Micro and Macro Lemmings Simulations Based on Ants Colonies Antonio Gonz´ alez-Pardo(B) , Fernando Palero, and David Camacho Computer Science Department, Universidad Aut´ onoma de Madrid, Madrid, Spain {antonio.gonzalez,david.camacho}@uam.es,
[email protected] http://aida.ii.uam.es
Abstract. Ant Colony Optimization (ACO) has been successfully applied to a wide number of complex and real domains. From classical optimization problems to video games, these kind of swarm-based approaches have been adapted, to be later used, to search for new metaheuristic based solutions. This paper presents a simple ACO algorithm that uses a specifically designed heuristic, called common-sense, which has been applied in the classical video game Lemmings. In this game a set of lemmings must reach the exit point of each level, using a subset of finite number of skills, taking into account the contextual information given from the level. The paper describes both the graph model and the context-based heuristic, designed to implement our ACO approach. Afterwards, two different kind of simulations have been carried out to analyse the behaviour of the ACO algorithm. On the one hand, a micro simulation, where each ant is used to model a lemming, and a macro simulation where a swarm of lemmings is represented using only one ant. Using both kind of simulations, a complete experimental comparison based on the number and quality of solutions found and the levels solved, is carried out to study the behaviour of the algorithm under different game configurations. Keywords: Lemmings video game · Micro and Macro simulations · Ant Colony Optimization algorithms
1
Introduction
Bio-inspired computation has been widely used in different areas from combinatorial optimization problems to stochastic search in a huge number of application domains. From industrial or engineering applications [10] to theoretical developments [13], they have been applied to study new bio-inspired approaches able to deal with NP-complete, or NP-hard, problems [1]. From the set of different methods and techniques that can be considered as bio-inspired: Artificial Neural Networks, Fuzzy Logic, Evolutionary Computation and Swarm Intelligence, this paper will be focused on the later. Swarm Intelligence (SI) algorithms are focused on the collective behaviour of self-organizing systems [14], where the iterations among individuals generate c Springer-Verlag Berlin Heidelberg 2014 A.I. Esparcia-Alc´ azar et al. (Eds.): EvoApplications 2014, LNCS 8602, pp. 337–348, 2014. DOI: 10.1007/978-3-662-45523-4 28
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collective knowledge based on social colonies [17]. