research on wave propagation phenomena in periodic media has been very active during many decades .the topics and applications include for example photonic crystals , meta - materials , bragg gratings of surface plasmon polariton waveguides , energy harvesting in piezoelectric materials as well as surface wave propagation in periodic channels , which is the subject of this paper .a standard mathematical approach consists of linearisation and posing a spectral problem for an elliptic , hopefully self - adjoint , equation or system .early on it was noticed that waves propagating in periodic media have spectra with allowed bands separated by forbidden frequency gaps .this phenomenon was first discussed by lord rayleigh .it has also attracted some interest in coastal engineering because it provides a possible means of protection against wave damages , for example by varying the bottom topography by periodic arrangements of sandbars .the existence of forbidden frequencies is conventionally related to bragg reflection of water waves by periodic structures . here , bragg reflection is an enhanced reflection which occurs when the wavelength of an incident surface wave is approximately twice the wavelength of the periodic structure .this mechanism works , if the waves are relatively long so that the depth changes can effect them .a similar phenomenon may also happen , when waves are propagating along a channel with periodically varying width .in , and later , the authors studied a channel , the wall of which had a periodic stepped structure . using resonant interaction theory they were able to verify that significant wave reflection could occur .these results are based on the assumption of small wall irregularities .gaps in the continuous spectrum for equations or systems in unbounded waveguides have been studied in many papers , and we refer to for an introduction to the topic . in authors studied the linear elasticity system and proved the existence of arbitrarily ( though still finitely ) many gaps , the number of them depending on a small geometric parameter ; the approach is similar to section 3.1 , below , and the result is analogous to corollary [ cor3.2 ] . in the setting of the linear water - wave problem ,spectral gaps have been studied in , , and , though the point of view is different from the present work .  in this paperwe consider surface wave propagation using the linear water wave equation with spectral steklov boundary condition on the free water surface , see the equations , which are called the original problem here .the water - filled domain forms an unbounded periodic channel consisting of infinitely many identical bounded containers connected by apertures of width , see figure [ fig1 ] .the first results , theorem [ th3.1 ] and corollary [ cor3.2 ] show that the essential spectrum of the original problem ( which is expected to be non - empty due to the unboundedness of the domain ) has gaps , and the number of them can be made arbitrarily large depending on the parameter .an explanation of this phenomenon can be outlined rather simply using the floquet - bloch theory , though a lot of technicalities will eventually be involved .namely , if , the domain becomes a disjoint union of infinitely many bounded containers , and the water - wave problem reduces to a problem on a bounded domain ( we call it the limit problem ) , hence it has a discrete spectrum consisting of an increasing sequence of eigenvalues . on the other hand , for , one can use the gelfand transform to render the original problem into another bounded domain problem depending on the additional parameter . for each fixed problem again has a sequence of eigenvalues .moreover , by results of , , theorem 3.4.6 , and , theorem 2.1 , the essential spectrum of the problem equals where the sets are subintervals of the positive real axis , or bands of the spectrum .( for the use of this so called bloch spectrum in other problems , see for example , or . ) in general , those bands may overlap making connected , but in theorem [ th3.1 ] we obtain asymptotic estimates for the lower and upper endpoints of : we show that for all and and for some constants . in view of this implies the existence of a spectral gap between and for small and such that .however , since the estimates depend also on , we can only open a gap for finitely many , though the number of gaps tends to infinity as .  the asymptotic position ( as ) of the gapsis determined more accurately in theorems [ th4.5 ] and [ cor4.6 ] : those main results state that where the numbers depend linearly on the three dimensional capacity of the set .this result also ensures that in case the bands do not degenerate into single points , which means that the spectrum of the original problem indeed has a genuine band - gap structure .facts concerning the numbers , are discussed after theorem [ cor4.6 ] .  as for the structure of this paper ,we recall in section [ sec:1.2 ] the exact formulation of the linear water - wave problem , its variational formulation as well as the parameter dependent problem arising from the gelfand transform , and the limit problem .section [ sec:3 ] contains the formal asymptotic analysis which relates the spectral properties of the original problem with the limit problem and which is rigorously justified in secion [ sec:4 ] .the main results , theorems [ th3.1 ] , [ th4.5 ] and [ cor4.6 ] as well as corollary [ cor3.2 ] are also given in section [ sec:4 ] .the proofs are based on the max - min principle and construction of suitable test functions adjusted to the geometric characteristics of the domains under study .the authors want to thank prof .sergey a. nazarov for many discussions on the topic of this work .let us proceed with the exact formulation of the problem .we consider an infinite periodic channel ( see ) , consisting of water containers connected by small apertures of diameter .the coordinates of the points in the channel are denoted by , and stands for the projection of to the plane .we choose the coordinate system in such a way that the axis of the channel is in -direction and the free surface is in the plane .we describe the geometric assumptions on the periodicity cell in detail , as well as some related technical tools including the cut - off funcions .let us denote by a domain with a lipschitz boundary and compact closure such that its intersections with - and -planes are simply connected planar domains with positive area and contain the points and with , respectively ; these points are fixed throughout the paper .then the periodicity cell and its translates are defined by setting ( see figure [ fig1 ] ) furthermore , we assume that the set is a bounded planar domain containing the origin and that the boundary is at least -smooth .we assume that is so small that the set is contained in and .we define the apertures between the container walls as the sets it is plain that for for all , by the choice of .we shall need at several places a cut - off function which is equal to one in a neighbourhood of the set and vanishes outside another compact neighbourhood of .more precisely , we require that ( this is possible by the specifications made on ) and vanishes , if or .we also assume that , when .furthermore , denoting , it follows from the above specifications that , if ; in particular vanishes on the free water surface .finally , we shall need the scaled cut - off functions it is plain that also vanishes on for and that for , .the periodic water channel is defined by and it will be the main object of our investigation . the free surface of the channel is denoted by , and the wall and bottom part of the boundary is .the boundary of the isolated container , the periodicity cell , consists of the free surface and the wall and bottom with two apertures and .[ rem1.1 ] we shall use the following general notation . given a domain , the symbol stands for the natural scalar product in , and , , for the standard sobolev space of order on .the norm of a function belonging to a banach function space is denoted by . for and , ( respectively , ) stands for the euclidean ball ( resp .ball surface ) with centre and radius .by ( respectively , , , etc . )we mean positive constans ( resp .constants depending on a parameter ) which do not depend on functions or variables appearing in the inequalities , but which may still vary from place to place . the gradient and laplace operators and act in variable , unless otherwise indicated .  in the framework of the linear water - wave theory we consider the spectral steklov problem in the channel , here is the velocity potential , is a spectral parameter related to the frequency of harmonic oscillations and the acceleration of gravity . by the geometric assumptions made above, the outward normal derivative is defined almost everywhere on .it coincides with on the free surface  .the rest of this section is devoted to presenting the operator theoretic tools which will be needed later to prove our results : gelfand transform , variational formulation of the boundary value problems , and max - min - formulas for eigenvalues .the spectral problem ( [ problem])([b - cond ] ) can be transformed into a family of spectral problems in the periodicity cell using the gelfand transform .we briefly recall its definition : where on the left while and on the right .as is well known , the gelfand transform establishes an isometric isomorphism between the lebesgue spaces , where is the lebesgue space of functions with values in the banach space endowed with the norm the gelfand transform is also an isomorphism from the sobolev space onto for .the space consists of sobolev functions which satisfy the quasi - periodicity conditions whereas is the sobolev space with the condition only .  applying the gelfand transform to the differential equation and to the boundary conditions , we obtain a family of model problems in the periodicity cell parametrized by the dual variable , here , is a new notation for the spectral parameter .more details on the use of the gelfand - transform can be found e.g. in , section 2 .the apertures disappear at so in that case the also quasi - periodicity conditions cease to exist . hence , we can consider the problem as a singular perturbation of the limit spectral problem with as a spectral parameter .our approach to the spectral properties of model and limit problems is similar to , sections 1.2 , 1.3 .. we first write the variational form of the problem ( [ model])([quasiperiodic-2 ] ) for the unknown function as and the corresponding variational formulation of the limit problem for reads as we denote by the space endowed with the new scalar product and define a self - adjoint , positive and compact operator using the problem is then equivalent to the standard spectral problem with another spectral parameter clearly , the spectrum of consist of 0 and a decreasing sequence of eigenvalues , which moreover can be calculated from the usual min - max formula where the minimum is taken over all subspaces of co - dimension . using and , we can write a max - min formula for the eigenvalues of the problem : on the other hand , the connection and the properties of the sequence mean that the eigenvalues form an unbounded sequence the eigenfunctions can be assumed to form an orthonormal basis in the space . the functions are continuous and -periodic ( see for example , ch.9 ) . hence the sets are closed connected segments , which may degenerate into single points ; their relation to the original problem was already mentioned in .  the spectral concepts of the limit problem can be treated in the same way as in . since the quasi - periodicity conditions vanish for , the space is replaced by ; the norm induced by is now equivalent to the original sobolev norm of .we denote by the operator defined as in . the limit problem has an eigenvalue sequence like , however , neither the eigenvalues nor the operator depend on ( cf . , section 3 ) .the first eigenvalue equals , and the first eigenfunction is the constant function .analogously to we can write where again is running over all subspaces of codimension .we denote by an -orthonormal sequence of eigenfunctions corresponding to the eigenvalues .[ rem1.6 ] for all there exists a constant such that for all , ( and hence for all , ) .let for example ( the other case is treated similarly ) , and define the domains with boundary such that and still so small that as a consequence , these domains are smooth enough so that we can use the local elliptic estimates , theorem 15.2 , to the solutions of the equation : this yields for every , a constant such that for . applying this first with and we get a bound for and then , with and , for .the standard embeddings and imply the result .to describe the asymptotic behaviour ( as ) of the eigenvalues of the problem - we consider first the case is a simple eigenvalue of the problem - for some fixed .let us make the following _ ansatz _ : where is a correction term and a small remainder to be evaluated and estimated . in this sectionwe derive the expression for , cf .also and , and the remainder will be treated in section [ sec:4.2 ]  the corresponding asymptotic ansatz for the eigenfunction reads as follows : where is as in .the functions and are of boundary layer type , and is given above .the boundary layers depend on the `` fast '' variables ( `` stretched '' coordinates ) they are needed to compensate the fact that the leading term in the expansion does not satisfy the quasi - periodicity conditions . by lemma [ rem1.6 ] and the mean value theorem , the eigenfunction has the representation near the points .we look for and as the solutions of the problems and in the half spaces and , respectively ; the meaning of the numbers will be explained below . both of the functions can be extended to even harmonic functions in the exterior of the set : furthermore , the problem ( [ harmonic ] ) admits a solution ( see ) where is the 3-dimensional capacity of the set and concerns large -behaviour .moreover , the solution has a finite dirichlet integral : for some constant .we aim to choose the coefficients such that satisfies the quasi - periodicity conditions .clearly , for each  which together with the asymptotic expansion yield the relations for the coefficients .hence ,   now we can write a model problem for the main asymptotic correction term : where we denote in addition to , the problem will also determine the number in a unique way for every and .this will follow by requiring the solvability condition to hold in the fredholm alternative , see lemma [ lem2.1 ] and its proof , below . indeed , using the green formula and the normalization in we write ( is the surface measure ) : taking into account that the last integral converges absolutely and using the green formula again yield see and remark [ rem1.1 ] for notation . according to we finally obtain   [ lem2.1 ]choosing as in , the problem has a solution .the variational formulation of the problem reads as we remark that the function is harmonic in , and since equals constant one in a neighbourhood of , the function vanishes there , hence , and are smooth as well as uniformly bounded everywhere in .moreover , .  using the definition of the operator ( cf . , and the remarks above ) we can rewrite as follows : which means that must be a solution of the equation notice that is the solution of the homogeneous problem , so , by the fredholm alternative , is solvable , if and only if the right hand side of it is orthogonal to the function .this condition is satisfied by choosing as above , since by and .this last identity follows from the first green formula , because the normal derivative of vanishes on due to the properties of the function , see below .   in this section we complete the asymptotic analysis by studying the behaviour of eigenvalues in the case some has multiplicity greater than one : we have the ansatz is used again .furthermore , as in we denote by an orthonormal system of eigenfunctions associated with the eigenvalue . any eigenfunction corresponding to can be presented as a linear combination analogously to we introduce the asymptotic ansatz using the same argumentation as in the previous section we construct the boundary layers , , which satisfy the conditions here the coefficients come from the equations , where is replaced by .the main asymptotic term is also treated in the same way as in section [ sec:3.1 ] . to use the fredholm alternative for finding , , we write andmaking use of the green formula as above we get where hence , is an eigenvalue of the matrix .this matrix has rank one , because it can be represented in the form , where is a vector with components , .this means that [ th3.1 ] for any there are numbers and such that for every and any dual variable , the eigenvalues of the problem and the eigenvalues of the limit problem are related as follows :   as mentioned in the introduction ( see the explanations around and ) , this result implies the existence of any prescribed number of gaps in the essential spectrum , since also establishes an estimate for the endpoints of the intervals . to prove that resultone needs to take enough many distinct eigenvalues and a small enough ._ proof of theorem [ th3.1 ] ._ we apply the max - min - principle described in section [ sec:1.2 ] and first prove the estimate indeed , we recall that in the equations and both and are arbitrary subspaces of co - dimension .since , each is contained in some , and thus the infimum in is smaller than that in .so we turn to the upper estimate in and fix a .let the eigenfunctions be as in and let be a subspace spanned by the functions , where and and are as in .we remark that the functions satisfy the quasi - periodicity condition in the definition of the space , since vanishes in a neighbourhood of the apertures , see the remarks around .moreover , the sequence , , , is still linearly independent , due to the -orthogonality in and the fact that equals 1 in the set .hence , the dimension of is .  if is an arbitrary subspace of of co - dimension ( cf . ) , the intersection contains a non - trivial linear combination by the remarks just above we have hence , from and we infer that to evaluate the first term on the right hand side notice that the functions satisfy so that the -orthogonality of implies where the last inequality follows from and the fact that the eigenvalues are indexed in increasing order .furthermore , we use lemma [ rem1.6 ] as well as the facts that the supports of , , have measure of order , are of order , and to estimate where . combining this with and yields the result .     [ neareig ] let be a selfadjoint , positive , and compact operator in a hilbert space .if a number and an element satisfy and , then the segment $ ] contains at least one eigenvalue of .  to apply lemma [ neareig ] to the operator of, we fix an arbitrary and , keeping in mind the formula , define the approximate :th eigenvalue and eigenvector of by where  is as in , and are as in .     taking into account the definition of the norm of , the formula and the fact that the functions and vanish on we find that is bounded by the sum of the expressions   first we use the observation that the supports of the functions and are contained in balls of radius and that and are uniformly bounded in these balls ( lemma [ rem1.6 ] ) , hence there .so , can be bounded by a constant times   we estimate the terms using the fact that the support of the function is contained in a set for some constants ( see above ) , hence , by the estimate , applying yields finally , by lemma [ lem2.1 ] , belongs to the space . for the terms we thus get the bound since , due to the measure of the support of .    [ th4.5 ] for every there exists a constant such that , for each , ( cf . ) in the case the eigenvalue is simple and is given by the formulas in the case is a multiple eigenvalue .  we apply lemma [ neareig ] to the operator with and as in . our aim is to show that of the lemma can be chosen as small as .the lemma then gives an eigenvalue of with the estimate using and this turns into an eigenvalue ( of ) satisfying in the place of .however , if is small enough , theorem [ th3.1 ] guarantees that in a neighbourhood of there is only one eigenvalue of the model problem , namely .so must coincide with it , and the estimate follows .we are thus left with the task of proving to this end we write , using , , , , , the supremum is calculated here over all functions with unit norm .the expression can be represented as a sum of the terms where is given by     the first term in can also be treated using the cauchy - schwartz inequality , the properties of the cut - off functions ( once again as in ) and lemma [ lem2.1 ] the surface integral in can be estimated simply by the trace inequality .  to provide an upper bound for we notice that by , hence , using we can write after integrating by parts and taking into account that are harmonic functions we obtain   in the second term , the support of the function is contained in a set for some constants , hence , by the estimate , hence , the first term in is treated with a similar argument , since still contains the support of and the estimate again holds , by .[ cor4.6 ] let the index be such that the eigenvalue is simple .then , the band of the continuous spectrum of the problem has the asymptotic form \ ] ] where   returning to the band - gap structure of the bloch spectrum , in general it may happen that a spectral band , closed interval , degenerates into a single point .in this case the band consists of a single eigenvalue of infinite multiplicity , and the band is thus contained in the essential but not in the continuous spectrum .however , by theorem [ cor4.6 ] , if is such that both and are nonzero and is small enough , the numbers and are distinct , and in this case the spectral band is indeed an interval with positive length .this obviously provides a way to construct examples where the spectrum of the linear water - wave problem has a genuine band - gap structure with proper intervals as bands and with at least a given number of spectral gaps , cf .theorem [ cor3.2 ] .agmon , s. , douglis , a. , nirenberg , l. , estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions .i. , comm .pure appl .12 , 623727 , 1959 .mciver , p. , water - wave propagation through an infinite array of cylindrical structures .journal of fluid mechanics , 424 , 101 - 125 , 2000 .mckee , w.d . , the propagation of water waves along a channel of variable width , applied ocean research , 21 , 145 - 156 , 1999 .nazarov s.a ., elliptic boundary value problems with periodic coefficients in a cylinder , izv .nauk sssr .ser . mat .45 ( 1 ) 101 - 112 , 1981 .( english transl . : math .18 ( 1 ) , 89 - 98 , 1982 )  nazarov , s.a . ,properties of spectra of boundary value problems in cylindrical and quasicylindrical domains , sobolev spaces in mathematics , vol .ii ( mazya v. , ed . ) international mathematical series 9 , 261309 , 2008 .  nazarov s.a . , opening gaps in the spectrum of the water - wave problem in a periodic channel , zh .fiz . , 50 , 6 , 10921108 , 2010 ( english transl .. math . and math .physics 50 , 6 , 10381054 , 2010 ) .

