Adv Syst Sci Appl 2020; 04:70–82 Published online at https://ijassa.ipu.ru. Multi-Valued Neural Networks II: A Robot Group Control Dmitry Maximov1∗ 1V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia Abstract: For the new concept of a multi-valued neural network introduced earlier, an analogue of the T-norm in fuzzy mathematics is considered. In the multi-valued neural network, all variables are elements of the lattice of linguistic variables, i.e., they are all only partially- ordered. The lattice operations are used to build the network output by inputs. However, a lattice elements’ multiplication may also be used to determine such operations in the case when not all of them are allowed by the lattice construction. In this paper, the lattice is assumed to be residuated, and the residual construction gives the analogue of a T-norm. The lattice elements’ multiplication determines the implication which is used, together with other lattice operations, in output determining of the neural network. Though, such a construction determines a multi-valued associative memory similar to the Brouwer lattice case considered earlier, this variant is more natural to use in the Kohonen-like networks that we demonstrate with the example of a robot group management. Keywords: multi-valued neural networks, fuzzy neural networks, robot group management, associative memory, linguistic variable lattice, system tasks’ lattice 1. INTRODUCTION The concept of multi-valued neural networks was introduced in [1], [2], and “Multi-Valued Neural Networks I” by D. Maximov, V. I. Goncharenko, and Yu. S. Legovich. It continues a line of research to assess the system state using elements of a lattice. The lattice may be a lattice of sets as in [3], [4] or, even, a lattice of graphs of system state configurations [5], [6]. Such an approach is used in different tasks when the situation is estimated by a set of linguistic variables along with degrees of significance/certainty of the values of these variables. Usually, in fuzzy systems, confidence levels take values in a numerical interval. However, the degree of fuzziness is determined by experts. In [7] it was demonstrated that it is not necessary to use numbers: linguistic variables (but already partially-ordered), which do not require a mandatory numerical evaluation, can again serve as an estimate. Such assessments are determined by the situation itself and do not need an expert opinion or, at least, the experts assessment of the situation is greatly facilitated. In this case, to compare the valuations and obtain control solutions in [7], the concepts of multi-valued logic are used in which the scale of truth values is a Brouwer lattice of a general form. Such scales of truth values generalize a linearly ordered scale (in particular, in fuzzy logic) and naturally define the implication as a lattice operation. In the case of a complicated lattice, the calculation of implications becomes a laborious task. In [1] and [2], it is proposed to use a multi-valued neural network to calculate implications and a multi-valued associative memory to obtain a control solution quickly. The ∗Corresponding author: jhanjaa@ipu.ru, dmmax@inbox.ru MULTI-VALUED NEURAL NETWORKS II 71 associative memory generalizes a fuzzy one of [8] to the case of using multi-valued logic operations instead of fuzzy ones. In “Multi-Valued Neural Networks I” the results of [8] on fuzzy associative memory with thresholds to the case of multi-valued one were expanded: the authors introduced the concept of such a network, investigated the properties of the associative memory and gave a generalization of the learning algorithm of fuzzy associative memory with thresholds to the multi-valued case. The inputs, outputs and connection weights of the network are linguistic variables (partially ordered), not numbers, which, however, made it possible to use such a network for processing control and diagnostic information of complex dynamic objects. However, a Brouwer lattice