Semantics of deductive databases with spiking neural P systems

Semantics of deductive databases with spiking neural P systems Daniel Díaz-Pernil a, Miguel A. Gutiérrez-Naranjo b, ∗ a CATAM Research Group - Department of Applied Mathematics I, University of Seville, Spain b Department of Computer Science and Artificial Intelligence, University of Seville, Spain Keywords: Membrane computing P systems Neural-symbolic integration a b s t r a c t The integration of symbolic reasoning systems based on logic and connectionist systems based on the functioning of living neurons is a vivid research area in computer science. In the literature, one can find many efforts where different reasoning systems based on different logics are linked to classic artificial neural networks. In this paper, we study the relation between the semantics of reasoning systems based on propositional logic and the connectionist model in the framework of membrane computing, namely, spiking neural P systems. We prove that the fixed point semantics of deductive databases without negation can be implemented in the spiking neural P systems model and such a model can also deal with negation if it is endowed with anti-spikes and annihilation rules. 1. Introduction Searching new ways for automating reasoning is one of the engines in computer science. In the last decades, the two main research areas devoted to this aim, namely, connectionist systems ans logic-based systems, have explored very different techniques, approaches and representation systems. On the one hand, logic-based systems are based on a symbolic representation of the knowledge and derivation rules which allow to obtain new formulae from previous ones. Many effort s have done in order to prove the correctness and soundness of the rules depending on the expressiveness of the language and the chosen semantics. On the other hand, the traditional connectionist systems used to handle automatic reasoning are based on artificial neural nets inspired in the network of biological neurons in a human brain. These nets have been also widely studied and provide a point of view where the data is encoded in real numbers and the synapses between neurons determine the flow of information. The integration of both paradigms is a vivid area in artificial intelligence (see, e.g., [1–3] ). In the framework of membrane computing, several studies have been presented where P systems are used for representing logic-based information and performing reasoning by the application of bio-inspired rules (see [4,5] ). These papers study approaches based on cell-like models, as P systems with active membranes, and deal with procedural aspects ∗Corresponding author. E-mail addresses: [email protected] (D. Díaz-Pernil), [email protected] (M.A. Gutiérrez- Naranjo). of the computation. The approach in this paper is different in both senses. On the one hand, the connectionist model of P systems is considered, i.e, the model of P system inspired by the neurophysiological behavior of neurons sending electrical impulses along axons to other neurons (the so-called spiking neural P systems or SN P systems for short). On the second hand, we consider the semantics of propositional deductive databases in order to show how SN P systems can deal with logic-based representing and reasoning systems. One of the key features of the integrate-and-fire formal spiking neuron models [6] (and, in particular, of the SN P systems) is the use of the spikes as support of the information. Such spikes are short electrical pulses (also called action potentials) between neurons and can be observed by placing a fine electrode close to the soma or axon of a neuron. From the theoretical side, it is crucial to consider that all the biological spikes of an alive biological neuron look alike. This means that we can consider a bio-inspired binary code which can be used to formalize logic-based semantics: the emission of one spike will be interpreted as true and the absence of spikes will be interpreted as false . As we will show below, SN P systems provide a natural way for dealing with this binary behavior in a connectionist model. SN P systems suffice for dealing with the semantics of propositional logic systems which do not use negation. The semantics of reasoning systems with negated information, even in the propositional case, needs to add new elements for dealing with the difference between negated information and absence of information (see [7] ). The literature of SN P system provides an efficient tool for dealing with such negated information: the use of anti-spikes and annihilation rules. SN P systems with anti-spikes were presented in [8] as a formalization of the idea of inhibiting in some way the communication between neurons. Such extended model has been widely studied (see, e.g., [9–11] ) and it has inspired the use of negative information (see, e.g., [12–14] ). Recently, the study of SN P systems has been extended with different features, see e.g., SN P systems with thresholds [15] , SN P systems with request rules [16] , SN P systems with structural plasticity [17] or cell-like SN P systems [18] . Among the recent contributions, the approaches shown in [19–21] deserve to be cited. In such papers, the interaction between reasoning systems and membrane computing is also studied. Instead of dealing with propositional logic, the authors consider fuzzy logic and the applications focus on fuzzy representation of the knowledge and fault diagnosis. The main result of this paper 1 is to prove that given a reasoning system based on propositional logic it is possible to find an SN P system with the same declarative