Theoretical frames for smart structures

MATERIALS SCIENCE & ENGINEERING ELSEVIER

Materials Scienceand EngineeringC 4 (1996) I43-148

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Theoretical frames for smart structures Octavian Iordache Circo Craft Inc., 205, Brunswick Blvd., Pointe Claire, Que., H9R 1A5, Canada

Abstract Structures able to perform cognitive tasks are studied here. We considered hierarchically organized structures on more conditioning levels, learning in stochastic conditions. Mathematical models for smart structures' activity, a "perfect mixing" and a "plug flow" model have been developed. The relation with classical designs of experiment and the role of Walsh-Hadamard waves for pattern recognition and classification are emphasized. A frame allowing such waves is proposed. Keywords: Smart structures; Polystochastic

1. Introduction A challenge for material science is to build "smart structures" able to perform cognitive tasks specific to living systems, e.g. data acquisition, transmission, classification and recognition, learning and oversight, computing, autonomy in various conditions, plasticity and creativity. At this stage, a project grouping in a device all these faculties is not realistic but one expects to increase smart structures' potentialities in small steps. Constructing a smart structure is at the same time a "hard" problem of materials and architecture, a " s o f t " and a " f i n e " problem of modeling. Molecular biology and neuroscience suggest that physical constructions allowing cognitive tasks to be performed are characterized by reversible self-organization, multiple time and space scales, memory, self-adaptability to stochastic conditions, multiple phase transition, etc. Architecture in levels is recognized as the principle that rules the building of living systems. Following such suggestions from biology, numerous biomimetic structures have been studied [ 1,2]. Paralleling the experimental research on the "hard" problem of smart structures, progress in " s o f t " and "fine" problems have also been reported. General features of hierarchically organized stochastic systems with memory have been revealed in cybernetics, in stochastic automata theory (see the modern approach to cellular automata [ 3,4] ), in genetic algorithms theory [5], in artificial neural network theory (see the adaptive resonance theory), etc. Less developed in the study of smart structures seems to be the "fine" part, namely modeling based on first principles and expressed by equations for cognitive processes. Specific 0928-4931/96/$15.00 © 1996 ElsevierScienceS.A. All rights reserved PIIS0928-4931(96)00153-1

models describing self-adaptive evolution on more conditioning levels have been developed recently under the name of polystochastic models (PM) ( [ 6,7 ] ). One of the starting points was the observation that existing complex systems prove their adaptability to ambient conditions by self-organizing in flexible hierarchical structures (the so-called conditioning levels of complexity). To take into account the hierarchy of conditioning levels one starts here from simple models, but, one introduces a hierarchical structures for parameters and specific operations. In PM approaches it is considered that a stochastic process proceeds at any level of complexity and that processes at one level are conditioned on process taking place at higher levels in the complexity hierarchy. On account of the fact that no cognitive task can readily be separated from its substrate, PM will be presented here as candidate frame for smart structures at the "fine" level and at the same time as a source of suggestions for "hard" implementation.

2. Constructions

Research on smart structures has focused on complex systems outlining more homogenous levels coupled by transitions as: structured~quasi-structured~disordered media, laminar ~ chaotic ~ turbulent media, sol ~ mesogel ~ gel media, crystals ~ quasi-crystals ~ collQids, etc. [2]. Some architectures for "hard" constructions are recalled in the followings. 1. An adaptive architecture can be composed by an association of parallel layers or films connected to the exterior by a sensor and an actuator. These layers needs to have