was used as the scale of truth value in both these cases, which restricts application possibilities. In [8], a general fuzzy associative memory with a T-norm is also considered. In residuated lattices, properties in the definition of the T-norm coincide with the same as in the residuated construction definition. In this paper we use this fact to expand the results of [8] to associative memories with variables taking values in residuated lattices that may not be Brouwerian in this case. We investigate the properties of such a multi-valued associative memory and give a generalization of the learning algorithm of fuzzy associative memory to the multi-valued case similarly to “Multi-Valued Neural Networks I” and [1]. However, the application of such a construction is natural in Kohonen-like neural networks rather than in associative memories due to the residuals’ intuitive meaning. We demonstrate such an application — with the example of a robot group management. 2. BACKGROUNDS Feedforward fuzzy neural networks in which internal operations are based on fuzzy operations of joins and meets ∨ − ∧, were proposed in [9]. Such networks are called ∨ − ∧ fuzzy associative memory, and fuzzy information in them represented by the elements of the interval [0, 1]. We will use elements of a general lattice in multi-valued associative memory instead of elements of the interval [0, 1]. 2.1. Lattices [10] Definition 2.1: A lattice is a partially-ordered set having, for any two elements, their exact upper bound or join ∨ (sup, max) and the exact lower bound or meet ∧ (inf, min). Definition 2.2: The exact upper bound of a subset X ⊆ P of a partially-ordered set P is the smallest P - element a, larger than all the elements of X: min(a) ∈ P : a ≥ x, ∀x ∈ P . Definition 2.3: The exact lower bound is dually defined as the largest P -element, smaller than all the elements of X . Definition 2.4: A complete lattice is a lattice in which any two subsets have a join and a meet. This means that in a non-empty complete lattice there is the largest “>” and the smallest “0” elements. If we take such a lattice as a scale of truth values in a multi-valued logic, then the largest element will correspond to complete truth (true), the smallest to complete falsehood (false), and intermediate elements will correspond to partial truth in the same way as the elements of the segment [0,1] evaluate partial truth in fuzzy logic. In logics with such a scale of truth values, implication can be determined by multiplying lattice elements, or internally, only from lattice operations. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) 72 DMITRY MAXIMOV Definition 2.5: Lattice elements, from which all the others are obtained by join and meet operations are called generators of the lattice. Definition 2.6: A lattice is called atomic if every two of its generators have null meets. Definition 2.7: A Brower lattice is the lattice that has internal implications. Definition 2.8: In such a lattice, the implication c = a⇒ b is defined as the largest c : a ∧ b = a ∧ c. Definition 2.9: The implication ¬a = a⇒ 0 is called the pseudo-complement of a. Distribution laws for join and meet are satisfied in Brouwer lattices. The converse is true only for finite lattices. 2.2. Residuated Lattices [11] In non-distributive lattices, the implication cannot be defined. However, we may introduce a multiplication of the lattice elements and use it to define an external implication. Definition 2.10: A residuated lattice is an algebra (L,∨,∧, ·, 1,→,←) satisfying the following conditions: • (L,∨,∧) is a lattice; • (L, ·, 1) is a monoid; • (→,←) is a pare of residuals of the operation ·, that means ∀x, y ∈ L : x · y 6 z ⇔ y 6 x→ z ⇔ x 6 z ← y In this case, the operation · is order preserving in each argument and for all a, b ∈ L both the sets {y ∈ L|a · y 6 b} and {x ∈ L|x · a 6 b} each contains a greatest element (a→ b and b← a respectively). The monoid multiplication · is distributive over ∨: x · (y ∨ z) = (x · y) ∨ (x · z). Also, x · 0 = 0 · x = 0. A special case of residuated lattices is a Heyting algebra, when the monoid multiplication coincides with ∧. In non-commutative monoids, residuals → and ← can be understood as having a temporal quality: x · y 6 z means “x then y entails z,” y 6 x→ z means “y estimates the transition had x then z,” and x 6 z ← y means “x estimates the opportunity if-ever y then z.” You may think about x, y, and z as bet, win, and rich correspondingly (Wikipedia). Though, the temporal quality refers to residuals, we refer the same meaning to any multiplication x · w 6 z. However, only the residual y estimates the transition x→ z. Definition 2.11: A residuated lattice A is cancellative if it satisfies the equations x · y = x · z =⇒ y = z and y · x = z · x =⇒ y = z [12]. A residuated lattice is cancellative if, and only if, it satisfies the equations x→ xy = y and yx← x = y [12]. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) MULTI-VALUED NEURAL NETWORKS II 73 Definition 2.12: A residuated lattice A is said to be integrally closed if it satisfies the equations x · y 6 x =⇒ y 6 1 and y · x 6 x =⇒ y 6 1, or equivalently, the equations x→ x = 1 and x← x = 1 [13]. Every cancellative residuated lattice is an integrally closed one, and any upper or lower bounded integrally closed residuated latticeL is integral, i.e., a 6 1, ∀a ∈ L [13]. Therefore, a finite calcellative lattice is integral. We will assume that the lattices used in a multi-valued neural network are complete, finite and residuated. We can compare the definition of a residuated lattice and the definition of a fuzzy operator and a T-norm [14], [8]: Definition 2.13: We call the mapping T : [0, 1]→ [0, 1] a fuzzy operator, if the following conditions hold: • T (0, 0) = 0, T (1, 1) = 1; • If a, b, c, d ∈ [0, 1], then a 6 c, b 6 d, ⇒ T (a, b) 6 T (c, d); • ∀a, b ∈ [0, 1], T (a, b) = T (b, a); • ∀a, b, c ∈ [0, 1], T (T (a, b), c) = T (a, T (b, c)). If T is a fuzzy operator, and ∀a ∈ [0, 1], T (a, 1) = a, we call T a T-norm. For a, b ∈ [0, 1], let us define a→ b ∈ [0, 1] : a→ b = sup{x ∈ [0, 1]|T (a, x) 6 b}. We see that the latter definition coincides with the residual construction definition in the case of a commutative monoid and L = [0, 1]. Excluding the condition of commutativity and the requirement L = [0, 1], we obtain the construction of a complete and residuated lattice (we demand also its finiteness). 3. MULTI-VALUED ∨ − ∗ ASSOCIATIVE MEMORY As in the case of fuzzy neural networks [8], a multiplication of variables is used together with lattice operations in ∨ − ∗ multi-valued associative memory, where ∗ stands for the multiplication. Let us suppose, an input signal is x ∈ Ln, and an output signal is y ∈ Lm, where L is the residuated lattice used. In this case, the input-output relationship in a two- layer multi-valued associative memory can be written as y = x ~ W, where ~ stands for the ∨ − ∗ composition operation, and W = (wij)n×m ∈ µn×m is the n×mmatrix of the weights of connections with elements from L, i ∈ N = {1...n}, j ∈M = {1...m} (Fig. 3.1). In [1], several of the simplest composition variants have been suggested for ∨ − ∧ multi-valued associative memory and the next two of them are considered: yj = ∨ i {xi ∧ wij}; (3.1) yj = ∧ i {xi ⇒ wij}. (3.2) In the theory of the simplest fuzzy associative memory, only option (3.1) is considered. More complex models use a combination of ∨ and t-norms [8], [15] and a combination with implication [15]. However, they all use a linear number segment as a set of variables, unlike this paper where a general non-number lattice with multiplying elements is used. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) 74 DMITRY MAXIMOV Fig. 3.1. Bilayer associative memory Thus, we consider the composition 3.1, but with the monoid multiplication ∗ instead of ∧ (Fig. 3.1), as in [8], however with all variables taking values in the residuated lattice L: yj = ∨ i {xi ∗ wij}. (3.3) In the vector notation, it can be written as: y = x ~ W. (3.4) We denote (X, Y ) = {xk, yk | k ∈ P}, P = {1...p} — the family of pairs multi-valued patterns with xk = (xk1, ..., x k n) and yk = (yk1 , ..., y k m). We establish the connection weight matrix W ∗ = (w∗ij)n×m: w∗ij = ∧ k∈P (xki → ykj ). (3.5) The definition generalizes the similar one from [8]. Then, let us define the sets [8]: Sij(W∗, X, Y ) = {k ∈ P | ykj 6 xki ∗ w∗ij}. Mw = {W ∈ µn×m | ∀k ∈ P, xk ~ W = yk}. This is the set of the pattern pair family and the connection matrix, which satisfy the equation 3.3. The next property follows from the residual operation definition and holds both in fuzzy [8] and residuated [11] cases: a ∗ (a→ b) 6 b. (3.6) Also, the following properties hold in residuated lattices [11]: c→ ( ∧ Y ) = ∧ {c→ y|y ∈ Y }; (3.7) b→ (a→ c) = (a ∗ b)→ c. (3.8) The property (3.6) is used in the following theorem proof exactly in the same way as in [8] due to the fact that the T-norm is a special case of the residuated lattice construction. However, such a theorem in [8] contains a necessity condition in point (iii) which does not hold in the case of a general non-chain lattice. In the general case, it may be ∨ixi = y, though ∀xi < y that contradicts the necessity. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) MULTI-VALUED NEURAL NETWORKS II 75 Theorem 1: Let W∗ = (w∗ij)n×m ∈ µn×m ∈ Lm. Then: (i) ∀k ∈ P, xk ~ W∗ ⊂ yk, and if the matrix W satisfies: ∀k ∈ P, xk ~ W ⊂ yk, then, W ⊂ W∗; (ii) If Mw 6= ∅, then, W∗ ∈Mw, and ∀W ∈Mw, W ⊂ W∗, i.e., ∀i ∈ N, j ∈M, wij 6 w∗ij; (iii) The set Mwcd 6= ∅ if ∀j ∈M, ⋃ i∈N Sij(W∗, X, Y ) = P . Theorem 1 tells us that W∗ is the largest solution of the equation (3.4), if such a solution exists, and we will use this fact in the next section when we go down from the largest lattice element to W in a learning algorithm. 3.1. Learning Algorithm The iteration scheme for learning the connection weight matrix W of the multi-valued associative memory is similar to [8] with changes as in [1] and “Multi-Valued Neural Networks I”. It generalizes the dynamic δ-learning algorithm of fuzzy associative memory, introduced in [16], [17], [18] for different fuzzy cases, to the ∨ − ∗ multi-valued case. However, we are bound by atomic cancellative (hence, integrally closed) residuated lattices in the following algorithm, since Theorem 2 is proved only in this case. Hence, 1 = > in the lattice used since it is finite. Thus, we get an Algorithm of wij iteration for i ∈ N and j ∈M : Step 1. Initialization: for i ∈ N , j ∈M let us put wij(0) = >, t = 0; Step 2. Let W(t) = (wij(t)); Step 3. Let us calculate the resulting output: y = x ~ W, i.e., ∀k ∈ P, j ∈M, ykj (t) =∨ i[x k i ∗ wij(t)]. Step 4. Weights selection. Denotation 1: From this place, we denote by {ykj } the set of generators contained in the lattice element ykj : {gl|gl 6 ykj } and by {w}ij the matrix of such sets. The matrix elements are the sets of generators of the weight matrix elements. Matrices wij of weights and {w}ij are one-to-one correspondent to each other in distributive lattices. In non-distributive lattices, not one set {w}ij may correspond to one wij . In this case, we take max{w}ij as corresponding set to wij .The set difference will be denoted by a minus sign. Then, {w}ij(t+ 1) = ⋂ k  {w}ij(t)− sym({ykj (t)} − {ykj }), if k : xki ∗ wij(t) > ykj ; {w}ij(t), otherwise. (3.9) Here operation sym means the symmetrized set difference: sym(A−B) = { A−B, A ∩B ⊂ A; B − A, A ∩B = A. (3.10) Step 5. For i ∈ N , j ∈M , let us check {w}ij(t+ 1) = {w}ij(t)? If this is true, then the Algorithm stops, otherwise t = t+ 1 and goes to Step 2. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) 76 DMITRY MAXIMOV Theorem 2: Let the matrix sequence {W(t) | t = 1, 2...} is obtained by the learning Algorithm. Then, (a) {W(t) | t = 1, 2...} is non-increasing sequence; (b) {W(t) | t = 1, 2...} converges; (c) {W(t) | t = 1, 2...} converges to W ⊆ W∗, where w∗ij is defined in (3.5). Proof (a) For i ∈ N , j ∈M , k ∈ P we get from (3.9) that {w}ij(t+ 1) 6 {w}ij(t). Therefore, for i ∈ N , j ∈M : W(t+ 1) ⊂W(t). Thus, {W(t) | t = 1, 2...