semantics. We prove it in both cases: when the reasoning system involves negation and when it does not. A declarative semantics for a rule-based propositional system is usually given by selecting models which satisfy certain properties. This choice is often described by an operator mapping interpretations to interpretations. In this paper we consider the so-called immediate consequence operator due to van Emden and Kowalski [22] . It is well-known that for a rule based system without negation KB , such operator is order continuous and its least fix point coincides with the least model of KB . We adapt the definition of the immediate consequence operator to a restricted form of SN P system and we prove that a least fix point, and hence a least model is obtained for the given reasoning system. The monotonicity is lost if negation is allowed, but even in this case, the immediate consequence operator can be computed with membrane computing techniques. The paper is organized as follows: firstly, we recall some aspects about SN P systems and the semantics of deductive databases. In Section 3 we prove that standard SN P systems can deal with the semantics of deductive databases if they do not involve negation. In Section 4 it is shown that endowing SN P systems with anti-spikes and annihilation, then such devices can deal with the semantics of deductive databases with negation. Finally, some conclusions and topics for further discussion are provided in the last section. 2. Preliminaries We assume the reader to be familiar with basic elements about membrane computing and the semantics of rule-based systems. Next, we briefly recall some definitions. We refer to [23] for a comprehensive presentation of the former and [7,24,25] for the latter. 2.1. Spiking neural P systems SN P systems were introduced in [26] with the aim of incorporating membrane computing ideas with spike-based neuron models. It is a class of distributed and parallel computing devices, inspired by the neurophysiological behavior of neurons sending electrical impulses ( spikes ) along axons to other neurons. In SN P systems the cells (also called neurons ) are placed in the nodes of a directed graph, called the synapse graph . The contents of each neuron consist of a number of copies of a single object type, called the spike . Every cell may also contain a number of firing and forgetting rules. Firing rules allow a neuron to send information to other neurons in the form of spikes which are 1 A preliminary version of this paper appeared in the Proceedings of the 14th Brainstorming Week on Membrane Computing (BWMC2016). accumulated at the target cell. The applicability of each rule is determined by checking the contents of the neuron against a regular set associated with the rule. In each time unit, if a neuron can use one of its rules, then one of such rules must be used. If two or more rules could be applied, then only one of them is non-deterministically chosen. Thus, the rules are used in the sequential manner in each neuron, but neurons function in parallel with each other. As usually happens in membrane computing, a global clock is assumed, marking the time for the whole system, and hence the functioning of the system is synchronized. Formally, an SN P system of the degree m ≥1 is a construct 2 = (O, σ1 , σ2 , . . . , σm , syn ) where: 1. O = { a } is the singleton alphabet ( a is called spike ); 2. σ1 , σ2 , . . . , σm are neurons , of the form σi = (n i , R i ) , 1 ≤i ≤m , where: (a) n i ≥0 is the initial number of spikes contained in σi ; (b) R i is a finite set of rules of the following two forms: (1) firing rules E / a p → a q , where E is a regular expression over a and p, q ≥1 are integer numbers 3 ; (2) forgetting rules a s → λ, with s an integer number such that s ≥1; 3. syn ⊆{ 1 , 2 , . . . , m } ×{ 1 , 2 , . . . , m } , with (i, i ) ∈ syn for 1 ≤i ≤m , is the directed graph of synapses between neurons. Rules of type (1) are firing rules, and they are applied as follows. If the neuron σi contains k spikes, k ≥p , and a k belongs to the language L ( E ) associated to the regular expression E , then the rule E / a p → a q can be applied. The application of this rule means removing p spikes (thus only k −premain in σi ), the neuron is fired, and it produces q spikes which are sent immediately to all neurons σj such that ( i, j ) ∈ syn . The rules of type (2) are forgetting rules and they are applied as follows: if the neuron σi contains exactly s spikes, then the rule a s → λfrom R i can be used, meaning that all s spikes are removed from σi . If a rule E / a p → a q of type (1) has E = a p , then we will write it in the simplified form a p → a q . In each time unit, if a neuron σi can use one of its rules, then a rule from R i must be used. Let us remark that it is possible that two or more rules can be applied in a neuron, and in that case only one of them is non-deterministically chosen regardless its type.. The j th configuration of the system is described by a vector C j = (t 1 , ... , t m ) where t k represents the number of spikes at the neuron σk in such configuration. The initial configuration is C 0 = (n 1 , n 2 , . . . , n m ) . Using the rules as described above, one can define transitions among configurations. Any sequence of transitions starting in the initial configuration is called a computation. A computation halts if it reaches a configuration where no rule can be used. Generally, a computation may not halt. If it halts, the last configuration is called a halting configuration. A useful extension to the model presented above was introduced by adding anti-spikes as a formalization of the idea of inhibiting in some way the communication between neurons (see [8] ). This extension leads to the definition of SN P systems with anti-spikes . In this extension a further object, a , is added to the alphabet O , and the spiking and forgetting rules are of the forms E / b p → b  q and b p → λwhere E is a regular expression over a or over a , while b, b  ∈ { a, a } and p, q ≥1. As above, if L (E) = b p , then we write the first rule as b p → b  q . The rules are used as in a usual SN P system, with the additional fact that a and a cannot stay 2 We provide a definition without delays, input or output neurons because these features are not used in this paper. 