The use of ideas of Information Theory for studying "language" and intelligence in ants

The use of ideas of Information Theory for studying “language” and in telligence in an ts Boris Ry abko ∗ , Zhanna Reznik ov a † Abstract In this review w e integrate results of long term experimental study on an t ”language” and in telligence whic h w ere fully based on fundamen tal ideas of Information Theory , such as the Shannon en tropy , the Kolmogorov complexit y , and the Shannon’s equation connecting the length of a message ( l ) and its frequency ( p ), i.e. l = − log p for rational comm unication systems. This approac h, new for studying biological comm unication systems, enabled us to obtain th e follo wing imp ortan t results on an ts’ communication and in telligence: i) to rev eal ”distant homing” in an ts, that is, their abilit y to transfer information ab out remote even ts; ii) to estimate the rate of information transmission; iii) to reveal that an ts are able to grasp regularities and to use them for ”compression” of information; iv) to rev eal that an ts are able to transfer to eac h other the information about the num b er of ob jects; v) to discov er that an ts can add and subtract small n umbers. The obtained results sho w that Information Theory is not only wonderful mathematical theory , but many its results may be considered as Nature laws. 1 In tro duction Since C.Shannon published his famous pap er “A mathematical theory of communication” [36] the fundamental role of Information Theory has b een appreciated not only in its direct appli- cations, but also in robotics, linguistics and biology . The comm unication systems and related cognitiv e skills of animals are a matter of special in- terest to ethologists, psyc hologists, linguists, and specialists in artificial in telligence and robotics. A ttempts to approach the question whether highly so cial and in telligent sp ecies can exc hange meaningful messages are based on a natural idea that the complexity of comm unication should b e connected with high levels of so cialit y , cognition and co op eration in animals’ so cieties. In the 1960s and 1970s, elegan t but am biguous experiments w ere conducted in which animals w ere ask ed to pass some piece s of information to eac h other. In Menzel’s [18] exp erimen ts, a group of c himpanzees living in an enclosure searc hed for hidden fo od. Menzel suggested that chimpanzees p ossess means for transferring information ab out b oth location and prop erties of ob jects, but it remained unclear ho w they did this. In other experiments the coop erativ e b eha vior of dolphins w as inv estigated, whic h could inv olv e intelligen t communication. T o get a fish, t wo dolphins, separated by an opaque barrier, had to press the paddles in the correct order. The obtained results enabled researchers to suggest that the dolphins can co-ordinate the actions of each other, probably , b y means of acoustic signals [5]. Despite these supp ortiv e experiments that ∗ Sib erian State Universit y of T elecommunications and Informatics; Institute of Computational T echnologies of Sib erian Branc h Russian Academy of Science, Nov osibirsk, Russia. b oris@ry abko.net † Institute for Animal Systematics and Ecology , Siberian Branch RAS and Nov osibirsk State Universit y . zhanna@reznik ov a.net 1 demonstrated that members of highly so cial in telligent species really hav e what to ”say” to eac h other, the question of existence of developed ”languages” in non-human b eings remained so far obscure. The main difficulties in the analysis of animal ”languages” app ear to b e metho dological. Man y researchers ha ve tried to directly decipher animal language by lo oking for ”letters” and ”w ords” and by compiling ”dictionaries” [32], for a review see [23]. How ever, only tw o cases of natural communications ha v e been deco ded up to the present. First, one of the most complicated of the known natural ”languages” in animals is the symbolic honey b ee ”Dance language”. Disco vered by K. V on F risch [6, 7] it was later in tensiv ely studied using differen t metho ds including rob otics and radars [19, 31]. The second case of a successful deciphering of several natural signals concerned alarm calls in v ervet monkeys which app eared to give different alarm calls to eagles, snakes and leopards [35, 4]. Later ”semantic” alarm calls and fo o d calls were describ ed for sev eral other species [2, 13]; for a detailed review see [23]. The fact that researc hers ha ve managed to compile such ”dictionaries” for a few sp ecies only , app ears to indicate not that other animals lack ”languages”, but that adequate metho ds are lacking. In b oth cases of communications that b ecame partly accessible for inv estigators, expressive and distinctiv e signals correspond to repeatable and frequen tly occurring situations in the context of animals’ life, and thus can serve as ”k eys” for deco ding their signals. The problem of cracking animals’ co des hav e b ecome esp ecially attractive since the great ”linguistic” p otential was disco vered in several highly so cial and intelligen t sp ecies by means of in termediary artificial languages. Being applied to ap es, dolphins and grey parrots, this metho d has rev ealed