} is a non-increasing sequence. (b) {W(t) | t = 1, 2...} converges, since the sequence wij(t) is bounded below by the smallest lattice element 0 ∀t = 1, 2.... (c) Let us prove ∀t ∈ {0, 1, ...}, wij(t) 6 w∗ij . If t = 0, then wij(0) = > ≡ 1. If wij(t) = > for t = 1, 2, ..., then xki ∗ wij(t) 6 ykj , or wij(0) = > is a decision, by (3.9). By the definition of the residual, ∀k ∈ P, xki → ykj = >, or wij(0) = w∗ij by Theorem 1. Therefore, w∗ij = > ≡ 1 = wij(t), ∀t ∈ {0, 1, ...}. Let us suppose limt{wij}(t) 6 w∗ij . Let t0 is the first step such that wij(t0) 6 w∗ij . Then, xki ∗ wij(t0) 6 xki ∗ w∗ij = xki ∗ ∧ k′∈P (x k′ i → yk ′ j ) 6 xki ∗ (xki → ykj ) 6 ykj by (3.6). Thus, by (3.9), we get wij(t0 + 1) = wij(t0) 6 w∗ij . Let limtwij(t) = wij(t0) = lij 6 w∗ij . Then, lim t ykj (t) = ∨ i∈N (xki ∗ lij) 6 ykj . (3.11) Therefore, by (3.9) and (3.11), we get: lim t {wij}(t) = {lij} = = ⋂ k [lim t {wij}(t)− sym({lim t ykj (t)} − {ykj })] = = ⋂ k [{lij} − ({ykj } − {lim t ykj (t)})]. (3.12) Thus, limt y k j (t) = ykj for ∀k ∈ P , and lij = wij(t0) = limtwij(t) 6 w∗ij is a decision of (3.4), or {lij} ⋂ ({ykj } − {limt y k j (t)}) = ∅. The latter case means, at least, that {lij} is incomparable with ({ykj } − {limt y k j (t)}). Thus, {ykj } ⋂ {limt y k j (t)} ⊇ {ykj } ⋂ {lij}. Then, {ykj } ⋂ {limt y k j (t)} = {ykj } ⋂ {lij}, since xki ∗ lij 6 lij in integral lattices, and, therefore, lim t ykj (t) = ∨ i∈N (xki ∗ lij) 6 lij. (3.13) Then, let us suppose limt{wij}(t) = lij > xki → ykj for ∀k ∈ P , hence lij > w∗ij . Then, we get by the definition of the residual: xki ∗ lij > ykj and, therefore, lim t ykj (t) = ∨ i∈N (xki ∗ lij) > ykj . (3.14) Hence, we obtain similarly (3.12) with (3.13) that limt y k j (t) = ykj that contradicts (3.14). Thus, limt{wij}(t) = lij ≯ xki → ykj for ∀k ∈ P that means limt{wij}(t) 6 {w∗ij}, or limt{wij}(t) is incomparable with {w∗ij}. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) MULTI-VALUED NEURAL NETWORKS II 77 However, limt{wij}(t) may not be incomparable with w∗ij . Indeed, let t0 is the first step such that wij(t0) is incomparable with w∗ij . Then, if xki ∗ wij(t0) is incomparable with ykj , we get limt{wij}(t) = lij = wij(t0) by (3.9), and limt y k j (t) = ∨ i∈N(x k i ∗ lij) is incomparable with ykj or limt y k j (t) = ∨ i∈N(x k i ∗ lij) > ykj . However, the latter case contradicts (3.12) with (3.10) similarly to (3.14) and (3.13). The same holds if xki ∗ wij(t0) > ykj , also similarly to (3.14) and (3.13). Let limt y k j (t) = ∨ i∈N(x k i ∗ lij) is incomparable with ykj . However, (3.12) and (3.10) demand then {limt y k j (t)} − {ykj } = ∅ since (xki ∗ lij) 6 lij , and, hence, {limt y k j (t)} ⊆ {lij}. Now, let us limt{wij}(t) = lij is incomparable with w∗ij , and xki ∗ lij 6 ykj . Then, xki ∗ lij is also incomparable with xki ∗ w∗ij , since the lattice is cancellative (otherwise, e.g., xki ∗ lij 6 xki ∗ w∗ij =⇒ lij 6 xki → xki ∗ w∗ij = w∗ij). Hence, it should be xki ∗ lij < ykj and xki ∗ w∗ij < ykj , otherwise, they are comparable. Let us consider lij → w∗ij = lij → ∧ k x k i → ykj . Thus, lij → w∗ij = ∧ k lij → (xki → ykj ) = ∧ k(x k i ∗ lij)→ ykj by (3.7) and (3.8). Therefore, (xki ∗ lij)→ ykj = 1 in integral lattices, since (xki ∗ lij) < ykj [13]. Hence, lij → w∗ij = 1, and lij 6 w∗ij which contradicts their incomparability. Thus, we obtain the unique result: limtwij(t) 6 w∗ij . 