3 In the general case, the restriction p ≥q is imposed. In this paper, we will also consider rules E / a → a 2 . A more complex design of our solutions satisfying p ≥q is possible, but we prefer to include rules E / a → a 2 for the sake of a simpler design. together: if in a neuron there are either objects a or objects a , and further objects of either type (maybe both) arrive from other neurons, such that we end with a r and a s inside, then immediately an annihilating rule of the form a a → λis applied in a maximal manner, so that either a r−s s or a s −r remain, provided that r ≥s or s ≥r , respectively. The mutual annihilation of spikes and antispikes takes no time, so that the neuron always contains either only spikes or anti-spikes. The j th configuration of the system is described by a vector C j = (t 1 , . . . , t m ) . If t i ≥0, it denotes that the neuron σi has t i spikes at the configuration C j . If t i = −s i with s i ≥1, it denotes that the neuron σi has s i anti-spikes at the configuration C j . 2.2. Semantics of rule-based deductive databases Given two pieces of knowledge V and W , expressed in some language, the rule V → W is usually considered as a causal relation between V and W . In this paper, we only consider propositional logic for representing the knowledge. Given a set of propositional variables { p 1 , . . . , p n } , a literal is a variable or a negated variable and a rule is a formula L 1 ∧ ···∧ L n → A, where n ≥0, A is a variable and L 1 , . . . , L n are literals. The variable A is called the head of the rule and the conjunction of literals L 1 ∧ ···∧ L n is the body of the rule. If L i = p i , it is said that p i occurs positively in the body of the rule. If L i = ¬ p i , it is said that p i occurs negatively in the body of the rule. If n = 0 , it is said that the body of the rule is empty. If there do not exist negated variables in the body of a rule, the rule is called definite . A finite set of rules KB is a deductive database and it is said that it is a definite deductive database if all the rules in KB are definite. An interpretation I is a mapping from the set of variables { p 1 , . . . , p n } to the set {0, 1}. As usual, we will represent an interpretation I as a vector (i 1 , . . . , i n ) with I(p k ) = i k ∈ { 0 , 1 } for k ∈ { 1 , . . . , n } . The set of all the possible interpretations for a set of n variables will be denoted by 2 n . Given two interpretations I 1 and I 2 , I 1 ⊆I 2 if for all k ∈ { 1 , . . . , n } , I 1 (p k ) = 1 implies I 2 (p k ) = 1 . We will denote by I ∅ the interpretation that maps to 0 every variable, I ∅ = (0 , . . . , 0) . The interpretation I is extended in the usual way, I(¬ p i ) = 1 −I(p i ) for a variable p i ; I(L 1 ∧ ···∧ L n ) = min { I(L 1 ) , ... , I(L n ) } and for a rule 4 I(L 1 ∧ ···∧ L n → A ) = 0 if I(L 1 ∧ ···∧ L n ) = 1 and I(A ) = 0 1 otherwise An interpretation I is a model for a deductive database KB if I(R ) = 1 for all R ∈ KB . Next, we recall the propositional version of the immediate consequence operator which was introduced by van Emden and Kowalski [22] . Definition 1. Let KB be a deductive database on a set of variables { p 1 , . . . , p n } . The immediate consequence operator of KB is the mapping T KB : 2 n → 2 n such that for all interpretation I, T KB ( I ) is an interpretation T KB (I) : { p 1 , . . . , p n } → { 0 , 1 } such that, for k ∈ { 1 , . . . , n } , T KB (I)(p k ) = 1 if there exists a rule L 1 ∧ ···∧ L n → p k in KB such that I(L 1 ∧ ···∧ L n ) = 1 ; otherwise, T KB (I)(p k ) = 0 . The importance of the immediate consequence operator is shown in the following theorem (see [27] ). Theorem 1. An interpretation I is a model of KB if and only if T KB ( I ) ⊆I. Since the image of an interpretation is an interpretation, the immediate consequence operator can be iteratively applied. 4 Let us remark that, from the definition, if n = 0 , I(L 1 ∧ ···∧ L n ) = 1 and, hence, for a rule with an empty body, we have I(→ A ) = 1 if and only if I(A ) = 1 . Definition 2. Let KB be a deductive database and T KB its immediate consequence operator. The mapping T KB ↑ : N → 2 n is defined as follows: T KB ↑ 0 = I ∅ and T KB ↑ n = T KB ↑ (T KB ↑ (n −1)) if n > 0. In the limit, it is also considered T KB ↑ ω = k ≥0 T KB ↑ k The next theorem is a well-known result which relates the immediate consequence operator of a definite deductive database with its least model of a definite deductive database (see [25] ). Theorem 2. Let KB be a definite deductive database. The following result hold •T KB ↑ ω is a model of KB •If I is a model of KB, then T KB ↑ ω ⊆I Example 1. Let us consider the following deductive database KB on the set of variables = { p 1 , p 2 , p 3 , p 4 } R 1 ≡→ p 1 R 2 ≡p 1 → p 2 R 3 ≡p 1 ∧ p 2 → p 3 R 4 ≡p 3 → p 4 R 5 ≡p 2 → p 4 and let us consider the interpretation I : → {0, 1} such that I(p 1 ) = 1 , I(p 2 ) = 0 , I(p 3 ) = 0 and I(p 4 ) = 0 . Such interpretation can be represented as I = (1 , 0 , 0 , 0) . The truth assignment of this interpretation to the rules is I(R 1 ) = 1 , I(R 2 ) = 0 , I(R 3 ) = 1 , I(R 4 ) = 1 , I(R 5 ) = 1 . Since I(R 2 ) = 0 , the interpretation I is not a model for KB . The application of the immediate consequence operator produces T KB (I) = (1 , 1 , 0 , 0) . We observe that T KB (I) ⊆ Iand hence, by Theorem 1 , we can also conclude that I is not a model for KB . Finally, if we consider I ∅ = (0 , 0 , 0 , 0) , the following interpretations are obtained by the iterative application of the immediate consequence operator T KB ↑ 0 = I∅ = (0 , 0 , 0 , 0) T KB ↑ 1 = T KB (T KB ↑ 0) = (1 , 0 , 0 , 0) T KB ↑ 2 = T KB (T KB ↑ 1) = (1 , 1 , 0 , 0) T KB ↑ 3 = T KB (T KB ↑ 2) = (1 , 1 , 1 , 1) In this case T KB ↑ 3 is a fix point for the immediate consequence operator and a model for the definite deductive database KB . 