astonishing mental skills in the sub jects [9, 10, 34, 21]. It is b ecome p ossible to demonstrate that animals are capable not only of decision making and using the exp erience gained in new situations, but also of using simple grammatical rules, visual symbols and num b er-related skills. F or example, language-trained c himpanzees w ere found to b e able to add and subtract small num bers [1] - the ability that was not av ailable for discov ery without the use of intermediary languages. Ho wev er, it is imp ortan t to note that this w ay to communicate with animals is based on adopted human languages. Y et surprisingly little is known ab out natural communication sys- tems of those sp ecies that were inv olved in language-training experiments. Explorers of animal language behavior thus ha ve met a complex problem of resolving the contradiction b et ween their kno wledge ab out significant ”linguistic” and cognitiv e p oten tial in some species and limitations in cracking their natural co des [17]. W e hav e suggested a principally new exp erimen tal paradigm based on concepts of Informa- tion Theory [26, 27, 33]. The main p oin t of our approach is not to decipher signals but to in vestigate the very pro cess of information transmission b y measuring time duration which the animals sp end on transmitting messages of definite lengths and complexities. Being applied to studying communication in highly so cial ant sp ecies, this approach enabled us to reveal basic prop erties of an t ”language” and estimate their cognitive skills. An ts are go od candidates for studying general rules of natural comm unication and cognition, b ecause these insects are known to com bine highly integrativ e colon y organization with sophis- ticated cognitive skills. Ants p ossess complex forms of communications, and they are known to b e able to use a large v ariet y of communication means for attracting their nestmates to a fo od source [12]. It has remained unclear for a long time whether an ts can use distan t homing, that is, whether they can transfer messages ab out remote even ts without other cues such as scen t trail or direct guiding. In this asp ect, the so-called tactile (or antennal) ”co de” has been discussed since 1899, when it was first hypothesised such an information transmission system in an ts [37]. Ho wev er, the n umerous attempts to decipher ants’ ”tactile language” hav e not giv en the desired results (for a review see [25]). At the same time, it is clear that highly so cial ant sp ecies p ossess the necessary prerequisites for complex communication. Exp erimen tal studies rev ealed sophisticated forms of so cial learning in ants [22]. How ever, metho dological limitations 2 ha ve hamp ered the progress of studying ”linguistic p oten tial” of an ts’ comm unication, and the problem of distant homing in these insects has not b een solv ed b efore our exp eriments. The exp erimen tal paradigm of our approach is simple. All we need to do is to establish a situation where an ts m ust transfer a specific amoun t of information to each other. The crucial idea of the first scheme of experiments is that w e know exactly the quan tit y of information to b e transferred. T o organize the pro cess of information transmission betw een an ts, a sp ecial maze has b een used, called a ”binary tree” [27], where the num b er and sequence of turns tow ards the goal corresp onds to the amount of the information to b e transferred. In another series of exp erimen ts ants had to transfer the information ab out the num b er of a branch in comb-lik e ”coun ting maz es”. It has been firstly demonstrated that group-retrieving F ormica sp ecies possess distant hom- ing, and they are able to pass meaningful messages. W e also succeeded in studying important prop erties of ants’ cognitiv e capacities, namely their abilit y to grasp regularities, to use them for co ding and ”compression” of information, and to add and subtract small n umbers to optimize their messages. The obtained results demonstrate what Information Theory can furnish for explorers of comm unication and intelligence in so cial animals. This new exp erimental paradigm provides a wa y for studying imp ortan t characteristics of animal communication which hav e not b een accessible to study b efore, suc h as the rate of information transmission and the p otential flexi- bilit y of comm unication systems. W e also succeeded in studying some important properties of an ts’ in telligence, namely , their ability to grasp regularities and use them for optimization their messages. 