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the possibility to self-interact and to self-associate in a hierarchy of assemblages. The architecture is in the socalled scaling region, where similar patterns appearing on different time scales have been studied. Other promising structures are the self-assembled films of semiconductors [2,8]. 2. An ordered hierarchy of filaments could be imposed from design using tree-like structures such as polymer chains with hierarchies of co-continuous phases. There is a practical limit of the number of phases that can be co-continuous, i.e. to the number of conditioning levels. The polymer chains should have in-between spaces allowing interactions. As examples, one considers the alkanoethiol electropolymerized layers on gold and conducting polymer trees [9], self-assembling nematic mesogels, hierarchically structured ceramics, etc. 3. The architecture could be formed by hierarchical porous blocks obtained by photostereo-lithography using lasers and computer-aided manufacturing tools. The resulting frames filled with conductive polymers should have neighboring interconnections. Examples are block frames filled with polymers, quasi-crystals ensuring plasticity, etc. 4. Biosystems are naturally characterized by self-organization in levels and in some cases by wavy multi-scale relaxation decays for luminescence, a useful phenomena for storage information. Bacteriorhodopsin, rhodopsin, Dafhis magna, Bryophyllum, Cucumis sativus, represents long studied opportunities ( [ 10-12] ). The external information is acquired by sensors embedded in the hierarchical filament's structure. They could be based on photoelectric, thermoelectric, piezoelectric effects, etc. A laser-based writing of data is described in Ref. [ 11 ]. The sensor needs to translate the input information in impulses determining an excitation of a number of hierarchy's levels according to the message (word) content. A natural request is to have a message codification that allows faster writing. On account of the physical framework of the sensor as a hierarchical structure, the translation task is facilitated by a hierarchical language in which the words are ordered vectors, expressing the hierarchy of characteristics or attributes of the described object. An example is the dyadic (binary) language. One associates with an object a word as: x = (xlx2...xk...) having the components 1 or O; 1 signifies the presence of an attribute and 0 its absence. The attributes are ranged in their order of significance. In the hardware, arbitrarily a state is assigned to binary state 0 and another one to binary state 1. At a molecular level this could corresponds to a polarized (biased) and nonpolarized (unbiased) state of the unit element. The information or excitation should be transmitted from sensor toward actuators along the layers, filaments, capillaries, etc. The data transmission could be facilitated by the continuous and spontaneous activity of the hierarchical configuration as a field of waves triggered by data acquisition. This implies that the unit elements of the filaments should have two states: an active and an inactive one corresponding

to 1 and to 0. Moreover, as will be established in the following, cyclic behaviors with periods in geometric progression are necessary. The reading of the information is accomplished by an actuator device having a function inverse to that of the sensor. It could be the same type of device as the sensor allowing the reverse processing of data. The logical values 1 and 0 (or - 1 ) of the exit signal could be ensured by adaptive thresholds. As established in genetic algorithm theory any program can be coded as strings of ones and zeros (the classifier systems). This means that, at least in theory, the information and its processing could be binarized. A p-adic representation, with p a prime number, is also useful if more than two states need to be considered.

3. Polystoehastic frames To explain the concept of polystochastic models one starts from the classical concept of stochastic learning system as introduced in cybernetics [ 13,14]. Such a system involves, in a simplified version, the system states S, the conditions of evolution C, the probabilities of the condition C conditional on the state S denoted by P (C/S), and the operators u, giving the new state S' as a function of the previous states and conditions: S ' = u(S, C). Notice that u could be a random operator and in some cases a conditional probability [ 14]. To give an example, consider a system that starts at the moment n = l from the state s ( t ) . With probability P ( c ( 1 ) / s ( 1 ) ) the initial condition c( 1 ) arises. On account of this condition the system evolves at n = 2 towards the new state: s ( 2 ) = u ( c ( 1 ) , s(1)) and so on. Notice that the Svalued process is markovian whereas the C-valued process is nonmarkovian, more exactly a stochastic process with complete connections [ 6,14]. In the case of PM, the spaces S, C and the operators u need to be considered at different levels of conditioning, corresponding to different time scales. At the conditioning level m, one needs to consider: the states Sm, the conditions C", the probabilities P(C"/S"', C '~- i, C,,-z . . . . ) of conditions Cm, conditional on the state S"' and on the established conditions on all previous levels m - 1, m - 2 , etc. One also considers the operators u', giving the new state S"" as a function of the previous states and conditions: S"" = um( s "', C% C"= t . . . . ). We are in fact faced with stochastic processes having two indices: n for real time and m for conditioning level. This looks complicated but specific tools such as nonarchimedean (i.e. p-adic or ultrametric) [ 15,16] and nonmarkovian theories [6,14] allow us to analyze particular classes of PM. To illustrate better the PM structure let us give an example showing the frame for memorizing data (see also Ref. [7] for different examples). Let us denote by sin(n) ~S, the matrix of memorized data in the nth step of recording data and by c ' ( n ) ~ C the corresponding condition, i.e. the new recorded vector (word) at the moment n. Suppose that