3. Semantics of deductive databases with SN P systems The semantics of deductive databases deals with interpretations, i.e., with mappings from the set of variables into the set {0, 1} (which stand for false and true ) and try to characterize which of these interpretations make true a whole deductive database which, from the practical side, may contain hundreds of variables and thousand of rules. The immediate consequence operator provides a tool for dealing with this problem and provides a way to characterize such models. In this section we will explore how this problem can be studied in the framework of SN P systems and prove that the immediate consequence operator can be implemented in this model and therefore, membrane computing provides a new theoretical framework for dealing with the semantics of deductive databases. Our main result for definite deductive databases claims that SN P systems can compute the immediate consequence operator and hence, its least model. Theorem 3. Given a definite deductive database KB and an interpretation I, an SN P system can be constructed such that (a) It computes the immediate consequence operator T KB ( I ) . (b) It computes the least model for KB in a finite number of steps. Proof. Let us consider a deductive database KB , let { p 1 . . . , p n } be the propositional variables and { r 1 , . . . , r k } be the rules of KB . Given a variable p i , we will denote by h i the number of rules which have p i in the head and given a rule r j , we will denote by b j the number of variables in its body. The SN P systems of degree 2 n + k + 3 KB = (O, σ1 , σ2 , . . . , σ2 n + k +2 , syn ) can be constructed as follows: •O = { a } ; •σj = (0 , { a → λ} ) for j ∈ { 1 , . . . n } •σn + j= (i j , R j ) , j ∈ { 1 , . . . , n } , where i j = I(p j ) and R j is the set of h j rules R j = { a k → a | k ∈ { 1 , . . . . . . , h j }} •σ2 n + j= (0 , R j ) , j ∈ { 1 , . . . . . . , k } , where R j is one of the following set of rules •R j = { a b j → a } ∪ { a l → λ| l ∈ { 1 , . . . , b j −1 } } if b j > 0 •R j = { a → a } if b j = 0 . For a better understanding, the neurons σ2 n + k +1 and σ2 n + k +2 will be denoted by σG and σT , respectively. •σG = (0 , { a → a } ) •σT = (1 , { a → a } ) •syn = { (n + i, i ) | i ∈ { 1 , . . . , n }} ∪ (n + i, 2 n + j) | i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and p i is a variable in the body of r j  ∪ (2 n + j, n + i ) | i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and p i is the variable in the head of r j  ∪ {( G, T ), ( T, G )} ∪ (T , 2 n + j) | j ∈ { 1 , . . . , k } and r j is a rule with empty body   Before going on with the proof, let us note that the construction of this SN P system is illustrated in the Example 2 . The next remarks will be useful: Remark 1. For all t ≥0, in the 2 t th configuration C 2 t the neuron σT contains exactly one spike and the neuron σG does not contain spikes. Proof. In the initial configuration C 0 , σT contains 1 spike and σG does not contain spikes. By induction, let us suppose that in the C 2 t the neuron σT contains exactly one spike and σG does not contain spikes. Since the unique incoming synapse in σT comes from σG and the unique incoming synapse in σG comes from σT and in both neurons occurs the rule a → a , then in C 2 t+1 the neuron σG contains exactly spike and σT does not contain spikes and finally, in C 2 t+2 the neuron σT contains exactly spike and σG does not contain spikes.  Remark 2. For all t ≥0 the following results hold: •For all p ∈ { 1 , . . . , k } the neuron σ2 n + p does not contain spikes in the configuration C 2 t •For all q ∈ { 1 , . . . , n } , the neuron σn + q does not contain spikes in the configuration C 2 t+1 Proof. In the initial configuration C 0 , for all p ∈ { 1 , . . . , k } , the neuron σ2 n + p does not contain spikes and each neuron σn + q contain, at most, one spike. Such spike is consumed by the application of the rule a → a and, since all the neurons with synapse to σn + q do not contain spikes at C 0 , we conclude that at the configuration C 1 , the neurons σn + q do not contain spikes. By induction, let us suppose that in C 2 t , for all p ∈ { 1 , . . . , k } , the neuron σ2 n + p does not contain spikes and for all q ∈ { 1 , . . . , n } , the neuron σn + q does not contain spikes in the configuration C 2 t+1 . According to the construction, the number of incoming synapses in each neuron σ2 n + j is b j if b j > 1 and 1 if b j = 0 . Such synapses come from neurons that send (at most) one spike in each computational step, so in C 2 t+1 , the number of spikes in the neuron σ2 n + j is, at most, b j if b j > 1 and 1 if b j = 0 . All these spikes are consumed by the corresponding rules. Moreover, at C 2 t+1 , all the neurons with outgoing synapses to σ2 n + p do not contain spikes, so we conclude that at C 2 t+2 , for all j ∈ { 1 , . . . , k } , the neuron σ2 n + j does not contain spikes. We focus now on the neurons σn + q with q ∈ { 1 , . . . , n } . By induction, we assume that they do not contain spikes in the configuration C 2 t+1 . Each neuron σn + q can receive at most h q , since there are h q incoming synapses and the corresponding neuron sends, at most, one spike. Hence, at C 2 t+2 , σn + q has, at most, h q spikes. All of them are consumed by the corresponding rule and, since all the neurons which can send spikes to σn + q do not contain spikes at C 2 t+2 , we conclude that, for all q ∈ { 1 , . . . , n } , the neuron σn + q does not contain spikes in the configuration C 2 t+3 .  Remark 3. For all q ∈ { 1 , . . . , n } , the neuron σq does not contain spikes in the configuration C 2 t . Proof. The result holds in the initial configuration. For C 2 t with t > 0 it suffices to check that, as claimed in Remark 2, for all q ∈ { 1 , . . . , n } , the neuron σn + q does not contain spikes in the configuration C 2 t+1 and each σq receives at most one spike in each computation step from the corresponding σn + q . Therefore, in each configuration C 2 t+1 , each neuron σq contains, at most, one spike. Since such spike is consumed by the rule a → λand no new spike arrives, then the neuron σq does not contain spikes in the configuration C 2 t . Before going on with the proof, it is necessary to formalize what means that the SN P system computes the immediate consequence operator T KB . Given a deductive database KB on a set of variables { p 1 , . . . , p n } , an interpretation on KB can be represented as a vector I = (i 1 , ... , i n ) with i j ∈ {0, 1} for j ∈ { 1 , . . . , n } . Let us consider that such values i j ∈ {0, 1} represent the number of spikes placed in the corresponding neuron σn + j at the initial 5 configuration C 0 . We will consider that the computed output for such interpretation is encoded in the number of spikes in the neurons σ1 , . . . , σn in the configuration C 3 . The main results of the theorem can be obtained from the following technical remark.  Remark 4. Let I = (i 1 , . . . , i n ) be an interpretation for KB and let S = (s 1 , ... , s n ) be a vector with the following properties. For all j ∈ { 1 , . . . , n } •If i j = 0 , then s j = 0 . •If i j  = 0 , then s j ∈ { 1 , . . . , h j } Let us suppose that at the configuration C 2 t the neuron σn + j contains exactly s j spikes. Then, the interpretation obtained by applying the immediate consequence operator T KB to the interpretation I, T KB ( I ) is (q 1 , . . . , q n ) where q j , j ∈ { 1 , . . . , n } , is the number of spikes of the neuron σj in the configuration C 2 t+3 . Proof. Firstly, let us consider k ∈ { 1 , . . . , n } and T KB (I)(p k ) = 1 . Let us prove that at the configuration C 2 t+3 there is exactly one spike in the neuron σk . If T KB (I)(p k ) = 1 , then there exists at least one rule r l ≡L d 1 ∧ ···∧ L d l →p k in KB such that I(L d 1 ∧ ···∧ L d l ) = 1 .  5 With a more complex design of the SN P system, it may be considered that these neurons do not contain spikes at the initial configuration and the vector I = (i 1 , ... , i n ) is provided as a spike train via an input neuron, but in this paper we have chosen a simpler design and focus on the computation of the immediate consequence operator. An analogous comment fits for the computed output. Case 1: Let us consider that there is only one such rule r l and the body of r l is empty. By construction, the neuron σ2 n + l has only one incoming synapse from neuron σT ; the neuron σn + j contains exactly s j spikes, j ∈ { 1 , . . . , n } and s j ∈ { 1 , . . . , h j } ; and according to the previous remarks: •In C 2 t the neuron σT contains exactly one spike. •For all p ∈ { 1 , . . . , k } the neuron σ2 n + p does not contain spikes in the configuration C 2 t •For all q ∈ { 1 , . . . , n } , the neuron σq does not contain spikes in the configuration C 2 t . In these conditions, the corresponding rules in σT and σn + k are fired and in C 2 t+1 , the neuron σ2 n + k contains one spike. In C 2 t+2 , the neuron σn + k contains one spike and σk does not contain spikes. Finally, in the next step σn + k sends one spike to σk , so, in C 2 t+3 , σk contain one spike. Case 2: Let us now consider that there exists r l ≡L d 1 ∧ . . . L d l → p k in KB such that I(L d 1 ∧ ···∧ L d l ) = 1 and d l > 0. We suppose that I(L d 1 ∧ ···∧ L d l ) = 1 and, since KB is definite, this means that I(L d 1 ) = ···= I(L d l ) = 1 and therefore, in C 2 t , the neuron σn + d j contains s d j spikes, with s d j ∈ { 1 , . . . , h d j } . All these neurons fire the corresponding rule, and σ2 n + k has at C 2 t+1 exactly b k spikes (since all the incoming synapses send the corresponding spike). The rule a b k → a is fired and in C 2 t+2 the neuron σn + k contains at least one spike. It may have more spikes depending on the existence of other rules with p k in the head, but in any case, the number of spikes is between 1 and h k . The corresponding rule fires and the neuron σk contains one spike in C 2 t+3 . Finally, we prove the statements claimed by the theorem: (a) The SN P system computes the immediate consequence operator T KB ( I ). Proof. It is directly obtained from Remark 4 . Let us note that one of the possible vectors S = (s 1 , . . . , s n ) obtained from the interpretation I is exactly the same interpretation I = (i 1 , . . . , i n ) . If we also consider the case when t = 0 , we have proved that from the initial configuration C 0 where i k represents the number of spikes in the neuron σn + k , then the configuration C 3 encodes T KB ( I ).  (b) The SN P system computes the least model for KB in a finite number of steps. Proof. Let us consider the empty interpretation as the initial one, i.e., T KB ↑ 0 = I ∅ . We will prove that (∀ z ≥1) T KB ↑z =C 2 z+1 [1 , ... , n ] where C 2 z+1 [1 , . . . , n ] is the vector whose components are the spikes on the neurons σ1 , . . . , σn in the configuration C 2 z+1 . We will prove it by induction.  For z = 1 , we have to prove that T KB ↑ 1 = T KB (T KB ↑ 0) = T KB (I ∅ ) is the vector whose components are the spikes on the neurons σ1 , . . . , σn in the configuration C 3 . The result holds from Remark 4 and it has been proved in the statement (a) of the theorem. By induction, let us consider now that T KB ↑ z = C 2 z+1 [1 , . . . , n ] holds. As previously stated, this means that in the previous configuration C 2 z the spikes in the neurons σn +1 , . . . , σ2 n can be represented as a vector S = (s 1 , ... , s n ) be a vector with the properties claimed in Remark 4 , namely, if the neuron σj has no spikes in C 2 z+1 , then s j = 0 and, if the neuron σj has spikes in C 2 z+1 , then s j ∈ { 1 , . . . , h j } . Hence, according to Remark 4 , three computational steps after C 2 z , T KB (C 2 z+1 [1 , . . . , n ]) is computed T KB ↑ z + 1 = T KB (T KB ↑ z) = T KB (C 2 z+1 [1 , . . . , n ]) = C 2 z+3 [1 , . . . , n ] Finally, it is well-known that for a definite database KB , T KB ↑ z ⊆T KB ↑ z + 1 and, since the KB has a finite number of variables and a finite number of rules, then there exist n ∈ N such that T KB ↑ n ⊆T KB ↑ ω and hence, T KB ↑ n is a model for KB .  T G 1 2 3 4 5 6 7 8 9 10 11 12 13 Fig. 1. Graphical representation of the synapses of the SN P system of Example 1 . Example 2. Let us consider the deductive database from Example 1 . The SN P system associated with this KB and the interpretation I ∅ is =(O, σ1 , σ2 , . . . , σ13 , σG , σT , syn ) and its graphical representation is shown in Fig. 1 . In such SN P system O = { a } , σ1 = (0 , { r 1 , 1 ≡a → a } )σ8 = 0 , r 8 , 1 ≡a → a r 8 , 2 ≡a 2 → a  σ2 = (0 , { r 2 , 1 ≡a → a } )σ9 = (0 , { r 9 , 1 ≡a → a } ) σ3 = (0 , { r 3 , 1 ≡a → a } )σ10 = (0 , { r 10 , 1 ≡a → a } ) σ4 = (0 , { r 4 , 1 ≡a → a } )σ11 = 0 , r 11 , 1 ≡a → λ r 11 , 2 ≡a 2 → a  σ5 = (0 , { r 5 , 1 ≡a → a } )σ12 = (0 , { r 12 , 1 ≡a → a } ) σ6 = (0 , { r 6 , 1 ≡a → a } )σ13 = (0 , { r 13 , 1 ≡a → a } ) σ7 = (0 , { r 7 , 1 ≡a → a } ) σG = (0 , { r G, 1 ≡a → a } and σT = (0 , { r T, 1 ≡a → a } with the synapses syn =  (5 , 1) , (6 , 2) , (7 , 3) , (8 , 4) , (5 , 10) , (5 , 11 ) , (6 , 11) , (6 , 13) , (7 , 12) , (9 , 5) , (10 , 6) , (11 , 7) , (12 , 8) , (13 , 8) , (G, T ) , (T , G ) , (T , 9)  The first steps of the computation are shown in Table 1 . Let us remark that T KB ↑ 0 = C 1 [1 , . . . , 4] = (0 , 0 , 0 , 0) T KB ↑ 1 = C 3 [1 , . . . , 4] = (1 , 0 , 0 , 0) T KB ↑ 2 = C 5 [1 , . . . , 4] = (1 , 1 , 0 , 0) T KB ↑ 3 = C 7 [1 , . . . , 4] = (1 , 1 , 1 , 1) 4. Dealing with negation The use of negation in reasoning systems is a complex task (see [28] ) and a detailed discussion is out of the scope of this paper. In order to study the semantics of deductive databases with negation in the body of the rules, we will use the immediate consequence operator as defined in Def. 1 . In this case, Theorem 1 still holds, but the immediate consequence operator is not monotonic (i.e., I 1 ⊆I 2 does not implies T KB ( I 1 ) ⊆T KB ( I 2 )), as the following example illustrates 6 . Example 3. Let us consider a set of variables V = { p 1 , p 2 , p 3 , p 4 } and the deductive database KB = { R 1 , R 2 , R 3 , R 4 } with R 1 ≡→ p1 R 2 ≡p 1 ∧ ¬ p 2 → p 3 6 In this section we will use −1 (instead of 0) to denote the value false . In this way, an interpretation will be represented by a vector I = (i 1 , . . . , i n ) with i j ∈ { 1 , −1 } . Table 1 First steps in the computation of the SNPS in Example 2 . R 3 ≡¬ p 3 ∧ ¬ p 4 → p 3 R 4 ≡¬ p 1 → p 4 Let us consider the interpretation T KB ↑ 0 = I ∅ = (−1 , −1 , −1 , −1) , then T KB ↑ 1 = T KB (T KB ↑ 0) = (1 , 1 , −1 , 1) T KB ↑ 2 = T KB (T KB ↑ 1) = (1 , −1 , −1 , −1) T KB ↑ 3 = T KB (T KB ↑ 2) = (1 , 1 , 1 , −1) T KB ↑ 4 = T KB (T KB ↑ 3) = (1 , −1 , −1 , −1) For k ≥1, •T KB ↑ 2 k = T KB ↑ 2 k + 2 •T KB ↑ 2 k + 1 = T KB ↑ 2 k + 3 . As we will show below, the use of anti-spikes allows to deal with negation. Theorem 4 claims that the immediate consequence operator can be implemented with SN P systems with anti-spikes. In such way, membrane computing also provides an efficient tool for handling with the semantics of deductive databases even if they include negative information. Theorem 4. For each deductive database KB on a set of propositional variables { p 1 , −, p n } and an interpretation I = (i 1 , ... , i n ) an SN P system with anti-spikes can be constructed such that •It computes the immediate consequence operator T KB ( I ) . •If I is the empty interpretation, it computes iteratively T KB ↑ k for k ≥0 . Proof. Let us consider a deductive database KB and let { p 1 , . . . , p n } be the propositional variables and { r 1 , . . . , r k } be the rules of the KB . Given a variable p i , we will denote as h i the number of rules which have p i in the head and given a rule r j , we will denote as b j the number of literals in its body. We will also denote by R 0 the number of rules with empty body. Let us define b KB = R 0 + j= k  j=1 b j The SN P systems of degree 2 n + k + 3 + b KB KB = (O, σ1 , σ2 , . . . , σ2 n + k +3+ b KB , syn ) can be constructed as follows: •O = { a } ; •σj = (0 , { a → λ, a → λ} ) for j ∈ { 1 , . . . , n } • σn + j = (i j , R j ) , j ∈ { 1 , . . . , n } , where R j is the set of rules R j = { a → a } ∪ { a k → a | k ∈ { 1 , . . . , 2 ×h j −1 }} •σ2 n + j= (0 , R j ) , j ∈ { 1 , . . . , k } , where R j is the following rules •R j = { a b j → a 2 } ∪ { a l → λ| l ∈ { 1 , . . . , b j −1 } } if b j > 0 •R j = { a → a 2 } if b j = 0 . For a better understanding, the neurons σ2 n + k +1 , σ2 n + k +2 and σ2 n + k +3 will be denoted by σT , σG and σH . •σG = (0 , { a → a } ) •σT = (−1 , { a → a } ) •σH = (0 , { a → a } ) As it will be shown below, the neurons σ2 n + k +3+1 , ... , σ2 n + k +3+ b KB have only an incoming synapse from a neuron σn + i , with i ∈ { 1 , . . . , n } (or σT ), and only one outgoing synapse to a neuron σ2 n + j , with j ∈ { 1 , . . . , k } . For a better understanding, we will denote by σi, j the neuron from σ2 n + k +3+1 , . . . , σ2 n + k +3+ b KB which has a synapse from a neuron σn + i and an outgoing synapse to a neuron σ2 n + j . The label of such neurons will be written as  i, j  and it will be said that such neurons are double-labelled . •σT,j = (0 , { a → a } ) for j ∈ { 1 , . . . , k } and r j is a rule with empty body. •σi,j = (0 , { a → a, a → a } ) for i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and the variable p i occurs positively in the body of the rule r j . •σi,j = (0 , { a → a , a → a } ) for i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and the variable p i occurs negatively in the body of the rule r j . Finally, the synapses of the SN P systems are •syn = { (n + i, i ) | i ∈ { 1 , . . . , n }} ∪ {( T, G ), ( G, H ), ( H, T )} ∪ { (H, n + i ) | i ∈ { 1 , . . . , n }} ∪ {( T ,  T, j  ) | r j is a rule with empty body} ∪ { ( T , j , 2 n + j) | r j is a rule with empty body} ∪ (n + i,  i, j ) | i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and l i is a literal in the body of r j  ∪ ( i, j , 2 n + j) | i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and l i is a literal in the body of r j  ∪ (2 n + j, n + i ) | i ∈ { 1 , . . . , n } , j ∈ { 1 , . . . , k } and p i is the variable in the head of r j  Before going on with the proof, let us note that the construction of this SN P system is illustrated in the Example 4 . As in Theorem 1 , we will prove different remarks which will be useful in the proof of the theorem.  Remark 1. For each k ≥0 •In the configuration C 3 k , the neuron σT has only one anti-spike and the neurons σG and σH have no spikes nor anti-spikes. •In the configuration C 3 k +1 , the neuron σG has only one antispike and the neurons σT and σH have no spikes nor antispikes. •In the configuration C 3 k +2 , the neuron σH has only one antispike and the neurons σT and σG have no spikes nor antispikes. Proof. This remark can be easily checked since σG has only an incoming synapse from σT , σH has only an incoming synapse from σG and σT has only an incoming synapse from σH . By construction, in C 0 , σT has one anti-spike in the initial configuration, σG and σH are empty and in the three neurons the unique rule is a → a . So in, C 1 , σG has one anti-spike in the initial configuration, σT and σH are empty and in C 2 , σH has one anti-spike in the initial configuration, σG and σT are empty. This situation is cyclic and it is repeated each three computational steps. Let us also note that the neurons σn + i with i ∈ { 1 , . . . , n } has an incoming synapse from σH , so these neurons receive one anti-spike at C 3 k +3 from σH .  Remark 2. Let I = (i 1 , . . . , i n ) an interpretation for KB and let S = (s 1 , ... , s n ) be a vector with the following properties. For all j ∈ { 1 , . . . , n } •If i j = −1 , then s j = −1 . •If i j = 1 , then s j ∈ { 1 , . . . , 2 ×h j −1 } Let us suppose that at the configuration C 3 t the neuron σn + j contains exactly s j spikes 7 . Then, the interpretation obtained by applying the immediate consequence operator T KB to the interpretation I, T KB ( I ) is (q 1 , . . . , q n ) where q j , j ∈ { 1 , . . . , n } is the number of spikes of the neuron σj in the configuration C 3 t+4 . Proof. Firstly, let us consider k ∈ { 1 , . . . , n } and T KB (I)(p k ) = 1 . Let us prove that at the configuration C 3 t+4 there is exactly one spike in the neuron σk . If T KB (I)(p k ) = 1 , then there exists at least one rule r l ≡L d 1 ∧ ···∧ L d l →p k in KB such that I(L d 1 ∧ ···∧ L d l ) = 1 . Case 1: Let us consider that there is only one such rule r l and the body of r l is empty. By construction, the neuron σ2 n + l has only one incoming synapse from neuron σ T, l  and σ T, l  has only one incoming synapse from σT . According to Remark 1 , at C 3 k the neuron σT has one anti-spike, so σ T, l  has one anti-spike at C 3 k +1 ; σ2 n + l has one spike at C 3 k +2 ; σn + k has one spike at C 3 k +3 and σk has one spike at C 3 k +4 . Case 2: Let us now consider that there exists r l ≡ L d 1 ∧ ···∧ L d l → p k in KB such that I(L d 1 ∧ ···∧ L d l ) = 1 and d l > 0. Without losing of generality, we can suppose that L d 1 ∧ ···∧ L d l = p d 1 ∧ ···∧ p d f ∧ ¬ p d f+1 ∧ ···∧ ¬ p d l and therefore •I(p d u ) = 1 if u ∈ { 1 , . . . , f} •I(p d u ) = −1 if u ∈ { f + 1 , . . . , l} According to the construction of the vector S = (s 1 , ... , s n ) , at the configuration C 3 k •σn + d u has an amount of spikes between 1 and 2 ×h d u −1 if u ∈ { 1 , . . . , f} •σn + d u has one anti-spike if u ∈ { f + 1 , . . . , l} By application of the corresponding rules, at C 3 k +1 •σ d u ,l has one spike and the rule a → a can be applied , if u ∈ { 1 , . . . , f} •σ d u ,l has one anti-spike and the rule a → a can be applied, if u ∈ { f + 1 , . . . , l} At C 3 k +2 , the neuron σ2 n + l has b l spikes and sends 2 spikes to σn + k . At C 3 k +3 , the neuron σn + k has received one anti-spike 7 If s j = −1 , then