2 Using Shannon en trop y and Kolmogoro v Complexit y to study comm unicativ e system in an ts: the binary tree ex- p erimen ts The exp erimen ts based on Shannon entrop y present a situation in which, in order to obtain fo od, ants hav e to transmit certain information which is quantitativ ely known to the researcher. This information concerns the se quence of turns tow ards a trough with syrup. The lab oratory maze ”binary tree” is used where each ”leaf” of the tree ends with an empty trough with the exception of one filled with syrup. The leaf on which to place the filled trough was chosen randomly b y tossing a coin for each fork in the path. The simplest design is a tree with t wo lea ves, that is, a Y-shap ed maze. It represents one binary choice which corresp onds to one bit of information. In this situation a scouting animal should transmit one bit of information to other individuals: to go to the right (R) or to the left (L) - see Fig. 1. In other exp erimen ts the num b er of forks of the binary tree increased to six. Hence, the n umber of bits necessary to c ho ose the correct wa y is equal to the n umber of forks, that is, turns to be tak en (Fig 2 sho ws a labyrin th with 3 forks). The use of ideas of Shannon Entrop y allo wed the presence of potentially unlimited num b ers of messages in ant ”language” to b e demonstrated and estimates of the rate of information transmission (approximately 1 bit/min.) to b e made. W e also succeeded in studying some prop erties of ants’ intelligence, namely , their abilit y to memorize and to use simple regularities, th us compressing the information av ailable. The latter exp erimen ts w ere based on the ideas of Kolmogoro v complexit y . 2.1 The exp erimen tal scheme In the ”binary tree” exp erimen ts ants were confronted with a rather complex life-or-death task: they could obtain fo od only in a ”binary tree” maze and only once every 2 - 3 days. Ants 3 Figure 1: The maze ”binary tree” with one fork and four forks. therefore were hungry and extremely motiv ated to obtain some fo od. They had to search for the fo o d placed on one of several ”lea ves” of the ”binary tree” maze (Figs. 1, 2). During differen t y ears three colonies of F ormic a p olyctena and tw o of F. sanguine a w ere used. Ants lived in the 2 × 1 . 5-meters lab oratory arena, in a transparent nest that made it p ossible for their activity to b e observed. The arena was divided into tw o sections: a smaller one con taining the nest, and a bigger one with an experimental system (Fig. 2). The t wo sections were connected b y a plastic bridge that was remov ed from time to time to mo dify the set-up or to isolate the ants. T o prev ent access to the fo od in the maze by a straight path, the set-up was placed in a bath of water, and the an ts reached the initial point of the binary tree by going ov er a second small bridge. The lab oratory colonies consisted of ab out 2000 individuals each. All actively foraging an ts w ere individually marked with colored paint. The lab oratory colonies were found to include teams of constan t membership which consisted of one scout and three to eigh t recruits (foragers): the scout mobilized only members of its team to the fo o d. The comp osition of the teams was rev ealed during sp ecial run-up exp erimen ts consisting of familiarization trials lasting as long as t wo or three weeks (see details in [25]). In total, 335 scouts along with their teams were used in all exp eriments with the binary tree. In eac h trial one of the scouts that were actively moving on the experimental arena at that moment was placed on a leaf of the binary tree that contained a trough with the fo o d, and then it returned to the nest by itself. All contacts b et ween the scout and its team w ere observed and time duration was recorded each time. All exp erimen ts w ere so devised as to eliminate all p ossible cues that could help the ants to find the fo o d, except information contact with the scout. T o av oid the use of an o dor track, the exp erimen tal set-up was replaced by an identical one when the scout w as in the nest or on the arena contacting its group. All troughs in the fresh maze contained only water to av oid the p ossible influence of the smell of syrup. If the group reac hed the correct leaf of the binary tree, they w ere immediately presen ted with the fo od. The scout had to make up to four trips b efore it was able to mobilize its group of foragers. Usually members of the team had already left the nest after the scout’s first trip and were w aiting on the arena for its return. Returning to the group, the scout con tacted one to four foragers in turn, sometimes tw o of them simultaneously . Con tacts were follo wed by n umerous antennal mo vemen ts. The exp eriments w ere designed to 4 Figure 2: The lab oratory arena with the maze ”binary tree”. in vestigate the characteristics of distant homing, so after the scout had contacted its team, it w as isolated for a while, and the foragers had to search for the fo o d by themselv es. This pro cess will b e describ ed in more details in the next section. Here it is imp ortan t to note that the comp osition of the working teams remained constant in each colon y from several days to sev eral w eeks, that is, during p eriods when a giv en scout was activ ely w orking (see detailed tables in [27]). It is notable that in both F. p olyctena and F. sanguine a , not all of the scouts managed to memorize the wa y to the correct leaf of the maze even after they had passed their ”final exams” during the run-up trials. The num b er of scouts that succeeded in memo-rising the w ay decreased with increasing complexit y of the task. In the case of tw o forks all active scouts and their groups (up to 15 per colon y) w ere successful whereas in the case of six forks, only one or t wo cop ed with the task. During the exp erimen ts each scout was placed on the trough containing fo o d, and after the scout had eaten it returned to the nest on its o wn. In all cases of mobilization of the group the duration of the contact b et ween the scout and the foragers w as measured in seconds. The con tact w as considered to begin when the scout touched the first forager an t, and to end when the first tw o foragers left the nest for the maze. When the scout rep eatedly returned to the trough alone, eac h of its con tacts with foragers was measured. Only the duration of the contact that was follow ed by the foragers’ leaving the nest was taken into account. These contacts were h yp othesized to b e ”informative”, and they differed sharply in duration from other contacts: all w ere more than 30 seconds. As a rule, all of the previous contacts b etw een scouts and foragers w ere brief (ab out 5 seconds) and were made for the exchange of fo od. During each series of exp erimen ts with the trough placed on the i − th leaf of the binary tree, all teams that were activ e on that day work ed successively . While the trial was going on, the bridge leading to the w orking part of the arena w as taken aw ay , so as not to let mem b ers of other teams to go there. While the scout w as inside the nest, the whole maze w as replaced by a fresh one with all troughs empt y . F oragers were presented with the syrup if they reached the correct leaf of the binary tree. 2.2 Information transmission b y distan t homing in an ts: statistical pro of Before analyzing ants’ ”linguistic p oten tial” and their ability to use rules for compression of information we consider the evidence of information transmission from the scouts to the foragers, 5 whic h came from tw o sets of data: first, from statistical analysis of the num b er of faultless findings of the goal by a group, and second, from a sp ecial series of control exp erimen ts with ”uninformed” (”naive”) and ”informed” foragers. The statistical analysis of the num b er of faultless findings of the goal was carried out by comparing the hypothesis H 0 (an ts find the leaf containing the fo o d by chance) with the hypothesis H 1 (they find the goal thanks to the information obtained), pro ceeding from the fact that the probabilit y of finding the correct wa y b y chance when the n um b er of forks is i is (1 / 2) i . W e analyzed differen t series of exp erimen ts (338 trials in sum), separately for 2, 3, 4, 5, and 6 forks. In all cases H 0 w as rejected in fa vor of H 1 , P < 0 . 001 (see [28]), thus unambiguously demonstrating information transmission from scouts to foragers. The control exp erimen ts were organized so as to compare searching results of the an ts that had and had not previous p ossibilit y to contact the scout (the ”informed” and ”naive” ants, resp ectiv ely). The ”naiv e” and ”informed” an ts w ere tested one by one. Each ant was allow ed to searc h for the fo o d for 30 min. In T able 1 the time sp en t on searching the trough by ”informed” and ”uninformed” F ormic a pr atensis are compared [30, 20]. F or every trial, Wilcoxon’s non- parametric test was used [11] to test the hypothesis H 0 (data from b oth samples follow the same distribution) against H 1 (they follow different distributions) at significance level 0 . 01. W e concluded that the duration of searching time is essentially smaller in those ants that had previously contacted the scout. These data demonstrate that scouts transfer information ab out the discov ered fo od to foragers b y means of distan t homing. T able 1: Comparison of duration of searc hing the trough by ”uninformed” (U) F. pr atensis ants and individuals that previously con tacted with the successful scout (”Informed”,I) Sequence of the turns An ts (U/I) Mean (second) Amounts of sampling P RRRR U 345.7 9 < 0 . 01 I 36.3 9 LLLL U 508.0 9 < 0 . 01 I 37.3 9 LRRL U 118.7 7 < 0 . 01 I 16.6 7 RRRR U 565.9 7 < 0 . 01 I 16.3 7 2.3 The rate of information transmission in ants W e now can ev aluate the rate of information transmission in ants. T o do this, observe that the quan tity of information (in bits) necessary to c ho ose the correct route in the maze equals i , the depth of the tree (the num b er of turns to b e taken), that is, log 2 n where n is the num b er of leav es. One can assume that the duration of the contacts b etw een the scouts and foragers ( t ) is ai + b , where i is the num b er of turns (the depth of the tree), a is the rate of information transmission (bits p er minute), and b is an introduced constant, since ants can transmit information not related directly to the task, for example, the simple signal ”foo d”. Besides, it is not ruled out that a scout an t transmits, in some wa y , the information on its route to the nest, using acoustic or some other means of communication. In this con text, it is imp ortan t that the route from the maze to the nest on the arena was in all exp erimen ts approximately the same. Being highly motiv ated, scouts h urried on to the nest in a b eeline, and, therefore, the time b efore they made an tennal contact with the foragers in the nest, which the scout could hypothetically use for message transmiss ion, was approximately the same and did not dep end on the num ber of turns 6 to b e tak en in the maze. F rom the data obtained, the