Complexity-Seeking Ants Howard Gutowitz ESPCI 10 rue Vauquelin 75005 Paris, France
[email protected]
November 10, 1994 Abstract
Deneubourg et al. [1] introduced a model of sorting behavior in ants. They found that simple model ants were able to sort into piles objects initially strewn randomly across the plane. The model is in qualitative agreement with the behavior of real ants. The model ants operate according to local strategic rules and possess only local perceptual capacities. Nonetheless, they are able to impose global order. What is the mechanism underlying this phenomenon? We hypothesize that it is the combination of two processes, one which decreases smallscale complexity, and another which couples small-scale decreases in complexity to larger scales. To test this hypothesis, we introduce variant ants which have a complexity-seeking strategy. These ants can "see" local complexity, and tend to perform actions (picking up and putting down objects) in regions of highest local complexity. Using this strategy, they are able to accomplish their task more eciently than the basic ants. For both basic and complexity-seeking ants, we nd that complexity-reduction begins at the nest scale and propagates out to ever-increasing scales. We quantify the eciency of various ant sorting strategies in terms of the amount of energy (in the form of random strategic choices) the ants require in order to e ect the reduction of global spatial entropy.
Submitted to ECAL '93, Brussels
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1 Introduction Organisms can collect energy and focus it on the reduction of entropy in their environment. We study this organizing capability of living things in the context of a simple model of collective behavior in ants introduced by Deneubourg et al. [1]. The ants of Deneubourg et al. are able to impose order in their environment using most rudimentary perceptual, motor, and strategic mechanisms. We will examine a number of variants to Deneubourg et al.'s basic ants. We shall see that some of these variants succeed better at their task than the basic ants do. However, our ultimate goal lies beyond the engineering of better ants, in the direction of connecting local and global criteria for the evaluation of autonomous-agent strategies (for some related work see [2] and [3]). The global measure we will use to evaluate the success of sorting strategies is a standard statistical-mechanical measure: the coarse-grained spatial entropy of the environment. As the ants can only perform local computations, they themselves cannot calculate the spatial entropy of their environment. They are unable to know how much contribution they make to the collective task. We will therefor attempt to build a local measure of success from quantities the ants themselves can perceive, quantities related to their own internal functions, and local e ects they can make in their environment. Though we will not directly consider learning in ants, it is clear that some such local measure will be required as the basis of any learning algorithm which is to operate at the individual, rather than the population level. We will discuss a number of simple local measures of sorting success. These measures derive from a particular philosophical stance regarding the algorithmic modeling of animal behavior. In brief, this is the view than any environment can be considered as a combination of rigid, predictable aspects with uid, unpredictable aspects. An ideally omniscient organism is one which is able to perceive and evaluate fully and correctly the predictable aspects its environment. A perfectly adapted organism is one which executes stereotyped strategies in the face of challenges presented by predictable aspects of its environment and reverts to more experimental, thought- and risk-intensive strategies in the face of challenges presented by unpredictable aspects of its environment. The content of this view is a measure of the quality of simulated autonomous agents. A simulated autonomous agent has some collection of determin2
istic strategies which may be comprised of one or more logical steps. Whether or not a given deterministic strategy is executed at a given instant in time depends on 1) the value of perceptual inputs, and 2) the value of some internally generated random bit or bits. The more inputs of type 1) are involved, the more re exive in nature is the action. The more inputs of type 2) are involved, the more the action can be characterized as " uid" or "re ective". The generation of a random bit represents an attempt to step outside of the agents rigid representation of the world contained in the structure of its set of stereotyped strategies. This attempt costs, and should be avoided if possible. A well-adapted agent will take such re ective steps if and only if required. Hence, we can quantify the relative adaptedness of an agent by measuring the number of random bits it generates in the course of accomplishing a unit of its task, and compare this measure with the degree of organization of the environment. This measure is especially cogent in the context of sorting behavior. In short, the task consists of eliminating entropy pumped into an environment in the form of objects randomly distributed on a square planar grid. The ants reduce this entropy by packing objects close together in space. Simple, effective, deterministic sorting strategies are easy to design for ants with global perceptual capabilities. However, if ants have only local perceptual abilities, they are capable mearly of estimating relevant environmental parameters, and must make some random choices according to these estimates. The uncertainty of the estimates is re ected in the number of random bits the ants use to act upon the estimates. Here we study a situation in which the perceptual power of an ant relates in a quantitative and clear way with the randomness of its actions.
1.1 The Basic Ants
We will refer to the model ants introduced by Deneubourg et al. as basic ants. Basic ants have 1) a nite memory, 2) an object-manipulation capacity, the capacity pick up or put down objects, 3) a function which gives the probability to manipulate an object as a function of the values in memory and a random variable, and 4) the capacity to execute a brownian motion. The memory is simply a register of length in which is recorded the presence or absence of objects at the ant's previous locations. Throughout the present work is set to 15. At each time step a basic ant generates a random number n
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between 0 and 1. It manipulates objects as a function of and a threshold calculated according to the following functions: p
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) = ( + + )2 (1) (2) ( ) = ( ? + )2 where is the fraction of memory locations registering the presence of an object (an estimate of the local density of objects), and + and ? are parameters, set by Deneubourg et al. to 0.3 and 0.1 respectively. ( ), ( ) are computed if the ant does, respectively does not currently hold an object. If exceeds then the action is taken, otherwise it is not. In the work of Deneubourg et al. the basic ants are used to collect objects strewn on a square grid into piles. These piles are such that only one object can occupy a given lattice site. A close variant of the basic ants is used to sort two di erent types of objects into piles of objects of a single type. In this paper we will only consider the collection of objects into piles without respect to type. For continuity with previous work, we will refer to this process as sorting. k
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2 Perception of Local Complexity The ability of basic ants to reduce global environmental entropy can be traced to their primitive capacity to perceive complexity locally and behave accordingly. Complexity and entropy are closely related concepts. While entropy can be reliably measured on a global scale, it is dicult to nd an adequate measure of entropy on a local scale. In this section we assert that the basic ants of Deneubourg et al. already have some implicit capacity to measure local entropy, and introduce a function, which we will call local complexity, which is more explicitly related to local entropy. Basic ants record in memory the presence or absence of an object at the last lattice sites they have visited. From this record they calculate the local density of objects on the lattice. Entropy is related to the density. At either very high or very low density the entropy is low, while at intermediate densities the entropy is high. Basic ants pick up or put down objects in a n
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probabilistic fashion according to density thresholds. If the density is very low or high, then the ant is likely to perform the correct action, picking up in regions of low density and putting down in regions of high density. If the estimated local density is close to a threshold, then the certitude of performing the appropriate action (picking up or putting down) is low. We build upon this insight by engineering ants which have an explicit capacity to sense complexity. Given a local con guration of states, complexity can be estimated in various fashions. We choose a simple method based on a 9-cell neighborhood about the ant's current location. A 9-cell neighborhood in the square lattice has 12 interior faces between cells. The "complexity" of the neighborhood is taken as the number of faces which separate cells of di erent type{containing or not containing an object. Thus, either an "all-empty" or "all-occupied" neighborhood will have complexity 0, while a checkerboard pattern will have complexity 12. Complexity-seeking ants can calculate the complexity of their current position, thus they can "see" the ambient complexity of their local environment. Complexity-seeking ants appeal to their sensory system to direct both their physical motion and their object-manipulation activities. At each step of simulation, each ant calculates its local complexity, . The information in the value can be exploited either deterministically or probabilistically. In a deterministic strategy, the value is compared with a threshold. If exceeds the threshold, then the ant randomly choses a new direction of motion and decides whether to pick up or put down an object at its current position. In a probabilistic strategy, the ant generates a random number between 0 and 12. If the number is greater than 12-( + ), where is a threshold, then the ant will randomly chose a new direction of motion and execute the object manipulation strategy used by the basic ants. If the number is less than 12-( + ), the ant will continue along its previous direction of motion, and will not attempt to manipulate an object. The result of these rules is that complexity-seeking ants tend to manipulate objects only in regions of high complexity. Similarly, the complexity-seeking ants execute the random-move generator of the basic ants only when the complexity is high. In regions of low complexity, such as the large spaces between piles in mature environments, the ants generally continue in the direction they were previously moving. The result is that complexity-seeking ants spend less time in futile search in the spaces between piles than their basic-ant counterparts, and spend correspondingly more time in careful processing at the borders of C
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3 Dependence on Parameter Values In this subsection we give an overview of the dependence of sorting behavior on the values of the parameters in the di erent strategies. More details are supplied below. Recall that + is a constant which determines the density threshold for picking up an object, and ? is a constant which determines the threshold for putting down an object, and is the memory length. In all these experiments is held constant at 15, a value used by Deneubourg et al.. In the experiments described below + = 0 43 and ? = 0 12 for the complexity-seeking ants, while + = 0 38 and ? = 0 15 for the basic ants. These parameter values were chosen by running an experiment with a population of ants, each having di erent values for the parameters. Each 5 104 time steps interval, the ant which had the greatest success (see below) during the interval was found. The parameter values leading to greatest success did not depend signi cantly on the maturity of the environment in which the ants were operating. The speed of sorting depends in general on the number of ants in the population. Throughout this paper, the population size is xed at 25. We will consider a number of di erent threshold values for both deterministic and probabilistic complexity-seeking strategies. For deterministic strategies values of 0, 3 and 6 are studied, while for the probabilistic strategies values of 0 and 0.5 are studied. The patterns resulting from running each of these strategies 107 time steps on a random initial pattern with density 0.1 is shown in gure 1. Note that periodic boundary conditions are used. The panels are arranged top to bottom and left to right in order of increasing determinism in the strategies. After 107 generations, the basic ants ( gure 1 top left) have transformed the uniformly random initial pattern into a mottled pattern of many small, loose piles. The probabilistic complexity-seeking ants, = 0 5 ( gure 1 middle left) have in the same time produced a pattern with one nearly continuous, loose pile. The probabilistic complexity-seeking ants, = 0 ( gure 1 bottom left) have collected the objects into two, tight piles. It would seem at this point that this strategy is by far the best, however see next section. The second column of gure 1 shows the results of the deterministic comk
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Figure 1: Patterns produced by some variant ant strategies. All patterns produced after 107 time steps on a 256 square grid. From top to bottom, column 1: Basic ants, probabilistic complexity-seeking = 0 5 and = 0. column 2: deterministic complexity-seeking = 0 3 6. Periodic boundary conditions. T
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plexity seeking strategies with = 0 3 6, in top, middle, bottom, positions respectively. While the deterministic strategy with = 0 produces a good initial sort, the = 3 and = 6 strategies are grossly inecient. The rectangular piles produced by the deterministic complexity-seeking = 3 ants are artifacts of the complexity measure we have adopted. When the threshold is 3 and the rule is deterministic, ants can only put down or pick up objects from the corner of a rectangular pile. Thus rectangular piles are stable against the activity of the ants. The creation of such artifacts in the case of highly deterministic strategies demonstrates that there is a lower bound for the rate of random input required for good sorting. While the introduction of a local complexity measurement has reduced the randominput requirements of the basic ants, it has not eliminated it. It is evident that any fully deterministic strategy will have some corresponding con gurations which it cannot sort. Let us now look at the rst two strategies, basic, and probabilistic complexityseeking = 0 5 in more detail. Figures 2 and 3 show a time series of patterns produced by the two strategies. In each gure the rst column contains snapshots of the evolution from time 0 to time 3 107 in increments of 107 , while the second column contains snapshots from time 4 107 to 1 108 in increments of 2 107. The evolution in the two cases is very similar, except that the complexity-seeking ants accomplish their task considerably more rapidly than the basic ants, and produce tighter clusters along the way. Indeed, even after 1 2 108 ant-update cycles, the basic ants are far from fully sorting the 256 256 lattice. The basic ants will, however, eventually produce a single pile, as can be demonstrated in experiments using smaller lattices, or longer evolution times. T
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Figure 2: Pattern evolution under the action of the basic ants Time runs from top to bottom, left to right, in increments of 107 time steps in the rst column, and 2 107 in the second column. Figure 3: Pattern of evolution under the action of probabilistic complexity-seeking ants = 0 5 Same format as the previous gure. T