o. Iordache/ MaterialsScienceand EngineeringC 4 (1996) 143-148 sin(n) are matrices of data (the text), that era(n) are always row vectors of the type (1 Xm) with m the number of conditioning levels. The number of rows in data matrix is n. Suppose that the experiment starts at n = 1 at the level m = l from the word: s 1 ( 1 ) = ( 1 ) . With probability p2 ( ca (1) / s 1(1) ), the word c a (1) is selected at the level m = 2. The index 2 refers to this level. Let ca(1) = ( 1 0). The new state vector s2(2) results by pasting c a (1) to theprevious state s 1(1). It results that the new text is sa(2) : 1 1

0

This means that sa(2) = s I ( 1 ) YIca (1). Here 1I denotes the pasting down operation. In operatorial form, sZ(2)= uZ(s I (1), ca(1) ). Notice that the elements of the text, i.e. the words, could be considered as real numbers in binary form and that Yi could be considered as real valued operators. The index 2 refers to the fact that the pasting is at the conditioning level m = 2 . Then with probability P3(c3(2)/sa(2)), the condition c 3 ( 2 ) = ( 1 0 1) is selected at the level m = 3. Obviously: s3(3)=ua(sa(2),c3(2)) where S 3 ( 3 ) = sZ(2)Yic3(2) is given by: 1 1 1

0 0

1

The elements of the PM, i.e. ( Sm, um, C% P"', ...), result. S~ are matrices of data containing the text, u m are pasting operators, C m are vectors of conditions, i.e. words, P " are probabilities of new words conditional on previous text. Properties of the S'~-valued and of the C"'-valued stochastic chains, on account of the properties of u m and P"', were established. The above proposed frame has similarities with simulation algorithms. A more fundamental approach to smart structure activity based on mathematical differential models will be presented in the following. Their PM structure will be emphasized.

4. M a t h e m a t i c a l m o d e l s o f c l a s s i f i c a t i o n

A basic step in data processing for both natural and artificial systems is classification and pattern recognition. Two PM of classification, both outlining the central role of Walsh-Hadamard waves, will be presented in the following. Such waves are similar to sinusoidal ones but take only the values 1 and - 1 (or 0) having a logical significance [17].

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by y(b) the output (answer) function for the classification process giving the response yes y(b) = 1 or no y(b) = - 1 (or 0), to the classification process at different degrees of classification. To give an example suppose that the trajectory followed by an input word x(b) in a classification scheme is another word for instance y = ( 1 1 - 1 - 1). This means that at successive dyadic values of the degree of classification b equal to 0.5, 0.75, 0.875 and 0.9375, the word was successively described by the sequence of answers yes, yes, no and no, allowing one to recognize the input word. One of the simplest classification processes is that in which small changes of the degree of classification db in the interval (b, b + d b ) determine small changes of the answer proportional to both the existing answer y(b) and to the change db of b. The model governing the evolution of the answer y(b) with the degree b is written by analogy with real linear processes: dy(b)/db= -ky(b) Notice that the rate of classification k is considered as a word, namely the dyadic expansion k--- ~jkj2J.' It corresponds to the sequency used in information theory where it is interpreted as half the number of zero-cros sings in the periodicity interval [171. Despite the similarity with real models we are faced with a new type of differential operations, dyadic ones ( [ 18,19] ). The solution of the differential equation, similar to that of the real linear equation but with different addition and product operations is [ 19] :