parameters of linear regression and the sample correlation co efficien t ( r ) can b e ev aluated. The rate of information transmission (a) derived from the equation t = ai + b w as 0.738 bits p er min ute for F. sanguine a and 1.094 bits p er minute for F. p olyctena . The rate of information transmission is relativ ely small in an ts. T o estimate the p otential pro ductivit y of ants’ ”language”, let us count the total num b er of differen t p ossible routes to the trough. In the simplest binary tree with one fork there are t wo lea ves and therefore tw o different routes. In a tree with tw o forks there are 2 2 routes, with three forks 2 3 routes, and with six forks, 2 6 routes; hence, the total num ber of different routes is equal to 2 + 2 2 + 2 3 + . . . + 2 6 = 126. This is the num ber of messages the ants must b e able to pass in order to pass the information ab out the fo od placed on any leaf of the binary tree with 6 forks. 2.4 The Kolmogoro v complexit y and data compression in the an ts language Another series of exp erimen ts with the binary tree was inspired by the concept of Kolmogorov complexit y and was designed to chec k whether highly so cial ant sp ecies p ossess such an im- p ortan t prop ert y of intelligen t communications as the ability to grasp regularities and to use them for enco ding and ”compressing” information. This concept is applied to words (or text) comp osed of the letters of any alphabet, for example, of an alphab et consisting of t wo letters: L and R. W e interpret a word as a sequence of left (L) and right (R) turns in a maze. Infor- mally the complexity of a word (and its uncertaint y) equates to its most concise description, according to Kolmogorov [15]. F or example, the word ”LLLLLLLL” can b e represented as ”8 L”, the word ”LRLRLRLR” as ”4LR”, while the ”random” word of shorter length ”LRRLRL” probably cannot b e expressed more concisely , and this is the most complex of the three. W e analyzed the question of whether ants can use simple regularities of a ”word” to com- press it. It is known that Kolmogorov complexity is not algorithmically computable. There- fore, strictly sp eaking, we can only chec k whether ants hav e a ”notion” of simple and complex sequences. In our binary tree maze, in human p erception, different routes ha ve differen t com- plexities. In one particular series of exp erimen ts, F. sanguine a ants were presented with differen t sequences of turns. T esting the hypothesis H 0 (the time for transmission of information do es not depend on the text complexity) against the h yp othesis H 1 (that time actually dep ends on it) allow ed us to reject H 0 , thus showing that the more time ants sp en t on the information transmission, the more information - in the sense of Kolmogoro v complexity - was contained in the message [33]. Let us test these h yp otheses formally . There are seven sequences of turns of equal length (lines 5-8 and 13-15 at the table 2). The total num b er of turn sequences orders, according to the duration of the transmission is 7! of which 2!2!3! are in the line with H 0 . The probabilit y of obtaining such an order according to H 0 is very small: (2!2!3!) / 7! = 1 / 210. Thus, we accept the hypothesis H 1 : the simpler the text the less time for information transmission. It is interesting that the an ts b egan to use regularities to compress only quite large ”words”. Th us, they sp en t from 120 to 220 seconds to transmit information about random turn patterns on the maze with five and six forks and from 78 to 135 seconds when turn patterns were regular. On the other hand, there was no essential difference when the length of sequences w as less than 4 (T able 2). These results enable us to suggest that ants not only pro duce a large num b er of messages but can use rule extraction in order to optimise their messages. The ability to grasp regularities and to use them for co ding and ”compression” of information can b e considered as one of the most imp ortan t prop erties of language and its carriers’ intellect. Thus w e can conclude that the an ts’ comm unication system is rational and flexible. 7 T able 2: Duration of transmitting information on the w ay to the trough b y F.sanguine a scouts to foragers (no.1-8 regular turn pattern; no. 9-15 random turn pattern) No Sequence Mean duration (sec) Num b ers of exp erimen ts 1 LL 72 18 2 RRR 75 15 3 LLLL 84 9 4 RRRRR 78 10 5 LLLLLL 90 8 6 RRRRRR 88 5 7 LRLRLR 130 4 8 RLRLRL 135 8 9 LLR 69 12 10 LRLL 100 10 11 RLLR 120 6 12 RRLRL 150 8 13 RLRRRL 180 6 14 RRLRRR 220 7 15 LRLLRL 200 5 3 The an ts n umerical comp etence Basic num b er-related skills, that is, kno wledge of quantities and their relations, is, perhaps, one of the highest prop erties of cognition. Recent studies hav e demonstrated some species as b eing able to judge ab out num b ers of stimuli, including things, and sounds, and ma yb e smells. F or example, lions can coun t roaring that comes from individuals who are not members of the pride [16]; honey b ees are able to use the num b er of landmarks as one of the criteria in searching for fo od sources [4]. There are many other examples that come from different animal sp ecies, from mealy b eetles [3] to elephants [14]; how ev er, we are still lacking an adequate ”language” for comparative analysis. The main difficulty in comparing numerical abilities in humans and other species is that our numerical comp etence is closely connected with abilities for language usage and for symbolic represen tation. W e elab orated an exp erimen tal paradigm for studying ants’ n umerical comp etence [29]. A scouting ant has to transfer the information about a n umber (in our case, the index n umber of a branch of a maze) to its nest mates. Quantitativ e characteristics of the an ts’ comm unications w ere used for in vestigating their ability to count. The main idea of this exp erimen tal paradigm is that exp erimen ters can judge how ants represen t num b ers by estimating how muc h time individual ants sp end on ”pronouncing” num b ers, that is, on transferring information ab out index num bers of branc hes. 3.1 The com b-like set - ups The experiments w ere based on a procedure similar to the binary tree study . Ant scouts were required to transfer to foragers in a lab oratory nest the information ab out whic h branch of a sp ecial ”counting maze” they had to go to in order to obtain syrup. ”Coun ting maze” is a collectiv e name for sev eral v ariants of set-ups. All of them serve to examine how ants transfer information ab out index num b ers of branc hes by means of distant homing. The first v arian t of the coun ting maze is a comb-lik e set-up consisting of a long horizontal 8 Figure 3: The comb-lik e set-ups for studying numerical competence in ants: a horizontal trunk, a v ertical trunk and a circle. 9 plastic trunk with 25 to 60 equally spaced plain plastic branches, each of them 6 cm in length (Fig.3). Each branc h ended with an empty trough, except for one filled with syrup. Ants came to the initial p oin t of the trunk ov er a small bridge. The second v arian t is a set-up with vertically aligned branches. In order to test whether the time of transmission of information ab out the n umber of the branch dep ends on its length as well as on the distance b et ween the branches, one set of exp erimen ts was carried out on a similar vertical trunk in which the distance b et ween the branches w as t wice as large, and the branches themselves w ere three times and five times longer (for different series of trials). The third v arian t was a circular trunk with 25 cm long branc hes. Similarly to the binary tree study , ants w ere housed in a lab oratory arena divided into tw o parts, one con taining a plastic nest with a lab oratory an t colony and another containing one of the v ariants of the coun ting maze. During differen t years t w o lab oratory colonies of F. p olyctena w ere used in this set of exp eriments. Each series of exp eriments was preceded by the run-up stage consisting of familiarization trials. In order to force a scout to transfer the information ab out fo od to its nest mates we sho wed it the trough con taining syrup (placing the scout directly on the trough) and then let it return to the nest. After allowing it to con tact the foragers within the nest, the scout w as remov ed and isolated for a while, so that the foragers had to search for the fo od b y themselves, without their guide. Again, similar to the binary tree study , the exp erimen ts with coun ting mazes w ere devised so as to eliminate all p ossible wa ys for the members of each foraging team to find a goal, except b y distan t homing, i.e., an information contact with their scout. The set-up was replaced with a fresh one, with all troughs filled with water, while the scout was in the nest; if the foraging team reached the correct branch in a b ody , then the water-filled trough was replaced with one with syrup; thus foragers had to rely solely on the information from the scout. 3.2 The an ts’ abilit y to transfer the information ab out num b ers of ob jects The findings concerning num ber-related skills in ants are based on comparisons of duration of information contacts b et ween scouts and foragers which preceded successful trips by the foraging teams. Duration of con tacts of the scout with its team was measured when the scout returned from the exp erimental set-up, loaded with b oth syrup and information. In total, 32 scout - foragers teams work ed in three kinds of set-ups. The teams left the nest after they w ere contacted b y scouts and mov ed to wards the trough by themselves 152 times (recall that the scouts were remov ed and the set-ups were replaced). In 117 cases the team immediately found the correct path to the trough, without making an y wrong trips to empty troughs. In the remaining cases, ants came to the empty troughs, and began lo oking for fo od by c hecking neigh b oring branches. Since all set-ups had no fewer than 25 branches, the probability of finding the correct trough b y chance is not more than 1 / 25. Thus, the success ratio which was obtained exp erimen tally can only b e explained b y information transmission from the scouts. The probabilit y of finding the fo od-containing trough by chance in 117 cases out of 152 is less than 10 − 10 . In addition, in con trol exp eriments ants, including scouts placed in the set-up, without information on which trough con tained fo od usually failed to find the fo o d, even though they activ ely searched for it. Data obtained on the vertical trunk are sho wn in T able 3 as an example. It turned out that the relation betw een the index num b er of the branch and the duration of the contact betw een the scout and the foragers is well describ ed by the equation t = ai + b for different set-ups whic h are characterized b y differen t shapes, distances b et ween the branches and lengths of the branc hes. The v alues of parameters a and b are close and do not depend either on the lengths of the branc hes or on other parameters. The correlation coefficient b et ween t and i w as high for differen t kinds of coun ting mazes; see T able 4. 