-This work introduces a model of fuzzy perception for BDI agents, to support the simulation of decision making processes in environments with imperfect information. An application to a fuzzy prey-predator environment was developed, as an example, where the process of deciding which prey a predator should attack is based on its fuzzy perception of the strength of the prey, and on the comparison of the preys' strengths with its own strength. Different simulations were realized for the comparative evaluation of different types of predator agents, in contexts with and without competition between predators. The quantitative analysis of the simulations shows that the fuzzy predator agent presents the best scores. However, the important result is that the fuzzy predator seems to behave more adequately in the environment, in the sense that it presents an apparently more natural, coherent and realistic behavior.
Fuzzy sets and Fuzzy Logic (FL) [
Zadeh [
In the context of Social Simulation (SS), Grüne-Yanoff [
Among the agent models commonly used in agent-based
simulation of decision processes in complex environments,
there are the ones of an intentional nature, whose behaviors
can be explained by attributing certain mental attitudes to the
agents, such as knowledge, belief, desire, intention, obligation,
commitment (see, e.g., [
A well-known intentional model is the BDI (Beliefs,
Desires and Intentions) architecture, introduced by Rao and
Georgeff [
Although Rao and Georgeff explicitly acknowledge that an agent’s model of the world is incomplete, the BDI model does not take into account the influence of the imperfect information (in the sense discussed above) acquired from the world in beliefs, desires and intensions. In particular, it does not consider that the agent could have a “fuzzy” perception of the world. Then, in this paper, we experiment with a BDI agent with fuzzy perception operating in a task environment with imperfect information, namely, a fuzzy prey-predator system.
Prey-Predator systems are an important theme in the area
of Population Dynamics, their modeling having achieved a
classical status through the formulation of the so-called
LotkaVolterra equations [
The paper is organized as follows. Section II discusses related work. Section III presents some concepts on the fuzzy inference system used in this work. The environment with imperfect information inspired by the Fuzzy Prey-Predator Model is introduced in Sect. IV, including our approach for the fuzzy perception module to be included in the BDI architecture of the predator agent. The results on simulations are discussed in Sect. V. Section VI is the Conclusion.
In the context of social simulation, FL has been playing an important role, and it is possible to find many interesting works using FL to deal with different problems that can not be solved with classical simulation models and tools. In this section, we briefly present some of those works, according to the different issues covered by them.
Hassan et al. [
In Ghasem-Aghaee and Ören [
Sabeur and Denis [
Fort and Pérez [
Concerning fuzzy perception in robots, Cuesta and
Ollero [
Notice that the agent architectures proposed so far mostly
deal with two-valued information. Casali et al. [
Hybrid models can be found in the literature, introducing
some kind of fuzziness to BDI architecture. Long and
Esterline [
However, neither the nice formalization by Casali et al. nor the other analyzed works have considered the influence of fuzzy perception on the operation of a BDI agent and its decision making.
Fuzzy set theory [
Fuzzy inference systems are non-linear models that aim to
describe the input-output relationship of a real system using a
family of linguistic If-then constructions and the inference
mechanisms of FL. Among the several methods available
for fuzzy inference, we adopt in this work the
Kang-TakagiSugeno (KTS) method [
If (x1 is A1,k) and . . . and (xN is AN,k) then z = fk(X), (1) where the linguistic terms An,k (n = 1, . . . , N ) in the rule antecedents represent fuzzy sets with membership functions μn,k, which are used to partition the domains of the input variables into overlapping regions. The functions fk in the rule consequents are usually first-order polynomials of the form: fk(x1, . . . , xN ) = b0,k + b1,kx1 + . . . , bN,kxN . (2) For a given input X = (x1, . . . , xN ), the degree of fulfilment of the kth rule evaluates the compatibility of the input X with the rule antecedent and determines the contribution of the rule’s response z = fk(x1, . . . , xN ) to the overall model’s output. The degree of firing of the kth rule is expressed as wk(x1, . . . , xN ) = T1(μA1,k(x1), . . . , μAN ,k(xN )), (3) where T1 is a t-norm (triangular norm). In this work, T1 is the Minimum t-norm (called Gödel t-norm), and then Eq. 3 becomes : wk(x1, . . . , xN ) = min{μA1,k(x1), . . . , μAN ,k(xN )}. (4) The overall output of a normalized first-order TSK fuzzy model with K rules is given by
K
P T2(wk(x1, . . . , xN ), fk(x1, x2, . . . , xN ))
z = k=1 , (5)
1The adoption of the KTS method is due to its better performance in some
applications, since it avoids the defuzzification step. See [
K
P wk(x1, . . . , xN ) · fk(x1, x2, . . . , xN ) z = k=1 . (6)
In [
In this paper, we informally build on the fuzzy prey-predator approach for an agent-based simulation in order to analyze the ability of a predator with fuzzy perception in surviving in an environment of imperfect information.2
In this environment, the age and the weight of a prey (and of a fuzzy predator itself) are vague information for the fuzzy predator. However, such information is crucial for a predator to evaluate the strength level of a certain prey in comparison with its own strength level, and, therefore, to estimate the probability of the success of its attack to such prey, which is given by:
RAP − RP P P rob = 50 + , (7) 200 where RAP and RP P are the predator’s and the prey’s strength levels, respectively.