y(b) = y ( 0 ) - k*b Observe that the real product kb was translated to a dyadic product k*b. k*b = ( ~ k j b j ) m o d 2 This is a logical gate denoted by (AND)mod 2. For any two words b = (bj) and b' = (b'j) one defines the dyadic sum:

b @ b ' = ( (bj+ bf)mod 2) The sum is the dyadic addition ® (equivalent to the dyadic difference and denoted by XOR). The defined operations for sum and product give in fact Walsh-Hadamard functions [20], WM(k, b), as solutions of the differential model since: (2y(b) - 1) =WM(k, b) Here M = 2 m, k = ~aikj2-/, b = Sjbi2 --/with 0 < j < m - 1, 0 <

4.1. Perfect mixing model Consider the classification as a process in which the usual time is the degree of classification b. It is a dyadic expansion of the type b = (blb2...b~...), with the digits bj = 0 or 1. This expansion may also be denoted as b = (bj). Notice that b varies from 0 to 1 by dyadic steps equal to different powers of 0.5. One associates to b the number, b = ~ b j 2 -J. Denote

k
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It is the type of coupling of digits corresponding to different transfer rates k of the word through the classification device, with the digits for different classification degrees b. Suppose that b increases by a unit digit bj = 1. Two types of transfer ( 1 and 0) become possible, one corresponding to kj = 1 and the other corresponding to kj = 0. After a new classification step at bj+l = 1 four possibilities result (11, 10, 01, 00). These four results cannot be stored within the imposed twovalued nature of the expected answer. It is useless to memorize if a yes response follows a previous yes or no. Therefore by the k*b product, 6ne lumps 11 and 00 in one class of answers and 10 and 01 in another one. The elements of the PM associated with the classification model are the states are S" = y ( b ) , i.e. the set of words, the conditions are C " = (k, m), i.e. the transfer rates and the level m, the probabilities are Pro(k, m) and the operators are um(y(b), (k, m) ). Here

um(y(b), (k, m ) ) - - y ( b ' )

y(b') = y ( b ) - k * ( b @ b ' )

This means that the system starts from a certain state, for instance a vcord, y(b). The condition choice involves the choice of both the classification rate k and of the level m related to the length of words and consequently to b'. According to the selected condition the new word y(b') wilt result as solution of the dyadic differential equation, Finally note that more complex dyadic models such as d y ( b ) / d b + ky(b) = x ( b ) where x(b) is the input signal and y (b) the output have been developed.

4.2. Plug flow model As in the case of perfect mixing models the degree of classification b plays the role of the usual time. One associates to any object a word such as z = (zlz2...zj...) having the components 1 or 0. Here 1 signifies the presence of an attribute and 0 its absence. The space of words is denoted by z. The physical interpretation of the plug flow model of classification is that the answer variation along b is proportional to the answer variation along the words' space coordinate z. The model of this classification process is, by analogy with plug flow models, the wave equation:

~u/Ob- kOu/Oz= 0 The initial condition is

u(z,O) =f(z)

b = (11>, the coordinate z should take the values DCBA in this order, to ensure that the initial condition f(00) is verified. This means for example that f(OO@OO)=f(O0); f(01 @01) =f(00) ;f(10 ® 10) =f(O0) ;f( 11 @ 11) =f(00). The resulting z-b diagram is a characteristic. Consider now a system starting at b = ( 0 0 ) from z = ( 0 1 ) = C . In this case, z should take the values CDAB in this order, to ensure that the solution f(01) is verified. This means that f ( 0 0 @ 0 1 ) = f ( 0 t ) ; f ( 0 1 @ 0 0 ) = f ( 0 1 ) ; f ( 10 (911 ) =f(01 ) ;f( 1 t @ 10 ) =f(O 1 ). Comp uting the characteristics for different starting positions, one obtained results in the form of a Latin square: DCBA CDAB BADC ABCD Interesting solutions of the wave equation are also