σn + j will contain one anti-spike. from σH , two spikes from σ2 n + l , and eventually, other spikes from neurons σ2 n + p if p k is also the head of a rule r p and the interpretation I satisfies the literals in their bodies of r p . Bearing in mind the annihilation of the anti-spike with one of the spikes, at C 3 k +3 , σn + k has an amount of spikes between 1 and 2 ×h p k −1 . Let us remark that if none of the rules with p k in its head has the literals in its body satisfied by I , then the neuron σn + k receives only one anti-spike from σH which is not annihilated. The spikes at C 3 k +3 fire the corresponding rule in σn + k and σk has one spike at C 3 k +4 . Let us consider now k ∈ { 1 , . . . , n } and T KB (I)(p k ) = −1 . In this case, all the rules r l ≡L d 1 ∧ ···∧ L d l → p k verifies I(L d 1 ∧ ···∧ L d l) = −1 . Firstly, let us note that the body of r l cannot be empty. Let us consider one of such rules r l ≡L d 1 ∧ ···∧ L d l → p k . Since I(L d 1 ∧ ···∧ L d l ) = −1 , there exists L d e in the body of the rule such that I(L d e ) = −1 . Let us suppose that L d e is a negative literal (the reasoning in case of positive literals is analogous), L d e = ¬ p d e and I(p d e ) = 1 . By construction, in C 3 k the neuron σn + d e has s d e spikes with s d e ∈ { 1 , . . . , 2 ×h d e −1 } . The corresponding rule is triggered and at C 3 k +1 , the neuron σ d e ,l has one spike. Since p d e occurs negatively in the body of r l , the rule a → a is applied and one anti-spike arrives to σ2 n + l at configuration C 3 k +2 . Since σ2 n + l has b l incoming synapses from neurons which send at most one spike, in C 3 k +2 , the number of spikes in such neuron does not reach b l and hence, no spike is sent in the next step. This happens in all neurons σ2 n + l and therefore, at C 3 k +3 , σn + k only has one anti-spike from σH , the rule a → a is applied and σk has one anti-spike at C 3 k +4 .  Example 4. Let us consider the KB in the Example 3 . The SNPS associated with this KB and the interpretation I ∅ is = (O, σ1 , σ2 , . . . , σ12 , σG , σT , σH , σT, 1 , σ1 , 2 , σ2 , 2 , σ3 , 3 , σ4 , 3 , σ1 , 4 , syn ) and its graphical representation is shown in Fig. 2 . In such SN P system O = { a } and the neurons 8 are σ1 = (0 , R 1 ) σ2 = (0 , R 2 ) σ3 = (0 , R 3 ) σ4 = (0 , R 4 ) σ5 = (−1 , R 5 ) σ6 = (−1 , R 6 ) σ7 = (−1 , R 7 ) σ8 = (−1 , R 8 ) σ9 = (0 , R 9 ) σ10 = (0 , R 10 ) σ11 = (0 , R 11 ) σ12 = (0 , R 12 ) σG = (0 , R G ) σT = (−1 , R T ) σH = (0 , R H ) σT, 1 = (0 , R T, 1 ) σ1 , 2 = (0 , R 1 , 2 ) σ2 , 2 = (0 , R 2 , 2 ) σ3 , 3 = (0 , R 3 , 3 ) σ4 , 3 = (0 , R 4 , 3 ) σ1 , 4 = (0 , R 1 , 4 ) with the following sets of rules R j = { a → a, a → a | j ∈ { 1 , . . . , 8 }} R j = { a → a | j ∈ { T , G, H}} R 9 = R 12 = { a → a 2 } R 10 = R 11 = { a → λ, a 2 → a 2 } R T, 1 = { a → a } R 1 , 2 = { a → a, a → a } R 2 , 2 = R 3 , 3 = R 4 , 3 = R 1 , 4 = { a → a , a → a } with the synapses syn = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (5 , 1) , (6 , 2) , (7 , 3) , (8 , 4) , (T, G ) , (G, H) , (H, T ) , (H, 5) , (H, 6) , (H, 7) , (H, 8) , (T ,  T , 1  ) , (5 ,  1 , 2  ) , (5 ,  1 , 4  ) , (6 ,  2 , 2  ) , (7 ,  3 , 3  ) , (8 ,  4 , 3  ) ( T, 1  , 9) , ( 1 , 2  , 10) , ( 2 , 2  , 10) , ( 3 , 3  , 11) , ( 4 , 3  , 11) , ( 1 , 4  , 12) (9 , 5) , (10 , 7) , (11 , 6) , (12 , 8) ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ Let us remark that T KB ↑ 0 = C 0 [1 , . . . , 4] = (−1 , −1 , −1 , −1) T KB ↑ 1 = C 4 [1 , . . . , 4] = (1 , 1 , −1 , 1) 8 We will represent by t that there are t spikes in the neuron and by −tthat there are t anti-spikes 9 10 11 G 1 2 3 4 T 5 6 7 8 T,1 1,2 2,2 3,3 4,3 1,4 H 12 Fig. 2. Graphical representation of the synapses of the SN P system of Example 3 . Table 2 First steps in the computation of the SNPS in Example 4 . T KB ↑ 2 = C 7 [1 , ... , 4] = (1 , −1 , −1 , −1) T KB ↑ 3 = C 10 [1 , . . . , 4] = (1 , 1 , 1 , −1) T KB ↑ 4 = C 13 [1 , . . . , 4] = (1 , −1 , −1 , −1) The first steps of the computation are shown in Table 2 . 5. Conclusions Biological neurons have a binary behavior depending on a threshold. If the threshold is reached, the neuron is triggered and it sends a spike to the next neurons. If it is not reached, nothing is sent. This binary behavior can be exploited in order to design connectionist systems which are able to deal with two-valued logic-based reasoning systems. The current efforts for bridging both methods try to one approach to the other one by the adaptation of some of the features. In this sense, SN P system is a bio-inspired model whose features seem to fit to the target of mixing both approaches. In this paper, we have proved that SN P systems are able to deal with the semantics of deductive databases even when such systems use negation. Namely, we have proved that the immediate consequence operator can be iteratively computed with such devices by using an appropriate representation. In the case of deductive databases without negation, it is suffices to characterize the fix point semantics, since the immediate consequence operator is monotonic and the least Herbrand model can be computed by a SN P system in a finite number of steps. In the case of databases which involve negation, the semantics is much more complex. In this paper we have shown that the immediate consequence operator can be computed with SN P systems, but it is not enough for dealing with a complex semantics that involves, e.g., supported, stable and perfect models (see [27] ). Considering the semantics of deductive databases in this bioinspired model open a new door to further research possibilities. 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