We assume that (i) predators and preys are initially randomly distributed in a grid; (ii) the food is always available for the different preys, and (iii) a predator loses weight for being looking around for preys and much more for each unsuccessful attack (on the contrary, it gains weight if its attack is successful). Then, the predator survival during the evolution of the time depends on its decision about attacking or not any prey it finds during its life. This decision is based on the imperfect information that the agent can perceive through its fuzzy perception mechanism, which uses a fuzzy inference system to determine the prey’s strength level and its own.
The predator is a BDI agent with beliefs3 on the following
parameters: age, weight and strength level. The age and weight
the agent can perceive through its perception mechanism.
The strength level can be estimated considering perceived
ages and weights. The abilities of the predator are: random
movement looking for preys, perception of preys’s age and
weight, estimation of prey’s strength level in comparison with
its own strength level at the current time, and decision on
attacks to preys, which considers if the probability of success
satisfies P rob > 0.25 (Eq. 7). The constraints of its life are:
2Notice that we did not study population dynamics, as it was done in [
3In this paper, we do not refer to the agent’s desires or intentions, only to its beliefs, since this is the component of the BDI model that is connected to the fuzzy perception mechanism.
Perceptions
Sensors BRF
Environment
Beliefs
Actions Actuators Deliber ations (i) at each movement it loses a fixed amount of weight (weight loss rate), and has its age incremented by a fixed value (aging rate); (ii) at each successful attack, it gains a fixed amount of weight (attack reward); otherwise, it loses a fixed amount of weight (attack punishment); (iii) there is a minimum weight that a predator can support; if it achieves a weight less than the minimum then it dies by weakness; (iv) there is a maximum age that a predator can achieve; after that it dies by ageing.
The Fuzzy Predator (FP) has a perception mechanism directly connected to the BRF (Belief Revision Function) of its BDI architecture, partially depicted in Fig. 1. This means that the fuzzy perception mechanism receives as input data the prey’s age and weight, as well as the predator’s own age and weight, all of which are perceived through the predator’s non precise sensors. Then, using the KTS inference system (Sect. III), the predator infers the prey’s strength level, and also its own strength level, updating its beliefs with the inferred information, in order to let this information be used in the decision process.
The linguistic variables age, weight and strength level are modeled as fuzzy sets with trapezoidal membership functions (Fig. 2). The analysis of those linguistic variables allowed the construction of a knowledge base composed by the linguistic rules presented in part in Table I. Table II shows part of the rule base for the KTS inference system of the perception model of the FP agent, each one with 2 inputs (age, weight) ∈ R2 and the output z ∈ R, where “young”, “adult”, “old”, “very light”, “light”, “average”, “heavy” e “very heavy” represent fuzzy subsets of R.
Example 1: In order to see how the inference system of the fuzzy perception mechanism operates, let us consider the following crisp input data: age = 16 and weight = 84. Those values are fuzzified, considering the membership grades in relation to the fuzzy subsets that define those linguistic variables, given in Fig. 2. Then, the age value age = 16 is considered “young” with grade μyoung(16) = 0, 4 and “adult” with grade μadult(16) = 0, 6. The weight value weight = 84 is evaluated as “heavy” with grade μheavy(84) = 0, 6 and “very heavy” with grade μvery−heavy(84) = 0, 4.
For each combination of those sets achieved by the input data, some of the rules of the knowledge base are activated. In this case, four rules are fired, namely, the rules R4, R5, R9 and R10 of the Tables I and II. Using Eq. 4, it is possible to find the degrees of firing of each one of those rules, as, e.g., w4 = min {μyoung(16), μheavy(84)} = 0, 4. Then, one has that w5 = 0, 4, w9 = 0, 6 and w10 = 0, 4. Using Eq. 6, we obtain the overall output of the process, where f4, f5, f9 and f10 are calculated using Table II: z= w4f4(16, 84)+w5f5(16, 84)+w9f9(16, 84)+w10f10(16, 84) =74, w4 + w5 + w9 + w10 which represents the predator’s strength level.