Abstract: we consider the linear water - wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width . motivated by applications to surface wave propagation phenomena , we study the band - gap structure of the continuous spectrum . we show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as : the endpoints are determined within corrections of order . the width of the first bands is shown to be . finally , we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity .   _ chebyshev laboratory , st . petersburg state university , 14th line , 29b , saint petersburg , 199178 russia _  university of oulu , department of electrical and information engineering , mathematics division , p.o . box 4500 , fi-90401 oulu , finland  university of helsinki , department of mathematics and statistics , p.o . box 68 , fi-00014 helsinki , finland .


the advent of multicore architectures and the slowdown of the processor s operating frequency increase are signs that cmos miniaturization is increasingly hitting fundamental physical limits .a key question is how computing architectures will evolve as we reach these fundamental limits .a likely possibility within the realm of cmos technology is that the integration density will cease to increase at some point , instead only the number of components , i.e , the transistors , will further increase , which will necessarily lead to chips with a higher area .this trend can already be observed with multi - core architectures . that in itself has implications on the interconnect architecture , the power consumption and dissipation , and the reliability .another possibility is to go beyond silicon - based technology and to change the computing and manufacturing paradigms , by using for example bottom - up self - assembled devices .self - assembling nanowires or carbon nanotube electronics are promising candidates , although none of them has resulted in electronics that is able to compete with traditional cmos so far .what seems clear is that the current way with build computers and the way we algorithmically solve problems with them may need to be fundamentally revisited , which this paper is all about .while the top - down engineered cmos technology favors regular and locally interconnected structures , future bottom - up self - assembled devices tend to have irregular structures because of the current lack of precise control over these processes .we therefore hypothesize that future and emerging computing architectures will be much more driven by manufacturing constraints and particularities than for cmos , which allowed engineers to implement a logic - based computing architecture with extreme precision and reliability , at least in the past .independent of the forthcoming device and fabrication technologies , it is generally expected that future nanoscale devices will be built from ( 1 ) vast numbers of densely arranged devices that ( 2 ) exhibit high failure rates .we take this working hypothesis for granted in this paper and address it from a perspective that focuses on the interconnect topology .this is justified by the fact that the importance of interconnects on electronic chips has outrun the importance of transistors as a dominant factor of performance .the reasons are twofold : ( 1 ) the transistor switching speed for traditional silicon is much faster than the average wire delays and ( 2 ) the required chip area for interconnects has dramatically increased .  in , zhirnov et al .explored integrated digital _ cellular automata _ ( ca ) architectures which are highly regular structures with local interconnects ( see section [ sec : ca])as an alternative paradigms to the von neumann computer architecture for future and emerging information processing devices . here , we are interested to explore and assess a more general class of discrete dynamical systems , namely _ random boolean networks _ ( rbns ) and _ random threshold networks _ ( rtns ) .we will mainly focus on rbns , but rtns are included in this paper because they offer an alternative paradigm to boolean logic , which can be efficiently implemented as well ( see section [ sec : fab ] ) .motivated by future and emerging nanoscale devices , we are interested to provide answers to the following questions :  * do rbns and rtns offer benefits over ca - architectures ?if yes , what are they ? * how does the interconnect complexity compare between rbns / rtns and cas ? *does any of these architectures allow to solve problems more efficiently ? *is any of these architectures inherently more robust to simple errors ? * can cmos and beyond - cmos devices provide a benefit for the fabrication of any of these architectures ?we will argue and illustrate that at least from a theoretical perspective random dynamical networks offer superior properties over classical regular ca - based architectures , such as inherent robustness as the system scales up , more efficient information processing capabilities , and manufacturing benefits for bottom - up fabricated devices , which motivates this investigation .we will present recent results on the dynamic behavior and robustness of such random dynamical networks while also including manufacturing issues in the assessment .  to answer the above questions , we will extend recent results on the complex dynamical behavior of discrete random dynamical networks , their ability to solve problems , and novel interconnect paradigms .the remainder of this paper is as following : section [ sec : rdn ] introduces random dynamical networks , namely random boolean and random threshold networks . section [ sec : ca ] briefly presents cellular automata architectures .damage spreading and criticality of cellular automata and random dynamical networks is analyzed in section [ sec : damage ] .section [ sec : complex ] analyzes the network topologies from a graph - theoretical and wiring - cost perspective .the task solving capabilities of rbns and cas are briefly assessed in section [ sec : problem ] , while section [ sec : fab ] looks into manufacturing issues .section [ sec : conclusion ] concludes the paper .a _ random boolean network _ ( rbn ) is a discrete dynamical system composed of nodes , also called _ automata _ , _ elements _ or _ cells_. each automaton is a boolean variable with two possible states : , and the dynamics is such that    where , and each is represented by a look - up table of inputs randomly chosen from the set of nodes .initially , neighbors and a look - table are assigned to each node at random .note that ( i.e. , the fan - in ) can refer to the _ exact _ or to the _ average _ number of incoming connections per node .a node state is updated using its corresponding boolean function :    these boolean functions are commonly represented by _ lookup - tables _ ( luts ) , which associate a -bit output ( the node s future state ) to each possible -bit input configuration .the table s out - column is called the _ rule _ of the node .note that even though the luts of a rbn map well on an fpga or other memory - based architectures , the random interconnect in general does not .we randomly initialize the states of the nodes ( initial condition of the rbn ) .the nodes are updated synchronously using their corresponding boolean functions .other updating schemes exist , see for example for an overview .synchronous random boolean networks as introduced by kauffman are commonly called _ nk _ networks or models . figure [ fig : rbn ] shows a possible _ nk _ random boolean network representation ( ) .   nodes and inputs per node ( self - connections are allowed ) .the node rules are commonly represented by _ lookup - tables _ ( luts ) , which associate a -bit output ( the node s future state ) to each possible -bit input configuration .the table s out - column is commonly called the _ rule _ of the node . ]   _ random threshold networks _ ( rtns ) are another type of discrete dynamical systems .an rtn consists of randomly interconnected binary sites ( spins ) with states .for each site , its state at time is a function of the inputs it receives from other spins at time :    with   the network sites are updated synchronously . in the following , the threshold parameter is set to zero .the interaction weights take discrete values or with equal probability .if does not receive signals from , one has ._ cellular automata _ ( ca ) were originally conceived by ulam and von neumann in the 1940s to provide a formal framework for investigating the behavior of complex , extended systems .cas are a special case of the more general class of random dynamical networks , in which space and time are discrete .a ca usually consists of a -dimensional regular lattice of lattice sites , commonly called _nodes _ , _ cells _ , _ elements _ , or _automata_. each cell can be in one of a finite number of possible states and further consists of a transition function ( also called _ rule _ ) , which maps the neighboring states to the set of cell states .cas are called _ uniform _ if all cells contain the same rule , otherwise they are _ non - uniform_. each cell takes as input the states of the cells within some finite local neighborhood . here , we only consider non - uniform , two - dimensional ( ) , folded , and binary cas ( ) with a radius- von neumann neighborhood , where each cell is connected to each of its four immediate neighbors only .figure [ fig:2dca ] illustrates such an ca .the boolean functions in each node must therefore define possible input combinations . to be able to compare cas with rbns, we do not consider self - connections .each node is connected to its four immediate neighbors ( von neumann neighborhood ) . ]as we have seen in section [ sec : rbn ] , rbns and their complex dynamic behavior are essentially characterized by the average number of incoming links ( fan - in ) per node ( e.g. , figure [ fig : rbn ] shows a network with 3 incoming links per node ) .it turns out that in the thermodynamic limit , i.e. , , rbns exhibit a dynamical order - disorder transition at a sparse critical connectivity  ( i.e. , where each node receives on average two incoming connections from two randomly chosen other nodes ) , which partitions their operating space into 3 different regimes : ( 1 ) , sub - critical , where , ( 2 ) complex , where , and ( 3 ) supercritical , where . in the sub - critical regime ,the network dynamics are too `` rigid '' and the information processing capabilities are thus hindered , whereas in the supercritical regime , their behavior becomes chaotic .the complex regime is also commonly called the `` edge of chaos , '' because it represents the network connectivity where information processing is `` optimal '' and where a small number of stable attractors exist .similar observations were made for sparsely connected random threshold ( neural ) networks ( rtn ) for . for a finite system size ,the dynamics of both systems converge to periodic attractors after a finite number of updates . at , the phase space structure in terms of attractor periods , the number of different attractors and the distribution of basins of attraction is complex , showing many properties reminiscent of biological networks .