u(z,b ) = WM( 1,z@ ( k ' b ) ) These solutions represents Walsh-Hadamard waves. The elements of the PM are the states S m= (y(b)), the conditions Cm= (k, m), the probabilities P"(k, m) and the operators um(y(b), (k, m) ) =y(b') given by:

y(b') =f(y(b) @ ( k * b ' ) ) The classification system starts from a state, e.g. the word

y(b). The condition selection means the selection of the classification rate k and of the level m. According to these, the new word y(b') will result from the partial differential equation. Note that, as a main result of modeling, classification methods parallel screening procedures in experimental design and diagnosis. Thus, for the perfect mixing procedures, one obtains a Walsh-Hadamard solution giving also the so-called Plackett-Burman matrix for the design of experiments while for the plug flow procedure the solutions are Walsh-Hadamard waves and the characteristics corresponds to a so-called Latin square in experimental design. As expected, models of cognitive processes need to have as solutions thinking methods such as those proposed in screening design of experiments. This represents a main argument for designing architectures exhibiting a field of Walsh-Hadamard waves as a potential candidate for smart structures. Note also that the polystochastic frames for both differential and partial differential models are similar.

4.3. Walsh-Hadamard waves

The general solution of the partial differential model is [21]

u(z,b) = f ( z ~ ( k ' b ) ) To give an example of a solution, let us consider k = 1 and denote the words by: A = ( l l ) , B=(10), C=(01), D = (00). Consider now a system starting at the classification level b = (00) from the word z = (00) = D. As b increases, taking in succession the values b = (00), b = (10), b = (01),

The Walsh-Hadamard waves represent the proposed support for word acquisition, transmission and processing in smart structures. The minimal elements necessary to build Walsh-Hadamard functions fields are summarized in the following. One starts from a periodic frame described by the function R, (t) = sign (sin 2" + 1 "rrt). This frame may be available in

O. Iordache/ MaterialsScienceand Engineering C 4 (1996)143-148 some systems exhibiting multi-scale wavy relaxation decays of luminescence. It can be artificially built using, for example, chromophores with laser induced photocycles as described by Birge et al. [ 11 ]. Obviously just one chromophore (one time scale in a photocycle) is not enough. As the b expansion shows, more photocycling periods are necessary [ 18]. Moreover, the lifecycles needs to be proportional to successive negative powers of 2. Suppose for instance that m - 3. The periods of the photocycles need to be proportional to 2 - i, 2 - 2, 2 - 3. One also needs the XNOR gate denoted here by ®. This gate is just the inverted XOR. The Walsh-Hadamard functions for m = 3 which result from Rn(t) and XNOR are as follows: w s ( o,t) = 1,

Ws(t,t) = Ro, Ws(2,t) =Ro®R~, Ws(3,t) =R1, Ws(4,t) =R1 ®R2, Ws(5,t) = Ro®RI ® R2, Ws(6,t) =Ro®R2,

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1 J

* the XNOR gate as used extensively in [7]. The process of reversible impregnation of the actuator structure with different correlations could allow the learning of several inputs and their recognition even when they are slightly distorted. The memory can be ensured by a hysteresis property of the architecture. This is related to the surrounding medium of the network of filaments, branches, capillaries, entering the smart structure building. Its effect on cycles is expected to allow learning and oversight. Based on the previous analysis the following strategy to build smart structures results: 1. to build a hierarchical adaptive framework allowing a field of Walsh-Hadamard waves (R~(t) and XOR); 2. to build multiple layers, branches and capillaries with XOR and AND gates; 3. to build hierarchical frames for XOR gates (allowing (AND) mod 2); 4. to implement writing and reading devices for binary (dyadic) information; 5. to train the structures obtained using specific distances and informational criteria [ 7 ].