For the comparative analysis of simulations, we implemented a Crisp Predator (CP), which is a BDI agent that does not consider that the information about itself and the one perceived from the environment are vague or incomplete. Its perception mechanism is inspired on the perception mechanism of the fuzzy predator, but, instead of using fuzzy subsets for the modeling of the input linguistic variables, we use classical sets with the usual characteristic functions into the set {0, 1}. For each set of input date, only one rule of the If R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R15 age young young young young young adult adult adult adult adult old old old old knowledge base is activated. The characteristic functions of the sets related to the linguistic variable age are: μyoung(x) = 10 iofthxer≤wi1se5; μadult(x) = 10 iofth1e5rw<isxe < 35;
The characteristic functions of the sets related to the linguistic variable weight are: μvery−light(x) = 10 iofthxer≤wi1se5; μlight(x) = 10 iofth1e5rw<isxe ≤ 35; μheavy(x) = μaverage(x) = 1 if65 < x ≤ 85 0 otherwise 1 if 35 < x ≤ 65; 0 otherwise μvery−heavy(x) = 1 if x > 85 0 otherwise (8) (9)
Example 2: Considering the same input data (age, weight) of Ex. 1 and the characteristic functions given in Equations 8 and 9, one has that weight = 84 and age = 16 are definitely evaluated as “heavy” (μheavy(84) = 1) and “adult” (μadult(16) = 1), respectively. In this case, only the rule R9 of the rule base of Tables I and II is activated. Obviously, the firing degree of this rule is w9 = 1. The general output, given by Eq. 6, results in the value of the straight level: w9f9(16, 84) 1 · 88 z = = = 88
w9 1
To enrich the possible comparisons, we have implemented a Greedy Predator (GP), which always attacks the preys it encounters, without considering any reasoning on strength levels and the probability of success of its attacks to preys.
The simulations were realized to obtain a general view
of the behaviors of the different predators4 in two kinds
of the Fuzzy Prey-Predator Environment: competitive
(SectV-A) and non-competitive environments (Sect. V-B). The
implementation was done in the Jason platform [
4Since we are not analyzing population behavior, in the simulations we only consider either 2 or 3 predators, in order to be able to compare directly their surviving abilities.
The results were obtained from a total of 100 simulation runs. In each run, the time grows in discrete units (1 time unit = one predator movement). In the beginning of each run, the predators present the following initial parameters: age = 1 and weight = 50. Those parameters change at each time instant according to the following fixed rates5: the weight loss rate (-0,1 kg for each movement/time), the aging rate (-0.05 year for each movement/time), the attach reward (+2 kg for each successful attack), and the attack punishment (-1 kg for each non successful attack). The simulation run ends when all the predators have died, either for weakness (weight less than 1 kg) or for ageing (age equal to 50 years).
The competitive environment consists of 2 kinds of predators (FP e CP) and different 250 preys. At each successful predator attack, the corresponding defeated prey dies. Considering that there is no prey reproduction, the prey population tends to decrease, increasing the probability of the predator not finding a prey as it moves in the environment, which may cause increasing weight losses. In this sense, both predators compete for the preys remaining in the environment.
5Variations of the initial parameters and rates are not considered here, since they affect only the agent’s deliberations, not its perceptions.
Fig. 3. The average of attacks (top), victories (middle) and defeats (bottom) at an age i, with 1 ≤ i ≤ 50, in a competitive environment.
Figure 3 (top) shows the average number of predators’ attacks at each year. Observe that the number of the CP’s attacks surges around the age of 15. This is so because, before 15, the agent thinks that it is young (with too low strength level), but, suddenly, as it achieves 15 years old, it concludes that it is already an adult (with too high strength level). The increase in the number of the FP’s attacks is more gradual, showing more coherence in its decisions. On the other hand, one might have expected that the high number of attacks would have lasted until around the age of 35, since it is only after this age that the CP considers itself old. However, due the prey population decreasing, the number of the attacks of both predators also decreases, even before the age 35. Around the age 35, the decrease in the number of the attacks of the CP is much more abrupt than the smooth decreasing of the number of the attacks of the FP, as it passes from young to adult/old.
Figure 3 (middle) presents the average of predators’ victories at each year. There is a significant increase in the number of victories when the CP is around 15, which is an expected result, since this is the period that, as it considers itself an adult by this age, it increases a lot the number of attacks until around the age of 35, when it considers itself old, as discussed in the previous paragraph. Also, due to the decrease in the prey population, and consequently, the decrease in the number of attacks, the number of victories also decreases, even before the age 35. Again, it is possible to observe that the graph corresponding to the FP increases and decreases smoothly, as the agent becomes old, whereas the one of the CP increases abruptly around 15 and decreases around 35, also drastically.
Analogous analysis can be done concerning the average of number of the predators’ defeats at each year, which is shown in Fig. 3 (bottom).
The non-competitive environment consists of 3 kinds of predators (FP, CP and GP), and 250 different preys. For each prey that dies in consequence of a predator attack, another prey with similar characteristics appears in the environment, at a random position. This means that the predators always have the same chance to find a prey to attack.