[ [ results ] ] results + + + + + + +  in we have systematically studied and compared damage spreading ( i.e. , how a perturbed node - state influences the rest of the network nodes over time ) at the sparse percolation ( sp ) limit for random boolean and threshold networks with perturbations . in the sp limit , the damage induced in a network ( i.e. , by changing the state of a node ) does not scale with system size . obviously , this limit is relevant to information and damage propagation in many technological and natural networks , such as the internet , disease spreading in populations , failure propagation in power grids , and networks - on - chips .we measure the damage spreading by the following methodology : the state of one randomly chosen node is changed .the damage is measured as the hamming distance between a damaged and undamaged network instance after a large number of system updates .we have shown that there is a characteristic average connectivity for rbns and for rtns , where the damage spreading of a single one - bit perturbation of a network node remains constant as the system size scales up .figure [ fig : dofk_rbnrtn_f ] illustrates this newly discovered point for rbns and rtns . for more details , see .after 200 system updates , averaged over 10000 randomly generated networks for each value of , with 100 different random initial conditions and one - bit perturbed neighbor configurations for each network .for both rbn and rtn , all curves for different approximately intersect in a characteristic point .,scaledwidth=47.0% ]  [ [ discussion ] ] discussion + + + + + + + + + +  both and are highly relevant for nano - scale electronics for the following reason : assuming we can build massive numbers of simple logic gates that implement a random boolean function , the above findings tell us that on average , every gate should be connected somewhere close to both and in order to ( 1 ) guarantee optimal robustness against failures for any system size and ( 2 ) optimal information processing at the `` edge of chaos . ''we are also hypothesizing that natural systems , such as the brain or genetic regulatory networks , may have evolved towards these characteristic connectivities .this remains , however , to be proved and is part of ongoing research .we have used the same approach as described above to measure the damage spreading in cellular automata . in order to vary the average number of incoming links per cell in a cellular automata ( e.g. , as pictured in figure [ fig:2dca ] ) ,we have adopted the following methodology : ( 1 ) for a desired average number of links per cell for a given ca size of cells , the total number of links in the automaton is given by ; ( 2 ) we then randomly choose possible connections on the regular ca - grid with uniform probability and establish the links .damage is induced in the same way as for rbns and rtns : the state of one ( or several ) randomly chosen node(s ) is changed .the damage is measured as the hamming distance between a damaged and undamaged ca instance after a large number of system updates , in our case .[ [ results-1 ] ] results + + + + + + +  figures [ fig : damage_average_k_d1 ] , [ fig : damage_average_k_d10 ] , and [ fig : damage_average_k_d20 ] show the average damage of both rbns and cas for different system sizes and for a damage size of 1 and 10 respectively . we have left out rtns for this analysis . as one can see , both the rbn and the ca average damage for different approximately intersect in the characteristic point .this point is less pronounced for the larger damage sizes ( figures [ fig : damage_average_k_d10 ] and [ fig : damage_average_k_d20 ] ) .the rbn curves confirm what was already shown above in figure [ fig : dofk_rbnrtn_f ] , and are merely plotted here for comparison with the ca architectures and their system sizes imposed by square lattices .interestingly , the cas show different damage propagation behavior for different system sizes and connectivities .first , we observe that the average damage for one - bit damage events ( figure [ fig : damage_average_k_d1 ] ) is independent of the system size for up to approximatively average incoming connections per cell .this behavior disappears completely for large damage sizes ( figure [ fig : damage_average_k_d20 ] ) .second , figure [ fig : damage_average_k_d1 ] shows that all curves intersect at .third , figure [ fig : damage_average_k_d20 ] suggest that for larger damage sizes , disappears for cas .fourth , the average damage for larger damage events , i.e. , 10 and 20 in our examples , converges to the same final values for both rbns and cas as approaches 4 .[ [ discussion-1 ] ] discussion + + + + + + + + + +  we hypothesize that the particular behavior can be explained by the percolation limit of the cellular automata . da silva et al . found that the link probability at the percolation limit is approximatively , which means that the average connectivity at the percolation limit in our ca topology with a maximum of 4 neighbors is given by .this value corresponds to the experimentally observed value where the damage spreading suddenly becomes dependent of the system size . because of the local ca connectivity , there are lots of disconnected components below the percolation limit . below this limit, the damage spreading is thus very slow and limited by the disconnected components , reason why it is essentially independent of system size . above the percolation limit ,the ca suddenly becomes connected and damage spreading becomes therefore dependent on the system size . for larger damage events , such as 10 or 20 , damage becomes more dependent on system size even below the percolation limit because there is a higher probability that damage is induced in several disconnected components at the same time .  in summary: for single - node damage events , cas offer system - size independent damage spreading for up to about ( which corresponds to the percolation limit ) , however , this particular behavior disappears for larger damage events .we conclude that in the general case , cas do not possess a characteristic connectivity , where damage spreading is independent of the system size .such a connectivity , however , exists for both rbns and rtns , which makes them particularly suitable as a computing model in an environment with high error probabilities or systems with low system component reliabilities .an example are logical gates based on bio - molecular components , where high failure rates can be expected .after 200 system updates , averaged over 100 randomly generated networks for each value of , with 100 different random initial conditions and a damage size of 1 node for each network .see text for discussion.,scaledwidth=47.0% ]   after 200 system updates , averaged over 100 randomly generated networks for each value of , with 100 different random initial conditions and a damage size of 10 nodes for each network .see text for discussion.,scaledwidth=47.0% ]   after 200 system updates , averaged over 100 randomly generated networks for each value of , with 100 different random initial conditions and a damage size of 20 nodes for each network .see text for discussion.,scaledwidth=47.0% ]most real networks , such as brain networks , electronic circuits , the internet , and social networks share the so - called _ small - world _ ( sw ) property . compared to purely locally and regularly interconnected networks ( such as for example the ca interconnect of figure [ fig:2dca ] ) , small - world networks have a very short average distance ( measured as the number of edges to traverse ) between any pair of nodes , which makes them particularly interesting for efficient communication .  the classical watts - strogatz small - world network is built from a regular lattice with only nearest neighbor connections .every link is then rewired with a _ rewiring probability _ to a randomly chosen node .thus , by varying , one can obtain a fully regular ( ) and a fully random ( ) network topology .the rewiring procedure establishes `` shortcuts '' in the network , which significantly lower the average distance ( i.e. , the number of edges to traverse ) between any pair of nodes . in the original model ,the length distribution of the shortcuts is uniform since a node is chosen randomly . if the rewiring of the connections is done proportional to a power law , , where is the wire length , then we obtain a _ small - world power - law network_. the exponent affects the network s communication characteristics and navigability , which is better than in the uniformly generated small - world network .one can think of other distance - proportional distributions for the rewiring , such as for example a gaussian distribution , which has been found between certain layers of the rat s neocortical pyramidal neurons .  in a real network , it is fair to assume that local connections have a lower cost ( in terms of the associated wire - delay and the area required ) than long - distance connections .physically realizing small - world networks with uniformly distributed long - distance connections is thus not realistic and distance , i.e. , the wiring cost , needs to be taken into account , a perspective that recently gained increasing attention . on the other hand, a network s topology also directly affects how efficient problems can be solved .teuscher has pragmatically and experimentally investigated important design trade - offs and properties of an irregular , abstract , yet physically plausible 3d small - world interconnect fabric that is inspired by modern network - on - chip paradigms .the results confirm that ( 1 ) computation in irregular assemblies is a promising and disruptive computing paradigm for self - assembled nano - scale electronics and ( 2 ) that 3d small - world interconnect fabrics with a power - law decaying distribution of shortcut lengths are physically plausible and have major advantages over local 2d and 3d regular topologies , such as ca interconnects .