4. EXAMPLE OF A ROBOT GROUP MANAGEMENT Such a network may be used as an associative memory or pattern classifier in the same manner as ∨ − ∧ networks in [2] and “Multi-Valued Neural Networks I”. It may be useful in the case of a non-distributive lattice of linguistic variables. However, a systematic method to introduce the monoid multiplication in such a lattice is absent, and we might use heuristics in every particular case. Moreover, the residual intuition treats the multiplication as action: we had something in the past, have something else now, that entails a third thing as a result of the operation. All these reasons lead us to the fact that the networks are more suitable for using, not with patterns (their storing or classifying), but in some leader output detecting. Thus, we consider here the problem of task distribution in a group of janitor robots. The group is fulfilling a set of tasks. Then, a new task arises. Robots must decide which one (or several) of them will perform the new task. Such a problem was considered in [5] based on linear logic. However, the structure of linear logic is rather intricate and difficult to implement programmatically . The residuated construction is much simpler and allows the task lattice to be determined more adequately. Let us consider a group of robots that can perform the following tasks: x1 — trash search, e — taking out the trash, x3 — sawing of the found garbage when it cannot be taken out, x2 — the stupor: it is the state of a robot when it tries, but cannot perform a regular task, e.g., the robot cannot pick up the trash, or cannot saw it, or can not move. All these main tasks have subtasks (Fig. 4.2): trash search x1 includes d1 — a move, and d4 — a video camera operation; trash removing e includes d1 and d3 — grabbing of the debris; sawing the garbage x3 includes d3 and d2 — sawing which is not depicted for simplicity. The stupor x2 includes only the grip: a robot has grabbed a thing, but can do nothing with it. Elements x2, x3, d1, and d4 may be taken as generators. All others are their meets and joins. Such tasks are, e.g., C42 — a robot in a stupor operates with a video camera, or C43 — the same during sawing. In this case, we consider the joined tasks as fulfilling in parallel. However, in other cases, we may consider joined tasks as performed sequentially. The lattice in Fig. 4.2 differs from the similar one in [5]: now, we include all joined tasks Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) 78 DMITRY MAXIMOV Fig. 4.2. A lattice of system task estimations which have meanings. However, including all of them may be unnecessary. Then, the lattice may be non-distributive. Thus, we consider, as in previous papers, the task lattice as the lattice of linguistic variables denoting tasks. Though, the tasks may also be associated with the sets of operations required for a task to be performed. As usually, the lattice partial order orders lattice elements by their values: the higher an element lies in the lattice diagram Fig. 4.2, the greater the value it has. Thus, maximal active system behaviour has the top value >, and activity absent has the bottom value 0. These values estimate transitions a→ b = c: the value c estimates the transition from the task a to the task b. Or, once more, “had a then c (arising) entails b.” 4.1. Monoid Definition There is no a general method to define the monoid operation in a residuated lattice. Every time we should use some heuristics. In our case, we demand C42 = 1, since we want to change the stupor x2 into the grip d3 if we obtain the stupor in two consequent iterations: x2x2 = d3 †, though in other cases, e.g., d1d1 = d1 ‡. Therefore, it may be x2 < 1. Subtasks d3 and d4 arising should not usually change the current task. Thus, we consider C42 = 1 as the only right neutral element of the monoid multiplication (correspondingly, we consider only the right multiplication and left residuals x→ y), since one can hardly give equally meaningful meaning to a left neutral element. Then, we want ex2 = d3, i.e., if we get stupor during debris pick up because the robot cannot lift the trash, only the grab remains from the trash remove. Therefore, d1x2 ∨ d3x2 = d3, and, hence, d1x2 = d3 or d1x2 = 0, and d3x2 = 0 or d3x2 = d3. We chose d1x2 = 0 (that is evident: the stupor during the move entails an activity cessation) and d3x2 = d3, because it should be for all lattice elements x1 = x: d11 = d1x2 ∨ d1d3 ∨ d1d4 = d1, and, therefore, †We omit the multiplication sign in this section for simplicity. ‡We may also demand x2x2 = 0 — stupor changes to activity absent. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) MULTI-VALUED NEURAL NETWORKS II 79 d1x2 6= d3 §. Thus, d1d3 = 0 and x2d3 = d3 by monotonicity, and, hence, d1d4 = d1. Also, we demand x1x2 = d4, i.e., if a robot cannot move during a debris searching, only camera functioning remains. Then, d4x2 = d4 — the stupor does not influence camera functioning. Similarly, x3x2 = d3 and x1d3 = d4d3 = d4. Then, evidently, x1d1 = x1, d4d1 = x1 (since, a move does not cancel the previous camera operation), x1d4 = x1 (since, the robot may stop during the search), and d4x1 = x1x1 = x1. Hence, x1e = x1, and ex1 = d3x1 = e since we suppose d3d1 = d1 and d3d4 = d3 — from d31 = d3. Thus, e.g., C42 = 1 = d4 → d4 and C1e = x1 → x1. Note, that the robot in search does not normally switch to trash removing, as in ants’ colonies, because x1e = x1 — “had x1, now e arises, that entails x1”. This result was obtained in [3] [3] from the linear logic structure definition, while here the conclusion is the consequence of our multiplication operation choice. Now, let us x2d1 = d1. Therefore, x2x1 = C2e, because x2d4 = x2 from x21 = x2. Then, let it be x3d1 = d1 and x3d4 = x3 — from x31 = x3 and evidently x3d3 = d3. Then, x3x1 = C3e. At least, let us consider x3 multiplication: d1x3 = x3, d3x3 = x3, d4x3 = C43. The first two are evident, in the latter case we establish that sawing does not cancel the camera operation. Also, we suppose x2x3 = > since we do not know if the stupor can switch to sawing at this place, or transition to any task (or again to the stupor) is possible. Then, U23e = x2 → >. Thus, we obtain the following basic definitions from the consideration above, though, our choice was sometimes not unique and, therefore, may be changed: d1d1 = d1 d3d3 = d3 d4d4 = d4 x2x2 = d3 or 0 x3x3 = x3 d1d3 = 0 d3d1 = d1 d4d3 = d4 x2d1 = d1 x3d1 = d1 d1d4 = d1 d3d4 = d3 d4d1 = x1 x2d4 = x2 x3d4 = x3 d1x2 = 0 d3x2 = d3 or 0 d4x2 = d4 x2d3 = d3 x3x2 = d3 d1x3 = x3 d3x3 = x3 d4x3 = C43 x2x3 = > x3d3 = d3 It is easy to verify that the (right) monoid definition satisfies these equations. 4.2. Neural Network Operation A variant of the Kohonen network may be used to determine the leading output or outputs in the problem of task distribution in a robot group. The Kohonen net [19] is a bilayer neural network in which the output layer consists of n neurons with m inputs in each neuron. The output of the j-th neuron is obtained according to the following formula: yj = dj + ∑ i∈[0;m] wijxi, where x = (x1, ..., xm) is an input vector, and dj is a threshold. The network operates by the rule “winner takes it all” which determines the leading output. Residuated lattice operations may be used in the formula as in ∨ − ∗ ((3.3)) and ∨ − ∧ [2], [1] multi-valued associative memories. Here, we consider only a variant without a threshold, with wij = 0 if i 6= j, and with a unique input x (this is the arising task in the robot group). The formula for neuron outputs is also different then: yi = Ti ∨ x > Ti(Ti → (Ti ∨ x)) (4.15) In the expression, Ti are tasks which the i-th robot supposes to perform (its intentions), and Ti ∨ x denotes the i-th robot intention to fulfil task x together with the other robot tasks. §If we chose d3x2 = 0, then ex2 = 0 that is also possible. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) 80 DMITRY MAXIMOV Fig. 4.3. The neural network topology for the system task choice The right part of (4.15) is a consequent of (3.3) and corresponds to the usual logic rule: [a ∧ (a⇒ b)]⇒ b. That means we obtain b if we have a, and a entails b. The formula has the same meaning in the residuated case: ac 6 b, where c 6 a→ b. Hence, we take Ti ∨ x as an output and we might have an extremal principal to choose the necessary output. We take here mini[Ti ∨ x] as the principal, because it is naturally to chose a robot with a minimal set of tasks which it is supposed to perform. Clearly, we take several such outputs in the absence of the minimum. Thus, we consider the neural network depicted in Fig. 4.3, in which the input variable x is an arising task in the robot group which is fulfilling tasks Ti at the current time. An arising task may be here x1, e, and x3, since stupor x2 refers to only one concrete robot, but not to all of them. We might choose now the necessary output for different input cases and tasks performed. Here, we will use an extended form for outputs instead of (4.15): Tix 6 Ti ∨ x (“there was Ti, now x arises, entails Ti ∨ x”), in the case of x 6 Ti → (Ti ∨ x). Let us suppose a standard situation: robots are looking for a trash and remove it. Then, if a search task x1 arises, some of the robots already performing x1 will include this new task into their intention set. Indeed, x1x1 = x1 is the minimal intention of joined tasks. A robot fulfilling subtask d4 of x1 (the camera operation) may also include the new task into its intentions, since d4x1 = x1. Such a state d4 may happen if a searching robot falls into a stupor: x1x2 = d4. A robot moving without any other task (subtask d1, e.g., which return after the trash remove) may also switch to this new one, since d1x1 = d1 < x1. If a task e of taking out the trash arises, then it will be performed by a robot already removing the debris: ee = e is the minimal value in the standard situation. Though, a moving robot (in d1) or a grabbing one (in d3) may also switch to taking out: d1e = d1 < e, d3e = e. The latter case arises after sawing when only grabbing remains. Thus, sawn-off pieces may be removed by the same robot that has sawn them off, or one robot can saw, and another remove the sawn-off pieces. If a task of garbage sawing x3 arises, then it will be performed by one of robots already fulfilling sawing (x3x3 = x3), or by a robot in grabbing: d3x3 = x3 — these values are minimal. The latter situation arises when the robot picking up the debris can not lift it: ex2 = d3 — only the grabbing remains. If the robot cannot saw the trash, it switches to grabbing: x3x2 = d3. However, in this situation, no regular task can be performed, and collective removing is required: d3e = e, where the task e is a part of the collective action. But this decision is external to the neural network application. Also, such an external decision Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) MULTI-VALUED NEURAL NETWORKS II 81 is needed if the robot cannot grab the trash: ex2 = d3, then, d3x2 = d3, and we obtain in two iterations x2x2 = d3 or x2x2 = 0. Thus, we see that the system of robots behaves quite reasonable, and the behaviour is due only to our choice of basic multiplications in the residuated lattice. 5. CONCLUSION A new concept of multi-valued neural networks with variables and weights taking values in a residuated lattice was introduced in this study. The concept continues a row of studies in which the state of a system is estimated not by numbers, but by elements of a partially ordered set, namely, the lattice. In our case the lattice is finite and residuated, and it may consist of some linguistic variables. Such an approach can facilitate the assessment of the situation by experts in cases requiring their participation. We have expanded the results on fuzzy neural networks to such a ∨ − ∗multi-valued case in which variables may take values in a non-distributive lattice. We found out the conditions under which it is possible to store in the multi-valued associative memory given pairs of network variable patterns. 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(2000) Novel neural algorithms based on fuzzy δ rules for solving fuzzy relation equations: Part I, Fuzzy Sets and Systems, 109, 355–362. 19. Kohonen T. (1989) Self-Organizing Maps, Berlin-New-York: Springer-Verlag. Copyright c© 2020 ASSA. Adv Syst Sci Appl (2020) Introduction Backgrounds Lattices Residuated Lattices Multi-Valued - Associative Memory Learning Algorithm Example of a Robot Group Management Monoid Definition Neural Network Operation Conclusion