Ws(7,t) =R2 Observe also that (AND)rood 2 results by applying XOR in successive steps to AND gates. Biomolecular versions of logic gates based on a protein photocycle have been proposed [ 10,11]. To have a photocycle with periods in geometric progression is an open problem. Notice that theoretically it is possible to exploit not only photocycles but also any series of structural changes that the material of the smart structures undergoes in response to excitations (light, pressure, temperature, etc.). The interaction of Walsh-Hadamard waves with the input signal gives the Walsh-Hadamard spectrum (or other correlation index) allowing the fast recognition of words. The word y(b) can be expressed as Fourier series of WalshHadamard functions.

y(b) = E q k W M ( k , b) k The coefficient qk depends on the contribution of the sequency (rate of classification) k to the word y(b). The spectrum qk is given by 1 q k = ~ r ~ ( b) WM(k, b) Simpler characteristics allowing classifications to be made are also • the product *: x*y = (~xiyi) mod 2 J • the dyadic correlation:

5. Example To take into account the randomness, the hierarchy and the complexity of input information, adaptive schemes for classification should be considered. These are nontrivial PM frames. Let us illustrate these by an example. Suppose that the smart structure starts the recognition at the degree of classification b ( 0 ) = (0). With probability P(0) the level m = 1 is selected. A b step at this level is performed from b (0) to b( 1 ) = (1). According to the perfect mixing differential model the response will be:

y(b(1) ) = y ( b (0)) - k ' b ( 1 ) If y(0) = 1 one obtains for different k, namely k = (0) and k = (1), the response y(b(1)): 1 1

1 0

A comparison of the resulting y(b) with standard ones, Ys, should be done. For any word y(b), the standard one Ys is in a binary form. Methods to classify based on digitized information and informational criteria have been presented in detail in Ref. [7]. A nonarchimedean valuation to evaluate the fitting is proposed here. The valuation is defined by:

V(y(b) - y s ) =r -j Here r > 1 is an arbitrary constant while j is the first rank showing differences between the corresponding elements of the resulting words y(b) and Ys. With probability P = r -j the next level for m is selected. With the probability 1 - P , the

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same level m = 1 is maintained. Suppose that now m = 2 is selected. In this case b varies from b ( 0 ) = ( 0 0 ) to b(2) = (11). The response will generally be according to perfect mixing model:

y(b) = y ( b ( 0 ) ) - k* (b(0) O b ) This gives for different k namely for (00), (10), (01), ( t 1) the matrix: 1 1 1 1

1 0 1 0

1 1 0 0

1 O 0 1

Recall that y(0) = 1. One observes that both the resulting matrices of type (2 × 2) and of type (4 X 4) show the structure of the classical Walsh-Hadamard design of experiment. The word Ys is compared with different y(b) (or its correlation with different words are compared) and the process continues towards recognition. Observe that in the above examples the states S' ' are matrices, the conditions C"' are words and I~qatafter the probabilistic selection of the level m the operatorial solution is generated for all possible sequences k.

6. Discussion The frames discussed here possess a number of properties that make them attractive, such as simplicity, fast calculus, learning potentialities, flexibility under actual conditions, etc. Surely the implementation of Walsh-Hadamard wave generators as proposed here, exploiting photocycles with well established periods, could allow operations of complex recognition to be done in real time, which would require hours to simulate on a computer. To develop smart structures, there is a reason to start from specific mathematical models of cognitive processes. The advantages of the models presented are more or less related to polystochastic frames based in turn on some significant concepts namely the hierarchy, the dyadic frame for cycles and the complete memory. Their basic roots are the nonarchimedean (p-adic) and the nonmarkovian theories. Recall that our method starts from very simple real and markovian

models but associates an enriched structure to parameters and performed operations to include levels, scales and memory in processes describing. Basically, the proposed strategy is that the difficulties encountered in characterizing the activity of smart structures impose a much simpler new type of approach. Imitating the PM on a hardware level is a different type of challenge.

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