10 T able 3: The results of exp erimen ts in the ”vertical trunk 1” with F. p olyctena. No Num b er of fo od- Duration of scout the scout’s ”name” con taining branch -forager con tact (sec) 1 10 42 I 2 10 40 I I 3 10 45 I II 4 40 300 I I 5 40 280 IX 6 13 90 I I 7 13 98 I 8 28 110 II I 9 28 120 X 10 20 120 X 11 20 110 I II 12 35 260 I II 13 35 250 X 14 30 160 I 15 30 170 I II T able 4: V alues of correlation co efficien t ( r ) and regression ( a, b ) co efficien ts for vertical trunk (v ert), horizontal trunk (horiz), and circle in the exp erimen ts with F. p olyctena T yp e of setup Sample size Num b ers of branc hes r a b V ert.1 15 40 0.93 7.3 -28.9 V ert.2 16 60 0.99 5.88 -17.11 Horiz.1 30 25 0.91 8.54 -22.2 Horiz.2 21 25 0.88 4.92 -18.94 Circle 38 25 0.98 8.62 -24.4 All this enables us to suggest that the ants transmit information solely concerning the index n umber of the branc h. It is interesting that quantitativ e c haracteristics of the ants’ ”num ber system” seem to b e close, at least outw ardly , to some archaic h uman languages: the length of the co de of a given n umber is prop ortional to its v alue. F or example, the w ord ”finger” corresp onds to 1, ”finger, finger” to the n umber 2, ”finger, finger, finger” to the n um b er 3 and so on. In modern human languages the length of the co de word of a num b er i is approximately prop ortional to log i (for large i ’s), and the mo dern numeration system is the result of a long and complicated dev elopment. 3.3 The an ts’ ability to add and subtract small n um b ers There are some exp erimental evidence that t wo years old human children, rhesus monkeys and c himpanzees can op erate with addition and subtraction of up to 4. The exp erimen ts are based on games where sub jects should demonstrate their abilities to judge ab out the num b er of items after the addition or remo v al [8]. W e elab orated a new exp erimen tal paradigm of studying ants’ ”arithmetic” skills based on a 11 fundamen tal idea of information theory , which is that in a ”reasonable” comm unication system the frequency of usage of a message and its length must correlate. The informal pattern is quite simple: the more frequently a message is used in a language, the shorter is the word or the phrase co ding it. This phenomenon is manifested in all known human languages. The main exp erimental pro cedure w as similar to other exp erimen ts with coun ting mazes. In v arious years four colonies of F. p olyctena were used in this set of experiments. The sc heme of the exp erimen ts w as as follows. Ants were offered a horizontal trunk with 30 branches. The exp erimen ts w ere divided into three stages, and at each of them the regularity of placing the trough with syrup on branches with different num b ers was changed. At the first stage, the branc h containing the trough with syrup was selected randomly , with equal probabilities for all branc hes. So the probabilit y of the trough with syrup b eing placed on a particular branch w as 1 / 30. At the second stage we chose tw o ”sp ecial” branc hes A and B (N 7 and N 14; N 10 and N 20; and N 10 and N 19 in different years) on which the trough with syrup o ccurred during the experiments m uch more frequently than on the rest - with a probabilit y of 1 / 3 for ”A” and ”B”, and 1 / 84 for each of the other 28 branches. In this wa y , t wo ”messages” - ”The trough is on branch A” and ”The trough is on branch B” - had a muc h higher probability than the remaining 28 messages. In one series of trials we used only one ”special” p oin t A (the branch N 15). On this branch the fo od appeared with the probabilit y of 1 / 2, and 1 / 58 for each of the other 29 branches. At the third stage of the exp eriment, the num b er of the branc h with the trough was chosen at random again. No w let us consider the relationship b etw een the time which the an ts spent to transmit the information about the branch containing foo d, and its n um b er. The information obtained at the first and third stages of the exp erimen ts are shown on the graph (Fig. 4) in whic h the time of the scout’s con tact with foragers ( t ) is plotted against the num b er ( i ) of the branc h with the trough. A t the first stage the dep endence is close to linear. At the third stage, the picture w as different: first, the information transmission time was v ery muc h reduced, and, second, the dep endence of the information transmission time on the branc h num b er is obviously non-linear: depression can b e seen in the vicinities of the ”sp ecial” p oin ts (10 and 20). So the data demonstrate that the