[ [ discussion-2 ] ] discussion + + + + + + + + + +  there is a trade - off between ( 1 ) the physical realizability and ( 2 ) the communication characteristics for a network topology .a locally and regularly interconnected topology , such as that of a ca , is in general easy to build ( especially for to - down engineered cmos technology ) and only involves minimal wire and area cost ( as for example shown by zhirnov et al . ) , but it offers poor global communication characteristics and scales - up poorly with system size . on the other hand , a random topology , such as that of rbns or rtns , scales - up well and has a very short - average path length , but it is not physically plausible because it involves costly long - distance connections established independently of the euclidean distance between the nodes .the rbn and rtn topologies we consider here as thus extremes , such as ca topologies , the ideal lies in between : small - world topologies with a distance - dependent distribution of the connectivity .such topologies are located in a unique spot in the design space and also offer two other highly relevant properties : ( 1 ) efficient navigability and thus potentially efficient routing , and ( 2 ) robustness against random link removals . for these reasons, we can conclude that small - world graphs are the most promising interconnects for future massive scale devices .in , mesot and teuscher have presented a novel analytical approach to find the local rules of random boolean networks to solve the global density classification and the synchronization task which are well known benchmark tasks in the ca community from any initial configuration .they have also quantitatively and qualitatively compared the results with previously published work on cellular automata and have shown that randomly interconnected automata are computationally more efficient in solving these two global tasks .in addition , preliminary results by the authors also suggest that rbn generalize better on simple learning tasks than sub - critical or supercritical networks , but more research will be necessary .[ [ discussion-3 ] ] discussion + + + + + + + + + +  to efficiently solve algorithmic problems with distributed computing architectures , efficient communication is key .this is particularly true for tasks such as the density or the synchronization task , which are trivial to solve if one has a global view on the entire system state , but non - trivial to solve if each cell only sees a limited number of neighboring cells .it is thus not surprising that cells interconnected by a network with the small - world property perform much better on such tasks because the information propagation is significantly better .this is a too often neglected fact for cas , in particular if one wants to use them as a viable mainstream and general purpose computing architecture .it is well - know that even simple cas are computationally universal ( and so are rbns ) , i.e. , they can solve any algorithmic problem , but due to their local non - small - world interconnect topology , that will only be possible in a highly inefficient way in the general case , i.e. , for a large set of different applications .this is well illustrated with the ( highly inefficient ) implementation of a universal turing machine on top of the _ game of life _naturally , there are exceptions to the general case , and it has been shown that cas can be extremely efficient for certain niche applications , such as for examle image processing .as chen et al . state , `` [ i]n order to realize functional nano - electronic circuits , researchers need to solve three problems : invent a nanoscale device that switches an electric current on and off ; build a nanoscale circuit that controllably links very large numbers of these devices with each other and with external systems in order to perform memory and/or logic functions ; and design an architecture that allows the circuits to communicate with other systems and operate independently on their lower - level details . ''  while we can currently build switching devices in various technologies besides cmos ( see for an overview ) , one of the remaining challenges is to assemble and interconnect these switching devices ( or logic functions ) to larger systems , and ultimately to design a computing architecture that allows to perform reliable computations .as mentioned before , there is little consensus in the research community on what type of technology and computing architecture holds most promises for the future .the motivation for investigating randomly assembled interconnects and computing architectures can be summarized by the following observations :  * long - range and global connections are costly ( in terms of wire delay and of the chip area used ) and limit system performance ; * it is unclear whether a precisely regular and homogeneous arrangement of components is needed and possible on a multi - billion - component or even avogadro - scale assembly of nano - scale components * `` [ s]elf - assembly makes it relatively easy to form a random array of wires with randomly attached switches '' ; and * building a perfect system is very hard and expensive  we have hypothesized in and that bottom - up self - assembled electronics based on conductive nanowires or nanotubes can lead to the random interconnect topologies we are interested in , however , several questions remain open and are part of a 3-year interdisciplinary research project at lanl .our approach consists in using a hybrid assembly ( as others explore as well , e.g. , ) , where the functional building blocks will still be traditional silicon in a first step , while the interconnect is made up from self - assembled nanowires .nanowires can be grown in various ways using diverse materials , such as metals and semiconductors .we have chosen a novel way to grow conductive nanowires , which wang et al . at lanl have pioneered and demonstrated : ag nanowires can be fabricated on top of conducting polyaniline polymer membranes via a spontaneous electrodeless deposition ( self - assembly ) method .we hypothesize that this will allow to densely interconnect silicon components in a simple and cheap way with specific distance - dependent wire - length distributions .we believe that this approach will ultimately allow us to easily and cheaply fabricate rbn - like computing architectures .random threshold networks , on the other hand , could be rather straightforwardly and efficiently implemented with resonant tunneling diode ( rtd ) logic circuits ( see e.g. , ) , and represent a very interesting alternative to conventional boolean logic gates .the reported results in this paper on random threshold networks can thus directly be applied to the implementation of such devices .there has been a significant body of research in the area of threshold logic in the past ( see e.g. , ) , but to the best of our knowledge , random threshold networks have not been considered as computing models for future and emerging computing machines .the central claim of this paper is that locally interconnected computing architectures , such as cellular automata ( ca ) , are in general not appropriate models for large - scale and general - purpose computations .we have supported this claim with recent theoretical results on the complex dynamical behavior of discrete random dynamical networks , their robustness to damage events as the system scales up , their ability to efficiently solve tasks , and their improved transport characteristics due to the short average path length .the arguments , in a nutshell , why we believe that cas are _ not _ promising architectures for future information - processing devices , are as following :  * their local interconnect topology is not small - world and has thus worse global transport characteristics ( than small - world or random graphs ) , which directly affects the effectiveness of how general - purpose algorithmic tasks can be solved ; * in terms of a complex dynamical system , they operate in the supercritical regime ( ) with the widely used von neumann neighborhood , which makes them sensitive to initial conditions ; * they do not generally have a characteristic connectivity , where damage spreading is independent of system size , which makes a system inherently robust ; and * it is unclear whether a precisely regular and homogeneous arrangement of components is possible at the scale of future information processing devices .we have assessed rbns and rtns as alternative models , however , as we have seen in section [ sec : complex ] they come at a serious cost : the uniform probability to establish connections with any node in the system independent of the euclidean distance between them is not physically plausible and too expensive in terms of wiring cost .the ultimate interconnect topology is small - world and has a distance - dependent distribution of the wires .we have preliminary evidence that , if we were to connect rbns and rtns by such a network topology , both and would still exist .research to clarify this question is under progress ,   naturally , there are a number of open questions and issues that we have not addressed because they are beyond the scope of this paper .in particular , an irregular topology with random logical functions makes the mapping of a given digital circuit much harder , if not impossible in certain cases . on the other hand, a regular interconnect topology clearly makes the mapping task easier .we believe , however , that this challenge can be addressed by automated design tools .after all , computation in random assemblies is not completely new and has been more or less successfully tried by others , e.g. , , however in different contexts and with a different perspective in mind than we have presented here .we have deliberately _ not _ focused on any particular application in this paper because our results are independent of the application .however , it is noteworthy that locally interconnected cas have been proven to outperform other general purpose architecture on very specific applications .a good example are cellular neural networks ( cnns ) , which , e.g. , allow to perform certain imagine processing tasks orders of magnitude faster than any other machine .  further , it is unknown at this point how exactly our findings fit into the interconnect predictions made by rent s rule , however , the rule may not be applicable to our non - traditional circuits since it is based on empirical results. further research on this is planned .last but not least , we would like to mention that , although we have only considered 2d arrangements and interconnects here for simplicity , the future is clearly 3d ( e.g. , see ) .the main reason is that the average wire length in 3d is shorter than in 2d interconnects .we believe that computation in random self - assemblies of simple components and interconnections is a highly appealing paradigm , both from the perspective of fabrication as well as performance and robustness .future work will focus on ( 1 ) the manufacturing issues , ( 2 ) appropriate design methodologies , ( 3 ) addressing the mapping issues , and ( 4 ) more realistic models , which will allow to better assess the performance and cost , and ( 5 ) specific applications .j. a. davis , r. venkatesan , a. kaloyeros , m. beylansky , s. j. souri , k. banerjee , k. c. saraswat , a. rahman , r. reif , and j. d. meindl .interconnect limits on gigascale integration ( gsi ) in the century . , 89(3):305324 , 2001 .c. gershenson .classification of random boolean networks . in r.k. standish , m. a. bedau , and h. a. abbass , editors , _ artificial life viii .proceedings of the eight international conference on artificial life _ , complex adaptive systems series , pages 18 . a bradford book , mit press , cambridge , ma , 2003 .c. teuscher , n. gulbahce , and t. rohlf .learning and generalization in random boolean networks . in _ dynamic days 2007 : international conference on chaos and nonlinear dynamics _ , boston , ma , jan 36 2007 .