Figure 4 (top) shows the average number of predators’ attacks at each year. For the same reasons discussed in Sect. V-A, the number of the CP’s attacks surges around the age of 15. However, since the prey population is constant along the time, the high number of attacks of the CP lasts until around the age of 35, and then it follows drastically. The behavior of the FP is much more natural and coherent, since it presents a gradual increase in the number of attacks as it becomes an adult, and a also a smooth decrease as it becomes old. The high number of attacks of the GP during its life was as expected. During adulthood the numbers of attacks of all aFtiga.n4a.geTih,ewaivthera1g≤eoif ≤att5ac0k,sin(toapn)o,nv-iccotomripeesti(tmivieddelnev)iraonndmdeenfte.ats (bottom) predators are similar.
Figure 4 (middle) presents the average number of predators’ victories at each year. There is an abrupt increase in the number of victories when the CP is around 15, due to the high predators are similar. The highest numbers of victories, for increase in the number of its attacks by this age. However, Crisp and Fuzzy predators, appear between the ages of 20 and since the prey population does not decrease, the number 33. Analogous analysis can be done concerning the average of of victories stays high until around the age 35. After that, number of predators’ defeats at each year (Fig. 4 (bottom)). it decreases radically. Again, it is possible to observe that Figure 5 (top) shows the average number of accumulated the graph corresponding to the FP increases and decreases attacks during the predators’ lives, until they reach a certain smoothly, as the agent becomes old. The higher number of age i, with 1 ≤ i ≤ 50. As expected, the GP had an average victories of the GP is due to its attack strategy. During number of accumulated attacks much higher than the other adulthood, the numbers of victories of the three kinds of two predators, which had a similar attack behavior. Fig. 5. Average accumulated number of attacks (top), victories (middle) and defeats (bottom) during the predator life until the age i, with 1 ≤ i ≤ 50.
Figure 5 (middle) presents the average number of accumulated victories during the predators’ lives, until they reach an specific age. As expected, the GP had an average number of accumulated victories much higher than the other two predators. However, this number for the FP is higher than that of the CP, as they become older.
Analogous analysis can be done concerning the average number of accumulated defeats during the predators’ lives, until they reach an specific age, shown in Fig. 5 (bottom).
Figure 6 (top) shows the average lifetime of the predators. As Fig. 6. Average lifetime (top), average weight at the end of the life (middle) and average of number of attacks/victories/defeats (bottom) of predators. expected, the Greedy and Fuzzy predators present the lowest and the highest average lifetimes, respectively.
Figure 6 (middle) presents the average weight of predators at the end of their lives. The average weight of the FP at the end of its life is the highest one.
Figure 6 (bottom) shows the average number of attacks, victories and defeats of predators during its whole life. The GP is the one that presents the highest averages in all categories, and it is the one that has the average number of defeats higher than that of victories. Its average numbers of attacks and victories are higher than the ones of the CP, whereas the average number of defeats is lower than that of the CP.
We conclude that the simulation of the fuzzy perception of the FP allowed for a more faithful simulation of naturally expected smoothness of the development of predation ability of predators in a Fuzzy Prey-Predator environment.
This paper introduces a model of fuzzy perception for BDI agents in task environments with imperfect information, which was inspired by an analysis of a particular Fuzzy PreyPredator model. The aim was to analyze the influence of fuzzy perception on the ability of BDI agents to simulate decision making processes in fuzzy environments. For that, we have defined a perception mechanism directly connected to the BDI agent’s BRF. The perception mechanism uses a KTS inference system, which is application dependant.
The simulations allowed us to obtain a general view of the behaviors of the different predators (CP, FP, GP) in two kinds of the Fuzzy-Predator Environment: a competitive environment and a non-competitive environment.
Although the difference between the results obtained by the Fuzzy and the Crisp predator agents were not so significant in the quantitative analysis that we have performed, it seems that the Fuzzy predator agent showed a more adequate simulated behavior in the environment with imperfect information, presenting a more natural, coherent and realistic behavior than the other agents.
Finally, two issues on the obtained results are important to point out. Firstly, the BDI agent with fuzzy perception seems to be a good model to be used in agent-based simulations in environments with imperfect information. Secondly, a fuzzy perception module can be a good alternative solution in the design of a BDI agent that can not perceive the information of the environment with accuracy.
Future work will consider a fuzzy perception mechanism for
a BDI agent that is more application independent. For that, we
are considering the use of the Mamdani inference method [
Acknowledgements. This work is part of a larger project (RSSOC: Rede Estadual de Simulação Social), being run under the FAPERGS/CNPq/PRONEX (Proc. 10/0049-7) context, where fuzzy perception of social interactions are considered. It is also supported by CAPES and CNPq (Proc. 483257/09-5, 307185/07-9, 304580/07