledge base, where their antecedent part or consequence part usually contain ‘‘and’’ or ‘‘or’’ connectors. Generally, composite fuzzy production rules can be classified as the following three types. Type 1 R i :IF p 1 and p 2 and ...and p k1 THEN p k ðCF ¼ s i Þð2Þ The type rule is a composite conjunctive fuzzy production rule, where p 1 ,p 2 ,...,p k1 are propositions in the antecedent part of the rule, and s i 2[0, 1] is a real number and it expresses the confidence factor of the rule. Suppose truth values of propositions p 1 ,p 2 ,...,p k1 are a 1 , a 2 ,..., a k1 respectively. Then, truth value of proposition p k can be evaluated as a k = min( a 1 , a 2 ,..., a k1 ) s i . p( ) (a) a a a a (b) p( ) Fig. 1. (a) A proposition neuron Pand (b) its simple form. Type 2 R i :IF p 1 THEN p 2 and p 3 and  and p k ðCF ¼ s i Þð3Þ where s i 2[0,1] is the confidence factor of the rule and p 1 is alone proposition in the antecedent part of the rule. Suppose the truth value of proposition p 1 is a 1 . Then, truth values of propositions p 2 ,p 3 ,...,p k can be evaluated as a 2 = a 1  s i , a 3 = a 1  s i ,..., a k = a 1  s i , respectively. Type 3 R i :IF p 1 or p 2 or  or p k1 THEN p k ðCF ¼ s i Þð4Þ The type rule is a composite disjunctive fuzzy production rule, where p 1 ,p 2 ,...,p k1 are propositions in antecedent part of the rule, and s i 2[0,1] is the confidence factor of the rule. Suppose truth values of propositions p 1 ,p 2 ,...,p k1 are a 1 , a 2 ,..., a k1 , respectively. Then, truth value of proposition p k can be evaluated as a k = max( a 1 , a 2 ,..., a k1 ) s i . 3.2. FRSN P system model for fuzzy production rules In order to model fuzzy production rules of a fuzzy diagnosis knowledge base, we need map them into a FRSN P system model. The basic principle can be described as follows. Each proposition in the fuzzy diagnosis knowledge base is represented by a proposition neuron, while each fuzzy production rule is expressed by a rule neuron (‘‘AND’’-type neuron or ‘‘OR’’-type neuron). At the initial stage, each input proposition neuron contains a spike and its pulse value is assigned to the truth value of the proposition associated with it. Moreover, value s i of each rule neuron is assigned to the confidence factor of the fuzzy production rule associated with it. Consequently, fuzzy production rules of a fuzzy diagnosis knowledge base can be modeled by FRSN P systems, and their dynamic reasoning process can be realized by the firing mechanism of neurons. According to the above principle, simple fuzzy production rule (1) can be modeled by the following FRSN P system P 0 . Fig. 4a shows the model P 0 and its reasoning result. P 0 =(A, r i , r j , r k ,syn,I,O), where (1) A={a} (2) r i is a rule neuron associated with rule R i with confidence factor s i . Its spiking rule is of the form E/a a ?a b , where b= a  s i . (3) r j and r k are two proposition neurons associated with propositions p j and p k with truth values a j and a k respectively. Their spiking rules are of the form E/a a ?a a . (4) syn ={(j,i), (i,k)}, I={ r j }, O={ r k }. A type-1 composite fuzzy production rule (2) can be modeled by the following FRSN P system P 1 .Fig. 4b shows the model P 1 and its reasoning result. P 1 =(A, r 1 , r 2 ,..., r k , r k+1 ,syn,I,O), where a a a AND (a) (b) n a AND min n R( ) R( ) Fig. 2. (a) A ‘‘AND’’-type rule neuron and (b) its simple form. a a a OR (a) (b) n a OR max n R( ) R( ) Fig. 3. (a) A ‘‘OR’’-type rule neuron and (b) its simple form. (1) A={a} (2) r j (j=1,2,...,k) are proposition neurons associated with propositions p j (j=1,2,...,k) with truth values a j (j=1,2,...,k) respectively. Their spiking rules are of the form E/a a ?a a . (3) r k+1 is a ‘‘AND’’-type rule neuron