Abstract: independent of the technology , it is generally expected that future nanoscale devices will be built from vast numbers of densely arranged devices that exhibit high failure rates . other than that , there is little consensus on what type of technology and computing architecture holds most promises to go far beyond today s top - down engineered silicon devices . cellular automata ( ca ) have been proposed in the past as a possible class of architectures to the von neumann computing architecture , which is not generally well suited for future massively parallel and fine - grained nanoscale electronics . while the top - down engineered semi - conducting technology favors regular and locally interconnected structures , future bottom - up self - assembled devices tend to have irregular structures because of the current lack of precise control over these processes . in this paper , we will assess random dynamical networks , namely random boolean networks ( rbns ) and random threshold networks ( rtns ) , as alternative computing architectures and models for future information processing devices . we will illustrate that from a theoretical perspective  they offer superior properties over classical ca - based architectures , such as inherent robustness as the system scales up , more efficient information processing capabilities , and manufacturing benefits for bottom - up designed devices , which motivates this investigation . we will present recent results on the dynamic behavior and robustness of such random dynamical networks while also including manufacturing issues in the assessment .


little research has been conducted into how haptic and auditory modalities can best be combined in human - computer interaction . however , in the non - computing world the two modalities have been partners for a long time .first hollywood gave us _movies _ ( the visual display in computing terms ) . then came the _ talkies _ ( and auditory display in computing ) . in_ brave new world _ ( 1932 ) aldous huxley offered us the notion of the _ feelies _ the arm rests in theatre seats would provide haptic stimulation during erotic features . whilst the feelies are not with us yet haptic displays are making inroads into the development of human - computer interfaces . as the auditory display community has discoveredthere are several hurdles that must be jumped before a new interaction modality is considered acceptable .haptic interaction also raises its own usability issues which are being dealt with by researchers .however , the combined use of auditory display and haptic input and output in a single application raises a new set of design challenges .the field is , perhaps , in an analogous position to that of cinema in the late 1920s .early film sound was largely causal and was thus a recording of the sonic events in the scene .the use of sound as a separate non - synchronous entity was advocated by russian filmmaker sergei eisenstein whose 1928 _ statement on sound _ suggested using sound as a counterpoint to the visuals .eisenstein wanted to make montages of sound just as he used visual montage to great effect in his earlier silent films .the french filmmaker ren clair was an early promoter of non - synchronous sound which has today developed into the disciplines of voice - overs / narration , sound effects & foley art , and film music .these early pioneers helped to make film a truly bimodal communication channel .today , we are now beginning to move away from direct causal haptics and sonification in which the touch and auditory modes are used to accentuate visual information and are seeing new true multi - modal interfaces in which the different senses are used for separate but complementary information streams .the larger questions of sonification design are concerned with issues of intrusiveness , distraction , listener fatigue , annoyance , display resolution and precision , comprehensibility of the sonification , and , perhaps binding all these together , sonification sthetics .there is a tension in auditory displays between the sonification being perceptible to its intended audience and being too intrusive or annoying . in their work on awareness support systems , hudson andsmith articulated the problem of intrusion in terms of awareness and privacy .they stated that this `` _ _ dual tradeoff is between privacy and awareness , and between awareness and disturbance _ _ '' .the more information an auditory display provides the richer the sonification yet the greater the potential for disturbance , annoyance , and an upset in the balance of the acoustic ecology .gutwin and greenberg claimed sonification is a tradeoff between being well informed and being distracted .kilander and lnnqvist noted the effect of such sonifications on people sharing the workspace:``__in a shared environment , one recipient may listen with interest while others find themselves exposed to an incomprehensible noise _ _ '' .indeed , commenting upon the design of their _ nomadic radio _ system , sawhney and schmandt cautioned that care must be taken to ensure that the auditory display intrudes minimally on the user s social and physical environment .in dealing with intrusiveness , pedersen and sokoler framed the problem as a balance between putting a low demand on attention versus conveying sufficient information .they studied this problem through an ` _ _ ecology of awareness _ _ ' thus acknowledging the importance of the acoustic ecology of a sonification .pedersen and sokola made the auditory , visual , and haptic representations of their aroma system highly abstract abstraction would allow useful information to be communicated without divulging too many details that would violate privacy .it was hoped that abstract representations would be better at providing `` _ _ peripheral non - attention demanding awareness _ _ '' .it was also noted that such abstract representations lend themselves to being remapped to other media ( what somers would call _ semiotic transformation _ similar to eisenstein s _ transference _ ) , or , in turn foster the accommodation of user preferences ( an important aspect of _ sthetic computing _ see fishwick ) .unfortunately , user studies showed that the abstraction led to users interpreting the representations in varied ways that were not always correct .furthermore , kilander and lnnqvist  warned that the `` _ _ monitoring of mechanical activities such as network or server performance easily runs the risk of being monotonous _ _ '' a finding observed by pedersen and sokola who reported that they soon grew tired of the highly abstract representations used in aroma .it is interesting that some of the blame was attributed to an impoverished sthetic , the feeling being that involving expertise from the appropriate artistic communities would improve this aspect of the work .cohen identified a general objection to using audio : people in shared office environments do not want more noise to distract them .buxton argued that as audio is ubiquitous it would be less annoying if people had more control over it in their environments .lessons from acoustic ecology would be helpful here .   the term _ acoustic ecology _ comes from work begun by r. murray schafer in the 1960s as part of his _ world soundscape project _ at simon fraser university ( see ) .schafer sees the world around us as containing ecologies of sounds .each soundscape possesses its own ecology , and sounds from outside the soundscape are noticeable as not belonging to the ecology . in schafer s worldviewwe are exhorted to treat the environments in which we find ourselves as musical compositions . by thiswe are transformed from being mere hearers of sound into active and analytic listeners exactly the characteristic needed to benefit most from an auditory display .when the environment produces noises that result from data and events in the environment ( or some system of interest ) we are able to monitor by listening rather than just viewing . in regard to auditory display the term acoustic ecology means the internal ecology of the various sounds within the sonification .that is , we treat the sonification both as a real - world soundscape in its own right , the acoustic ecology of which is jumbled , and as part of the wider real - world soundscape in which it is situated .again , its sonic components may sit uneasily within the acoustic ecology of the host soundscape .cohen defined an acoustic ecology as `` _ _ a seamless and information - rich , yet unobtrusive , audio environment _ _ '' .kilander and lnnqvist  tackled this problem in their fuseone and fusetwo environments with the notion of a _ weakly intrusive ambient soundscape _ , or wisp . in this approachthe sound cues for environmental and process data are subtle and minimally - intrusive .minimal- or weak - intrusion is achieved in kilander and lnnqvist s scheme by drawing upon the listener s expectation , anticipation , and perception ; anticipated sounds , say kilander and lnnqvist , slip from our attention .for example , a ticking clock would be readily perceived and attended to when its sound is introduced into the environment ( assuming it is not masked by another sound ) . as the steady - state of the ticking continues and the listener expects or anticipates its presence the perceived importance drops and the sound fades from our attention ) .however , a change in the speed , timbre , or intensity of the clock tick would quickly bring it back to the attention of the listener .intrusiveness can thus be kept to a minimum by using and modulating sounds that fit well with the acoustic ecology of the sonification s environment .the sonification is discriminable from other environmental sounds ( either by deliberate attentiveness on the part of the listener , or by system changes to the sounds ) yet is sufficiently subtle so as not to distract from other tasks that the listener ( and others in the environment ) may be carrying out .to increase the quality of the acoustic ecology further , kilander and lnnqvist used real - world sounds rather than synthesized noises and musical tones .they concluded that `` _ _ easily recognisable and natural sounds _ [ stand ] _ the greatest chance of being accepted as a part of the environment . in particular , a continuous background murmur is probably more easily ignored than a singular sound , and it also continuously reassures the listener that it is operative _ _ '' .the audibility of sonifications is an important factor and is tightly coupled to the