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4 Spatial Entropy To quantitatively study the sorting phenomenon shown in gures 1 through 3, we compute the evolution with time of the coarse-grained entropy of the patterns. The coarse-grained spatial entropy of the environment is measured at intervals during the sorting activity of the ants. Several di erent levels of coarse graining are used in order to track the organization e ected by the ants across length scales. In typical simulations a 256x256 square 2D lattice is used, with periodic boundary conditions imposed. 5 levels of coarse graining are computed, varying from a 8x8- to 128x128-point grains. To control inaccuracies due to clusters which span grain boundaries, the location of the entropy-computation grid is shifted relative to the lattice, and a minimum over shifts of the entropy is taken. We see that in all cases, entropy measured at the smallest grain sizes decreases quickly to an equilibrium value. The smaller the grain size, the faster the approach to equilibrium. This indicates that organization begins at the smallest scales and then propagates to larger scales. Stochastic uctuations due to probabilistic strategy elements aids in the di usion of organization from small to large spatial scales. In gure 4 the evolution of the coarse-grained entropy at all space scales is shown for basic and probabilistic complexity-seeking = 0 5 ants. For both basic and probabilistic complexity-seeking ants, the reduction in entropy e ected by the ants is seen rst in the small scales and later at larger scales. Small-scale entropy quickly reaches an equilibrium value while the large-scale entropy continues to gradually decrease. The complexity-seeking ants achieve lower values of equilibrium spatial entropy, at all scales, than the basic ants. The reduction in entropy occasionally proceeds by jumps. This is especially clear in the evolution of the coarsest-grained entropy under the action of the complexity-seeking ants. These jumps follow plateaus during which similarly T
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Figure 4: Evolution of the Coarse-Grained Entropy. Top: basic ants. Bottom:probabilistic complexity-seeking ants, = 0 5. The coarse-grained entropy is measured every 2 104 ant-update cycles for a total of 1 2 108 ant-update cycles. The line thickness increases as a function of grain size. Note the di erence in vertical scale between the panels. T
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sized piles compete for deposition of objects by the ants. In every case, stochastic uctuations will eventually lead some of the piles to suciently dominate the others in size so that the ants may act to amplify the di erence. This destroys the smaller piles and leads to a new metastable state during which competition occurs at a larger space scale. We now examine the evolution of the coarse-grained entropy for four of the variant strategies considered in this paper. In gure 5, the entropy at the largest scale of coarse-graining is shown for each of these variant strategies. This gure should be compared to gure 1. The probabilistic strategy with = 0 is initially a much better sorter than the other strategies. The probabilistic strategy with = 0 5 wins over the deterministic strategy, but less impressively. However, in the end game as the environment matures, the situation becomes more complicated. When the number of piles has been reduced to 2, which occurs between 10 and 30 million time steps for all complexity-seeking strategies in this gure, the more deterministic strategies cannot cope well with the task of choosing which pile to favor with object deposition. The probabilistic = 0 5 strategy also enters a meta-stable state consisting of two piles, as we saw in gure 4, but it is able to gradually break the symmetry between the piles. When is set to 0, the coarse-grained entropy may actually signi cantly increase for long periods, as happens in this run at approximately 2 4 107 time steps into the simulation. Part of the reason for the failure of the more deterministic strategies can be seen by considering the action of the ants in a mature environment with large spaces between piles. For the deterministic strategy, and the probabilistic strategy with = 0, no changes of direction occur in the empty spaces between piles. Ants thus continue in a straight line through regions between piles. If there are few piles, some ants will be trapped into orbits which never intersect a pile. These ants perform no work. This "neurotic" behavior which only shows itself after a large time has positive value at short times and at the population level. A population of neurotic ants produces tighter piles than a T
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Figure 5: Evolution of the Coarse-grained Entropy for the Variant Strategies. This gure shows the evolution of the coarse-grained entropy for each of the variant strategies: basic, probabilistic complexity-seeking = 0 and = 0 5, and deterministic complexity-seeking with = 0. Note that the probabilistic complexity-seeking = 0 strategies initially achieves the fastest reduction in entropy, but the probabilistic strategy with = 0 5 overtakes it eventually. T
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population of their well-adapted counterparts, and do so in a comparatively short time (see gure 1).
5 Strategic Eciency Bioenergetics is the study of the energy expenditures of an organism as it executes a process or accomplishes a task. In the present context, a bioenergetic analysis would be concerned with the amount of movement through space, picking up and putting down of objects etc. that each ant does as it goes about reducing the complexity of its world. As our main interest is in strategic rather than physical eciency of ants we will monitor strategic rather than physical energy expenditure in our model. Our main measure of strategic energy expenditure will be the number of bits of random information an ant consumes as it acts. The reasoning behind this is simple: a hard-wired strategy, no matter how many steps it entails, requires little thinking in order to execute. If the individual senses the appropriate releaser in its environment, it response re exively with no engagement of its "higher" mental capacities. A soft-wired strategy, one in which various options must be weighed in the absence of complete information, requires thoughtful analysis to ensure the highest probability of success. "Thoughful analysis" for a simulated ant is the estimate of aspects of its environment beyond its direct perception by means of input random information. In general, for a xed repertoire of strategies, the more randomness there is in the environment, the more random input will be required for the e ective execution of strategies. Use of randomized strategies in the face of environmental regularities indicates inaccurate or incomplete perception of these regularities. On the other hand, stereotypic action in the face of a shifting environment 10
indicates non-adapted, "neurotic" behavior. We will consider a number of di erent measures of ant strategies. These measures were selected with the general aim of understanding the model ant's behavior. We are concerned in particular with nding local i.e. ant-based measures of success which correlate with the environment-based measure of success discussed above, namely the evolution of the spatial entropy. That is, we consider measures of success which might be obtained by the ant itself from observation of its own actions and the changes it e ects in the local environment. For instance, if we give the ant the ability to measure the local complexity of a site both before and after it puts down an object, as well as an integer counter, then it can track its rate of success in reducing the local complexity when it puts down an object (see below). For the experiments described in this section, the basic ants and each
avor of complexity-seeking ants were run 107 time steps on a 256 256 grid.
5.1 Random Input
As suggested above, one potentially relevant local measure of success is the amount of "thinking"{the amount of random number generation|that each ant uses as it works. Ants use random numbers to direct their motion, and also to direct their actions. Probabilistic complexity-seeking ants use random numbers in addition to interpret the information from their complexitysensing equipment.