patterns of dep endence of the information transmission time on the n umber of the fo od-containing branch at the first and third stages of exp erimen ts are considerably different. Moreo ver, in the vicinities of the ”sp ecial” branches, the time taken for transmission of the information ab out the num b er of the branch with the trough is, on the av erage, shorter. F or example, in the first series, at the first stage of the exp eriments the ants to ok 70 - 82 seconds to transmit the information ab out the fact that the trough with syrup was on branch N 11, and 8 - 12 seconds to transmit the information ab out branch N 1. A t the third stage it to ok 5 - 15 seconds to transmit the information ab out branch N 11. These data enable us to suggest that the ants hav e changed the mo de of presenting the data ab out the num ber of the branch con taining fo o d. What ab out ants’ ability to add and subtract small n umbers? Analysis of the time duration of information transmission by the an ts raises the p ossibilit y that at the third stage of the exp erimen t the scouts’ messages consisted of tw o parts: the information ab out which of the ”sp ecial” branc hes was the nearest to the branch with the trough, and the information ab out ho w man y branc hes a w ay is the branch with the trough from a certain ”sp ecial” branc h. In other w ords, the an ts, presumably , passed the ”name” of the ”sp ecial” branch nearest to the branch with the trough, and then the num ber whic h had to be added or subtracted in order to find the branc h with the trough. That ant teams wen t directly to the ”correct” branch enables us to suggest that they p erformed correctly whatever ”mental” op eration (subtraction or addition) w as to be made. In order to v erify this statistically , the co efficien t of correlation w as calculated b etw een the time required for transmission of information about the trough b eing on the branch i and the distance from i to the nearest ”special” branc h. The results confirmed the hypothesis that the 12 Figure 4: Dep endence of the time ( t ; measured in seconds) of transmission of information ab out the n umber of the branch having foo d on its ordinal num ber ( i ) in the first and the third series of exp erimen ts in the ant F ormic a p olyctena . Diamonds, the time taken for transmission of information at the first stage; Squares, the same it the third stage. 13 T able 5: Dep endence of the time of information transmission ( t ) on the distance from the branch with a trough to the nearest ”special” branch (special branches are 10 and 20) The n umber of the branch having Distance to the nearest Times of transmission of fo od (exp erimen ts in different da ys, sp ecial branc h information ab out the branc h consequen tly) n umber for different scouts (sec) 26 6 35,30 30 10 70,65 27 7 65,72 24 4 58,60,62 8 2 22,20,25 16 4 25,8,25 16 4 25 22 2 15,18 18 2 20,25,18,20 15 5 30,28,35,30 20 0 10,12,10 6 4 25,28 16 4 30,25 15 5 20,25,20 14 4 25,28,30,26 17 3 17,15 11 1 10,12 time for transmission of a message ab out the num b er of the branch is shorter when this branch is closer to any of the ”special” ones. F or this purp ose, the data obtained at the third stage of the experiment w ere transformed to present them in the form sho wn in T able 5 where data of one year are given as an example. In this table we do not include branc hes that are close to the starting p oin t of the set-up (N 1 - 4) b ecause there is no need to use ”arithmetic” for an ts where rewarded branc hes are very close to the first one (in fact ants s pent roughly the same time transmitting information ab out these branches: from 10 to 20 seconds). T able 6: V alues of correlation co efficien t (r) in the exp erimen ts with differen t ”sp ecial” branc hes Sample size Num b ers of ”sp ecial” r for the first stage r for the third stage branc hes of the exp erimen ts of the exp erimen ts 150 10,20 0.95 0.80 92 10,19 0.96 0.91 99 15 0.99 0.82 It can b e seen from T able 6 that the coefficients of correlation b et w een the transmission time and the distance to the nearest sp ecial p oint hav e quite high v alues and they differ significan tly from zero (at the confidence level of 0.99). So the results supp ort the hypothesis that the time for transmission of a message ab out the num b er of the branch is shorter when this branch is close to either of the sp ecial ones. This, in turn, sho ws that at the third stage of the exp eriment the ants used simple additions and subtractions, achieving economy in a manner reminiscent of 14 the Roman numeral system when the num bers 10 and 20, 10 and 19 in different series of the exp erimen ts, play ed a role similar to that of the Roman n umbers V and X. Our interpretation is that an ts of highly so cial group-retrieving sp ecies are able to add and subtract small n umbers. This also indicates that these insects hav e a communication system with a great degree of flexibility . Until the frequencies with which the fo o d was placed on differen t branches started exhibiting regularities, the ants w ere ”enco ding” each num b er ( i ) of a branc h with a message of length prop ortional to i , whic h suggests unitary co ding. 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