issue of intrusiveness .the comprehensibility of sonifications depends on many factors including the production quality of the sounds , the quality of the playback system , and cultural and metaphoric associations .many data require metaphoric or analogic mappings for audio representation as they do not naturally possess their own sound. the choice of metaphor may determine how learnable and comprehensible the mapping is .for example , kilander and lnnqvist found that the sound of a golf ball dropping into a cup was difficult for listeners to recognize `` _ _ except possibly for avid golfers _ _ '' whilst the sound of a car engine was easy to identify .this highlights the fact that when using real - world sounds it is important to assess the cultural attributes of those sounds . investigating musical tones for the monitoring of background processessrsen found that sudden onset or disappearance of a timbre is easier to detect than changes in the rhythm and melody of that timbre .he concluded : `` _ _ changes within one single instrument should be very carefully designed to represent non - binary changes in state or modus _ _ '' .vickers and hogg argued that the sthetics and composition approaches of electroacoustic and musique concrte may potentially lead to great success in sonification design given their dependence upon the gestural encoding present in sounds . according to smalley s spectro - morphological classification we hear the physical , gestural qualities in sounds , and these in and of themselves ( though usually in combination with timbre and volume ) carry sufficient information regarding movement , atmosphere , size , material quality , and so forth to offer information to the perceiver that serves to generate meaning akin to that generated by musical harmonic / tonal systems .it has the added advantage of being arguably less culture specific , that is it is not classical , or pop , or anything we already recognize it is rather a system that is more open to reading than it is a musical style that is recognised as such .criticisms of cultural imperialism are often raised when sonifications based upon tonal ( or even atonal or schoenbergian serialist ) structures are presented at conferences .the potential offered by electroacoustic musical forms to avoid cultural stereotyping means much more serious consideration should be given to their use in auditory display research .  to help make the link between auditory display and electroacoustic music clearer ,let us consider emmerson s _ language grid _ in which he classifies electroacoustic music into a nine - sector space on two axes : the level of syntactical abstraction of the music and the use of mimetic reference vs. aural discourse ( see fig .[ fig : grid ] ) .mimetic sound imitates or represents nature and aspects of human culture .    in the language grid pitch - oriented music ( both tonal and atonal )occupies grid sector 1 ( abstract musical syntax with aural discourse dominant ) .we have shaded positions 7 , 8 , and 9 because they bound the region occupied by sonification .sonifications are by definition mimetic in that their goal is to represent objects , events , or data of the domain in question .some sonifications ( sector 7 ) are grounded in the tonal music / melodic paradigm and so use _ abstract _ syntax ( e.g. vickers and alty s caitlin program sonification system whilst others ( sector 9 ) rely more on sounds _ abstracted _ from real objects ( e.g. auditory icons . in region 8 are those sonifications that use both abstract and abstracted syntax ( such as barra et al.s webmelody ) .sonification , then is concerned with mimetic discourse along the abstract - syntax / abstracted - syntax dimension .vickers and hogg introduced the idea of _ indexicality _ to sonification discourse .indexicality is associated with mimesis as it is a measure of how strongly a sound sounds like the thing that made it . in sonification practice indexicalityis related to whether sonifications make more use of direct data - to - sound mappings ( high indexicality the sound is derived directly from the data ) or more use of metaphoric or interpretive mappings ( low indexicality ) . auditory seismograms are an example of the former in which seismographic data were scaled and frequency - shifted until they lay in the human audible range .vickers and alty s program sonifications are metaphoric : tonal musical motifs were used to stand for data and objects . from this, we can adapt emmerson s language grid to the sonification domain to give fig .[ fig : indexgrid ] in which emmerson s abstraction dimension has been retained but only the mimetic sectors have been carried over .in addition , the polarity of indexicality is indicated .note that sonifications relying predominantly on abstract syntax ( e.g. vickers and alty s program sonifications ) possess lower indexicality than those making use of predominantly abstracted syntax ( e.g. hayward s seismograms ) .    there is a mapping , then , between sonifications and music compositions in that direct sonifications and concrete music possess high indexicality and metaphorical sonifications and abstract music possess low indexicality .that is , ` direct ' ( abstracted syntax ) and ` metaphorical ' ( abstract syntax ) in the sonification domain map to ` concrete ' and ` abstract ' respectively in the music domain .musicologists would argue that music is as much a construct of the listener s mind as of the composer s ; if the listener perceives something as music then it _ is _ music ( though smalley claims that the listener must `` _ _ discover a perceptual affinity with its materials and structure _ _ '' in order for this to happen ) .thus , vickers and hogg offer us :    kramer notes the similarity in structure between sonification and music creation : sonification renders data in sound to allow a human listener to detect and comprehend patterns and structures in that data , whilst a musician renders a musical score so as to make it audible and thus make perceptible the music s structure and even give clues as to the composer s and the musician s emotional states .thus , a piano is a sophisticated auditory display machine : through the agency of the musician , the piano renders in sound ( albeit in a highly complex and abstract way ) the score , the technique of the musician , the physics of the piano , the emotional state of the musician and the composer , and even the musician s response to the feedback loop offered by his own ears .this gives us :    thus , vickers and hogg argue that sonification and music are mutually implicated and thus we get the logical biconditional :    that is , if something is music then it is also a sonification and vice versa . as an illustration of the link between music and sonificationconsider barra et al s webmelody system .drawing on the ideas of futurist composer luigi russolo ( 1885 - 1947 ) , principles of pierre schaeffer s musique concrte , and inspired by edgard varse s _ pome lectronique _ ( 1958 ) and john cage s aleatoric compositions ( e.g. _ music of changes _ ( 1951 ) ) , barra et al tried to construct sonifications for monitoring a web server that were `` _ _ neutral with respect to the usual and conventional musical themes . _ _ '' they attempted to move away from the idioms of tonal and atonal ( serialist ) music and towards the more concrete compositions found in the musique concrte and electroacoustic traditions .one device employed very successfully by webmelody ( and many other sonification systems ) is _acousmatic sound_. the term was introduced by pythagoras who reputedly taught his students whilst standing behind a screen .acousmatic sound , then , is that which one hears without the originating cause being visible to the listener .auditory display research is replete with acousmatic sound as sonifications are often designed to highlight unseen or hidden data or events .we may also use haptic feedback to communictate information about unseen ( or unheard ) objects , data , or events ( e.g. notification of incoming email ) . asthis is an analogue of acousmatic sound we shall call it _ haptmatic _ sensation .an example of an existing haptmatic display technique is brewster and brown s _ tacton _ a haptic icon ( c.f .auditory icons and earcons ) .in fact , there is a very direct relationship between sonification and the kind of haptic feedback found in tactons and other similar approaches .sound is simply a transverse wave with properties of frequency , phase , timbre , and amplitude .the vibro - tactile display offered by the tacton possesses similar properties indeed , if a tacton emitter is placed on a resonant surface it becomes a loud speaker .sound and touch are very closely related in our everyday experience , especially in the region of low frequency sounds which are easily conducted through floors and other surfaces . at night clubswe can feel the bass frequencies in our chest ; on the street we can feel heavy lorries approaching .the relationship between touch and sound is not only in this direction . in the next sectionwe look at the relationship between haptic ( gestural ) input and sonification .smalley s classification of sounds via spectro - morphology provides us with an approach to sound and musical structures that focuses on the spectrum of available pitches and frequencies and how they are shaped ( morphed ) over time . in smalley s modelevery sound contains gestural information that codes for the identity of movement , atmosphere , size , material quality , and so forth of the sound object in question .we are used to listening to sounds and decoding such gestural information .we can tell how fast a car is approaching us ; we can estimate how hard a drum was hit and what size the drum is .indeed , there is a very strong association between gesture and interpretation .this is especially true of music performance in which very complex and subtle gestural control is used to shape the music we hear and experience .one of the challenges faced by audiences of electroacoustic music concerts is the lack of visible gestural interaction between the musician and the instrument .when the instrument is tape decks and laptop computers the familiar frame of reference of the direct relationship between a performer bowing strings , striking keys , moving sliders , and banging drums and the resultant sound is lost .it is replaced by a visible gesture set that often , at best , has a seemingly indirect relationship to the sonic experience and , at worst , no perceptual link at all ( a reiteration of eisenstein s contrapuntal sound , perhaps ) .the ` traditional ' musical instrument has been with us so long that it has become the natural mechanism for making music and we are well attuned to watching musicians physically play their instruments .we note that for every input gesture there is a corresponding output sound .the expert listener and observer notes the subtlest of gestures but even the neophyte can observe the link between gross gestures ( such as strumming a guitar ) and the overall sonic output .how , then , does this inform us about the haptic control of interfaces that also use sonification ?spectro - morphology tells us that we listen for and can identify gestures in a sound .experience shows us that we are used to associating physical gestures with consequent sound .if we are so sensitive to gesture , it might be that the haptic gestures become associated with the system s auditory output whether they are directly related or not .might the subtle gestures needed for fine control of systems such as sensable s phantom be translated in the mind of the listener as being somehow related to any auditory display ?we have shown above vickers and hogg s assertion that sonification and music are in a mutual implication relationship .if a sonified interface is thus a music playing system and the interface also provides gestural input in the form of haptic control , we suggest :  auditory display + haptic input = musical instrument  that is , because of the ubiquitous frame of reference from musical instruments in which input gestures result in musical output we argue that a device / application interface that combines auditory display with haptic input might be viewed or interpreted as a musical instrument in the _ mind _ of the user .it does not necessarily matter that the sonification produced by the system is not related either directly or indirectly to the haptic input ( though a lack of even an indirect relationship is questionable ) as one would still naturally seek links between the gestural inputs and the sonic outputs . where there is no direct relationship it becomes quite possible that artificial cause - effect relationships will be constructed in the user s mind which may cause usability problems .a user moves a lever , grabs a virtual object , or makes a circular gesture to a mobile phone s camera , and the system happens to emit a sonified data stream immediately afterwards , it is very possible that a causal link will be established in the user s mind even if no link is present .   in other words ,a musical instrument is a system in which manual gestures result ( however visibly indirectly ) in musical output .a multimodal system that combines haptic input with auditory display may thus be considered to be a musical instrument .if causal relationships between the haptic control and the auditory display are intended , then system designers might find it useful to draw lessons from the musical instrument design community for ways of improving the interaction . if causal relationships between gestures and audio are not intended or desired ( eisenstein again ) , then interface designers also need to be aware of the dynamics of instrument design in order to try to ` design out ' any user perceptions of causality that arise through familiarity with the musical instrument paradigm .where systems combine auditory display with haptic output , other possible interactions between the two modalities arise .we commented above about the relationship between sound and touch , how certain sounds also generate vibrations that are perceived physically .again , the question arises whether causal relationships between the haptic and auditpory outputs are inferred by the user .where no causal relationship exists such inference would likely be detrimental to the user s interpretation of the system s outputs .again , the system might be considered a musical instrument .researchers who have reported the most success with their sonifications also tended to deal directly with the issue of the sthetics and acoustic ecology of their sonifications . as the role of sthetics is increasingly entering the consciousness of designers of computing systems ( e.g. ,see fishwick ) so it needs to inform the work of the auditory display community .it has been proposed that sonifications be viewed as works of musical art as they could then benefit from the application of the sthetic practices employed by artists . when haptics are added to the mixthere is a great potential for causal associations between the sound and the touch to be created in the user s mind .what is needed then , is research that explores the cognitive and artistic issues surrounding our haptic - sonification musical instruments . in what waysdoes haptic control affect the way we perceive an auditory display ?if force feedback and sonification are used in tandem , does the brain automatically assert a link between the two data streams even when none is present ? in the world of television and the movies , it is usually the case the the soundtrack affects the perceived meaning of the visual track and not the other way around .will sound also have such an influence on our perception of force feedback ?it would be instructive to look at the semiotics of haptic - auditory interfaces .whilst much has been written about the semiotics of visual and auditory messages less attention has been paid by semioticians to touch . however , as musicologists and designers of new interfaces for musical expression are very interested in the relationships between physical gesture and sound with the machine as intermediary , it may benefit the hci community to temporarily cast their multimodal interfaces as musical instruments and see what design lessons can be learnt by studying these systems as a music researcher might . _ speech act theory _ studies how people use language ( rather than establishing the truth - value of statements ) and offers three dimensions of act : _ locutory _ , _ illocutory _ , and _ perlocutory_. the locutory dimension deals with material aspects of an act s generation ( e.g. strength of a physical movement ) .the illocutory aspects are to do with the intention behind an act .the percolutory dimension deals with the effect an act has upon the receiver . by drawing together the skills of the hci practitioner , the music researcher , and the semiotician ,we may be better placed to understand the locutory , illocutory , and perlocutory nature of haptic and auditory signals and thus able to explore the rich interactions and the acousmatic and haptmatic effects that will result in the new generation of multimodal systems .10 eisenstein , s. , pudovkin , v. , alexandrov , g. : statement on sound .in taylor , r. , christie , i. , eds . : the film factory : russian and soviet cinema in documents , 1896 - 1939 . harvard university press , cambridge , massachusetts ( 1988 ) 234235  hudson , s.e . , smith , i. : techniques for addressing fundamental privacy and disruption tradeoffs in awareness support systems . in : cscw 96 : proceedings of the 1996 acm conference on computer supported cooperative work , new york , ny , usa , acm press ( 1996 ) 248257  gutwin , c. , greenberg , s. : support for group awareness in real time desktop conferences . in : proceedings of the second new zealand computer science research students conference , hamilton , new zealand , university of waikato ( 1995 )   sawhney , n. , schmandt , c. : nomadic radio : scaleable and contextual notification for wearable audio messaging . in : proceedings of the chi 99 conference on human factors in computing systems : the chi is the limit , new york , ny , acm press ( 1999 ) 96103   pedersen , e.r . ,sokoler , t. : aroma : abstract representation of presence supporting mutual awareness . in : chi 97 : proceedings of the sigchi conference on human factors in computing systems , new york , ny , usa , acm press ( 1997 ) 5158  somers , e. : a pedagogy of creative thinking based on sonification of visual structures and visualization of aural structures . in brewster ,s.a . , edwards , a.d.n . , eds . :icad 98 fifth international conference on auditory display .electronic workshops in computing , glasgow , british computer society ( 1998 )   pedersen , e.r . : people presence or room activity supporting peripheral awareness over distance . in : chi 98 : chi 98 conference summary on human factors in computing systems , new york , ny , usa , acm press ( 1998 ) 283284  cohen , j. : monitoring background activities . in kramer , g. , ed . : auditory display .volume xviii of santa fe institute , studies in the sciences of complexity proceedings .addison - 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visual information display . in : proceedings of australasian user interface conference , dunedin , new zealand , austalian computer society ( 2004 ) 1523  mcgee , k. , harup , a. : contact expressions for touching technologies . in mitchell , g. , ed .computational semiotics in games and new media , university of teesside , uk , university of teesside ( 2003 ) 6876  magnusson , t. : screen - based musical interfaces as semiotic machines . in schnell , n. , bevilacqua , f. , lyons , m. , tanaka , a. , eds . : proceedings of the 2006 international conference on new interfaces for musical expression ( nime06 ) , paris , france , ircam centre pompidou ( 2006 ) 162167

Abstract: in this paper we look at some of the design issues that affect the success of multimodal displays that combine acoustic and haptic modalities . first , issues affecting successful sonification design are explored and suggestions are made about how the language of electroacoustic music can assist . next , haptic interaction is introduced in the light of this discussion , particularly focusing on the roles of gesture and mimesis . finally , some observations are made regarding some of the issues that arise when the haptic and acoustic modalities are combined in the interface . this paper looks at examples of where auditory and haptic interaction have been successfully combined beyond the strict confines of the human - computer application interface ( musical instruments in particular ) and discusses lessons that may be drawn from these domains and applied to the world of multimodal human - computer interaction . the argument is made that combined haptic - auditory interaction schemes can be thought of as musical instruments and some of the possible ramifications of this are raised .