5.1.1 Random Input for Motion and Object Manipulation
Ants may move in any one of 8 directions on the lattice. The choice of direction thus requires 3 bits. Basic ants constantly consume this many bits at each time step to direct their motion. Complexity-seeking ants consume fewer random bits for motion since they do not change direction each time. Since complexity sensing lends a degree of adaptability to changes in the environment, the amount of random-number consumption changes over time for the complexity-seeking ants. This is shown in gure 6. Here we see that all kinds of complexity-seeking ants consume fewer random numbers for motion than the basic ants. Further, as time goes on the amount of random number generation decreases. The probabilistic ants continue to adapt to their environment longer than the deterministic ants, as evidenced by the longer 11
Figure 6: Random Number Generation for Motion The log of the number of calls to the random number generator used for motion is show for each of 4 variant strategies: basic, deterministic complexity-seeking = 0, and probabilistic complexity-seeking = 0 and = 0 5. This measure correlates with global success in the initial iterations. T
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Figure 7: Success Rate The rate of success achieved by each of 4 sorting strategies. transients toward a stable rate of random number consumption. Initially, i.e. for the rst 107 time steps, the rate of random number generation for motion correlates well with the ecacy of the strategy. The faster sorters use random numbers at a slower rate to direct ant motion. The rate of random number consumption for object manipulation follows closely the rate of random number consumption for motion (not shown). This is as is to be expected, as the impact of the complexity-seeking strategy on random number consumption is the same for both motion and objectmanipulation systems. In principle these e ects could be decoupled, by using for instance di erent thresholds of attention to the complexity measurement for the motion and object-manipulation systems.
5.1.2 Rate of Success
If an ant puts down an object and thereby decreases the local complexity, it is considered to have scored a success. The rate of success computed over intervals of 20,000 times steps is shown as a function of time in gure 7 for four sorting strategies. Remarkably, the raw number of successes in an interval indicates little concerning the success of the strategy evaluated at a global level. Indeed, the strategy which has the greatest initial success globally, the probabilistic strategy with threshold 0, scores the lowest rate of success measured locally. Therefor, to nd a local measure of success which correlates with global success, we must relate the raw local success rate to other locally measured quantities.
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Figure 8: Random Number Consumption per Success. The random number consumption per success is plotted as a function of time for each of 4 variant strategies. Note that in the case of probabilistic complexity-seeking = 0 ants, random number consumption per success actually increases after an initial decline. This appears to be due to the maturing of the environment in which the ants operate. T
5.1.3 Random Input per Success
Rather than mearly measuring the total random number consumption of ants and the total rate of success, we can determine how much random-number consumption is required to obtain each success. This is done by dividing the total random number consumption in the interval by the number of successes. The results are shown in gure 8. All strategies initially require a large dose of random input for each success. This dose decreases toward an equilibrium value for all strategies but for the probabilistic complexity-seeking = 0 strategy. In this case, random number consumption per success actually increases, after an initial decline. It also show sharp bursts toward large values. These bursts appear to correlate with similar bursts in the large-scale coarse-grained entropy. It would seem that these ants attempt to adapt to their maturing environment by investing increasingly more random input in each success. The rate of random number usage per success does not correlate well with the global success measure. However, modulation of this rate as a function of the environment is indicative of the exibility of the strategy. T
5.2 Time and E ort per Success
We considered above one measure of the e ort each ant puts into achieving a successful reduction in the local complexity, the consumption of random numbers. Here we will examine two related quantities: the time required to achieve each success, and the ratio between the number of successful putdowns and total put downs. This later measure is well-correlated with the rate of global entropy reduction. We rst turn the the time interval between successes (see 9). The time to achieve each success is strongly correlated with the random number consumption per success. It is not obvious that this should be the case. For 13