associated with rule R i with confidence factor s i . Its spiking rule is of the form E/ a a ?a b , where b= a  s i . (4) syn = {(1, k+ 1), (2, k+ 1), ...,(k1, k+ 1), (k+1,k)}. (5) I={ r 1 , r 2 ,..., r k1 }, O={ r k }. A type-2 composite fuzzy production rule (3) can be modeled by the following FRSN P system P 2 .Fig. 4c shows the model P 0 and its reasoning result. P 2 =(A, r 1 , r 2 ,..., r k , r k+1 ,syn,I,O), where (1) A={a} (2) r j (j=1,2,...,k) are proposition neurons associated with propositions p j (j=1,2,...,k) with truth values a j (j=1,2,...,k) respectively. Their spiking rules are of the form E/a a ?a a . (3) r k+1 is a rule neuron associated with rule R i with confidence factor s i . Its spiking rule is of the form E/a a ?a b , where b= a  s i . (4) syn = {(1, k+ 1), (k+ 1,2), (k+ 1,3), ...,(k+1,k)}. (5) I={ r 1 }, O={ r 2 , r 3 ,..., r k }. A type-3 composite fuzzy production rule (4) can be modeled by the following FRSN P system P 3 .Fig. 4d shows the model P 0 and its reasoning result. P 3 =(A, r 1 , r 2 ,..., r k , r k+1 ,syn,I,O), where (1) A={a} (2) r j (j=1,2,...,k) are proposition neurons associated with propositions p j (j=1,2,...,k) with truth values a j (j=1,2,...,k) respectively. Their spiking rules are of the form E/a a ?a a . (3) r k+1 is an ‘‘OR’’-type rule neuron associated with rule R i with confidence factor s i . Its spiking rule is of the form E/ a a ?a b , where b= a  s i . (4) syn = {(1, k+ 1), (2, k+ 1), ...,(k1, k+ 1), (k+1,k)}. (5) I={ r 1 , r 2 ,..., r k1 }, O={ r k }. (a) j k pj(j)pk(k) k=j*i i Ri(i) k=min( 1,2, , k-1)* i 2k p2(2) pk(k) Ri(i) 1 p1(1) k-1 pk-1(k-1) AND k+1 (b) 3=1*i 2 k p2(2) p3(3) k+1 Ri(i) 1 p1(1) 3 pk(k) k=1*i 2=1*i (c) k=max( 1,2, , k-1)* i 2k p2(2) pk(k) Ri(i) 1 p1(1) k-1 pk-1(k-1) OR k+1 (d) Fig. 4. Fuzzy production rules modeled by ERSN P systems and their rule reasoning: (a) simple rule; (b) type-1 composite rule; (c) type-2 composite rule; (d) type-3 composite rule. 4. Reasoning algorithm based on FRSN P systems In this section, we will present a fuzzy reasoning algorithm based on FRSN P systems. Suppose all fuzzy production rules in a fuzzy diagnosis knowledge base have been modeled by a FRSN P system model P . The model P consists of mneurons, where r ={ r 1 , r 2 ,..., r n } are its nproposition neurons,  r ¼f  r 1 ; r 2 ;... ; r k gare its krule neurons (‘‘AND’’ type neurons or ‘OR’’ type neurons), m=n+k. Usually, we should provide the fuzzy truth values for a part of the fuzzy propositions before reasoning, and proposition neurons associated with the part of fuzzy propositions are indeed input neurons of P . The goal of the reasoning algorithm is to reason out the fuzzy truth values of other unknown fuzzy propositions (proposition neurons) from known fuzzy propositions (input neurons). These unknown fuzzy propositions are often associated with output neurons of P . The reasoning algorithm developed here is based on neuron’s firing mechanism and uses matrix operation. As a result, The reasoning algorithm can give full play to the advantages of parallel computing of FRSN P systems. In the following, in order to succinctly illustrate the fuzzy reasoning algorithm, we introduce some matrix and vector notations as follows. (1) U=(u ij ) nk is a binary matrix, where u ij 2{0,1}. u ij = 1 if there is a directed arc (synapse) from proposition neuron r i to rule neuron  r j .u ij = 0 if there is no directed arc (synapse) from proposition neuron r i to rule neuron  r j . (2) V=( v ij ) nk is a binary matrix, where v ij 2{0,1}. v ij = 1 if there is a directed arc (synapse) from rule neuron  r j to proposition neuron r i . v ij = 0 if there is no directed arc (synapse) from rule neuron  r j to proposition neuron r i . (3) K = diag( s 1 , s 2 ,..., s k ) is a diagonal matrix, where s i represents confidence factor of ith production rule associated with rule neuron  r i ;16i6k. (4) H 1 = diad(h 1 ,h 2 ,...,h k ) is a diagonal matrix. If the ith rule neuron is an ‘‘AND’’-type neuron, h i = 1; otherwise h i =0. (5) H 2 = diad(h 1 ,h 2 ,...,h k ) is a diagonal matrix. If the ith rule neuron is an ‘‘OR’’-type neuron, h i = 1; otherwise h i =0. (6) a p =( a p1 , a p2 ,..., a pn ) T is a truth value vector, a pi 2[0,1]. a pi represents the truth value of ith proposition neuron. a r =( a r1 , a r2 ,..., a rk ) T is also a truth value vector, a rj 2[0,1]. a rj represents the truth value of jth rule neuron. (7) a p =(a p1 ,a p2 ,...,a pn ) T is an integer vector, where a pi represents the number of spikes received by ith proposition neuron. a r =(a r1 ,a r2 ,...,a rk ) T is also an integer vector, where a rj represents the number of spikes received by jth rule neuron. (8) k p =(k p1 ,k p2 ,...,k pn ) T is an integer vector, where k pi represents the number of spikes required by firing ith proposition neuron. k r =(k r1 ,k r2 ,...,k rk ) T is also an integer vector, where k rj represents the number of spikes required by firing jth rule neuron. (9) b p =(b p1 ,b p2 ,...,b pn ) T is a truth value vector, b pi 2[0,1]. b pi represents truth value exported by ith proposition neuron after firing. b r =(b r1 ,b r2 ,...,b rk ) T is also a truth value vector, b rj 2[0,1]. b rj represents the truth value exported by jth rule neuron after firing. (10) b p =(b p1 ,b p2 ,...,b pn ) T is an integer vector, where b pi 2{0,1} represents the number of spikes exported by ith proposition neuron after firing. b r =(b r1 ,b r2 ,...,b rk ) T is also an integer vector, where b rj 2{0,1} represents the number of spikes exported by jth rule neuron after firing. In addition to above notations, we also introduce the following several operators and functions. (1) :C=AB, where A,Band Care all rsmatrices, such that c ij = max{a ij ,b ij }. (2) :C=AB, where A,Band Care rs,stand rtmatrices respectively, such that c ij ¼max 16r6s fa ir b rj g. (3) :C=AB, where A,Band Care rs,stand rtmatrices respectively, such that c ij = min 16r6s {a ir b rj }. (4) b=fire( a ,a,k), where b=(b 1 ,...,b r ) T , a =( a 1 ,..., a r ) T ,a=(a 1 ,...,a r ) T ,k=(k 1 ,...,k r ) T . The function is defined as follows: b i ¼ a i ;if a i ¼k i 0ifa i <k i (; where i=1,2,... ,r. (5) b=update( a ,a,k), where b, a ,aand kare vectors described above. The function is defined as follows: b i ¼ 0ifa i ¼0 b i þ a i ;if 0 <a i <k i 0ifa i ¼k i 8 > < > : ; where i=1,2,... ,r. (6) D=diag(b), where D=(d ij )isarrdiagonal matrix and b=(b 1 ,...,b r ). For 1 6i6r,d ii =b i , while d ij = 0 for i–j. Then, the developed fuzzy reasoning algorithm based on FRSN P system is described as follows. Fuzzy reasoning algorithm based on FRSN P system. INPUT: parameter matrixes U,V, K ,H 1 ,H 2 ,k p ,k r , and initial inputs a 0 p ;a 0 p . OUTPUT: The fuzzy truth values of propositions associated with the neurons in O. Step 1) Let a 0 r ¼ð0;0;...;0Þ T ;a 0 r ¼ð0;0;...;0Þ T . Step 2) Let t=0. Step 3) (1) Process the firing of proposition neurons. b t p ¼fire a t p ;a t p ;k p  ;b t p ¼fire 1;a t p ;k p  ; a t p ¼update a t p ;a t p ;k p  ; a t p ¼update a t p ;a t p ;k p  ;B t p ¼diag b t p  : (2) Compute the truth values of rule neurons and the number of received spikes. a tþ1 r ¼ a t r H 1 ðB t p UÞ T b t p  þðH 2 B t p U  T b t p  ; a tþ1 r ¼a t r þB t p U  T b t p  : (3) Process the firing of rule neurons. b tþ1 r ¼fire K  a tþ1 r ;a tþ1 r ;k r  ;b tþ1 r ¼fire 1;a tþ1 r ;k r  ; a tþ1 r ¼update a tþ1 r ;a t p ;k p  ;a tþ1 r ¼update a tþ1 r ;a t p ;k p  ;B tþ1 r ¼diag b tþ1 r  : (4) Compute the truth values of proposition neurons and the number of received spikes. a tþ1 p ¼ a t p VB tþ1 r  b tþ1 r hi ;a tþ1 p ¼a t p þVB tþ1 r  b tþ1 r hi : Step 4) If a tþ1 p ¼ð0;0;...;0Þ T and a tþ1 r ¼ð0;0;...;0Þ T (computation halts), the reasoning results are obtained; otherwise, t=t+ 1, go to Step 3). 5. Application to fault diagnosis As discussed above, fuzzy production rules can be modeled by the proposed FRSN P system and the developed reasoning algorithm is a parallel fuzzy reasoning algorithm. In this section, a practical example is used to illustrate application of the proposed method to a transformer fault diagnosis system. The following fuzzy production rules are from knowledge base of a transformer fault diagnosis system. Rule 1(Confidence = 0.8) Symptom: (1) Total hydrocarbon is little high (p 1 ). (2) C 2 H 2 is low (p 2 ). Anticipated Fault: General overheating fault occurs (p 11 ). Rule 2(Confidence = 0.8) (1) Total hydrocarbon is rather high (p 3 ). (2) C 2 H 2 is too high (p 4 ). (3) H 2 is high (p 5 ). (4) C 2 H 2 in total hydrocarbon occupies a too low proportion p 6 Anticipated Fault: Serious overheating fault occurs (p 11 ). Rule 3(Confidence = 0.8) (1) Total hydrocarbon is little low (p 7 ). (2) H 2 is high (p 5 ). (3) CH 4 in total hydrocarbon occupies a large proportion p 8 (4) CH 4 in total hydrocarbon occupies a higher proportion than C 2 H 2 p 9 Anticipated Fault: The partial discharge occurs (p 13 ). Rule 4(Confidence = 0.8) (1) Total hydrocarbon is rather low (p 10 ). (2) C 2 H 2 is too high (p 4 ). (3) H 2 is high (p 5 ). Anticipated Fault: The spark discharge occurs (p 14 ). These fuzzy production rules can be modeled by the following FRSN P system P 4 , shown in Fig. 5. P 4 =(A, r 1 ,..., r 14 , r 15 ,..., r 18 ,syn,I,O), where (1) A={a}. (2) r 1 ,..., r 14 are proposition neurons associated with propositions p 1 ,...,p 14 respectively. (3) r 15 ,..., r 18 are ‘‘AND’’-type rule neurons associated with production rules R 1 ,...,R 4 respectively. (4) syn = {(1,15), (2,15), (3,16), (4,16), (4,18), (5,16), (5,17), (5,18), (6,18), (7,17), (8,17), (9,17), (10,18), (15,11), (16,12), (17,13), (18,14)}. (5) I={ r 1 , r 2 , r 3 , r 4 , r 5 , r 6 , r 7 , r 8 , r 9 , r 10 }, O={ r 11 , r 12 , r 13 , r 14 }. According to the definition of matrices and vectors given above, U,V, K ,H 1 and H 2 are follows: U¼ 11000000000000 00111100000000 00001011100000 00011000010000 2 6 6 6 4 3 7 7 7 5 T H 1 ¼ 1000 0100 0010 0001 2 6 6 6 4 3 7 7 7 5 V¼ 00000000001000 00000000000100 00000000000010 00000000000001 2 6 6 6 4 3 7 7 7 5 T H 2 ¼ 0000 0000 0000 0000 2 6 6 6 4 3 7 7 7 5 K ¼ 0:8000 00:80 0 000:80 0000:8 2 6 6 6 4 3 7 7 7 5 k p ¼ð1;1;1;1;1;1;1;1;1;1;1;1;1;1Þ T k r ¼ð2;4;4;3Þ T In on-scene information detection of transformer, total hydrocarbon content is high (confidence = 0.8), C 2 H 2 content is high (confidence = 0.8), H 2 content is high (confidence = 0.9), C 2 H 2 content in total hydrocarbon content is little (confidence = 0.8), CH 4 content in total hydrocarbon content is little (confidence = 0.1). Thus, initial truth value vector a 0 p ¼ð0:8;0:2;0:8;0:8;0:9;0:8;0:2;0:9;0:1;0:2;0;0;0;0Þ T and initial spike vector a 0 p ¼ð1;1;1;1;1;1;1;1;1;1;0;0;0;0Þ T . Let a 0 r ¼ð0;0;0;0Þ T and a 0 r ¼ð0;0;0;0Þ T . According to reasoning algorithm described above, we have (1) a 1 p ¼ð0;0;0;0;0;0;0;0;0;0;0;0;0;0Þ T ; a 1 r ¼ð0:16;0:64;0:08;0:16Þ T , a 1 p ¼ð0;0;0;0;0;0;0;0;0;0;0;0;0;0Þ T ;a 1 r ¼ð2;4;4;3Þ T . a 1 p1(0.8) a 2 p6(0.8) 11 R1(0.8) p2( ) p11 15 a p3(0.8) a 6 12 R2(0.8) 16 a 4 p4(0.8) a 5 p5(0.9) p12 a p7(0.2) 13 R3(0.8) 17 a 8 p8(0.9) a 9 p9(0.1) p13 14 R4(0.8) p14 18 p10(0.2) a 10 Fig. 5. An example of a transformer fault diagnosis modeled by the FRSN P system model P 4 .