Figure 9: Time per Success The time taken to achieve a success under each of 4 strategies. Figure 10: Rate of Success per Put Down. The ratio between the number of success and number of put downs performed in an interval is shown for four strategies. This rate corresponds well and in some detail with the global measure of success. instance, basic ants may spend a long time between success when they are caught in large spaces between piles. During this time they will consume random numbers at their usual (large) rate. In such an environment, the deterministic strategy will also spend a long time in transit between piles, but will consume no random numbers while doing so. The most notable di erence between gures 8 and 9 is that basic ants consume more random numbers per success than the probabilistic = 0 5 ants, but require less time per success. This may be attributed to the fact the basic ants have a large basal rate of object manipulation and random number consumption. They achieve successes more rapidly, but also more rapidly produce failures, events in which putting down an object increases the local complexity. The nal local measure we will consider is the rate of successes per put down. This is shown in gure 10. Here we see that the probabilistic = 0 ants rapidly reach an equilibrium rate of success per put down of nearly 40 percent. Recall that these are the best sorters initially as judged by the rate of decrease of the spatial entropy of the lattice. Probabilistic ants = 0 5 eventually reach the same rate of success, but take longer to do so. This again correlates with the global success measure. The deterministic = 0 strategy initially has a local rate of success per put down similar to but somewhat below the probabilistic = 0 5 ants. The deterministic strategy appears to reach an asymptotic rate of local success which is lower than the probabilistic strategies. Finally, the basic ants have a very low rate of success per put down compared to the complexity-seeking ants. This too correlates well with the global measure of success. T
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6 Discussion The sorting behavior discovered by Deneubourg et al. is "emergent" in the sense that ants are not explicitly programmed to sort. Sorting results from the dependence of the ant's behavior on the perceived local density. In the same way, the complexity-seeking behavior described here is emergent from from the dependence of the ant's behavior on the perceived local complexity. As with sorting itself, the complexity-seeking behavior inherits robustness to perturbations from the implicit means by which it is programmed. Sorting is representative of a large class of biological activities which lead to structure in an environment. One may expect that appropriate adaptation of the complexity-seeking strategy could lead to new methods from improving the eciency of many types of collective computation. To this end, it could be of interest to study how sorting depends on the topology of the lattice, the number of types of objects to be sorted, their relative concentrations etc. We can summarize our experimental results as follows: Of the local measures of success considered, that which correlates best with global success is the ratio of successful put downs to total put downs in an interval. The best strategy in this game seems to be to invest increasingly more time and energy into achieving each success as the environment matures. Churning the success counter by inconsiderately picking up and putting down objects does not lead to e ective global behavior. While complexity-seeking increases the effectiveness of sorting and reduce the amount of random input required, there appears to be a lower bound to the amount of random input needed to adapt to randomness in the environment. It is likely that environments which will foil any overly deterministic strategy can be designed. Such work would help establish this lower bound on random input requirements for given sensory and motor capacities. Especially in view of the philosophy of autonomous-agent modeling outlined in the introduction, it would be valuable to consider the sorting behavior of ants in time-dependent environments, i.e. environments in which objects may appear, disappear and move by means other than direct action by the ants. It may well be that strategies well adapted to sorting in a closed environment become ine ective in environments a ected by e.g. external stochastic perturbations. By including such in uences, one may hope to better understand how the random elements of sorting strategies interact with random environmental in uences. 15
The complexity-seeking model explored in this paper could be compared with biological experiments. It makes predictions concerning the statistics of activity of real ants as they go about sorting. For instance, it should be possible to obtain laboratory measurements of the amount of time real ants spend in various environments as a function of local complexity. It would be of value to measure the probability for an ant to change direction of motion as a function of the local environmental complexity. While it is unlikely that real ants compute the particular, ad-hoc, local complexity function employed in these simulations, it is reasonable that they would have some means to evaluate the general simplicity or complexity of their immediate sensory environment. In general, the ant sorting problem would seem to be a good means to connect issues in computational sociology and ethology with the tools of statistical mechanics and dynamical systems theory. On one hand it presents entities whose activities have a clear biological interpretation. On the other hand, it exhibits phenomena, such as a type of order/disorder transition, which are familiar from physics. One may hope that this connection could help bring some organizing perspective to the myriad modeling e orts currently under way in the study of autonomous agents.
References [1] J.L. Deneubourg, S. Goss, N.Franks, A. Sendova-Franks, C. Detrain, and L. Chretien, The Dynamics of Collective Sorting, Robot-Like Ants and Ant-Like Robots, page 356 in [2] (1991). [2] J.-A. Meyer and S.W. Wilson, Eds. From Animals to Animats, MIT Press, Bradford Books (1991) [3] B.A. Huberman, Ed. The Ecology of Computation, North-Holland, Amsterdam, (1988)
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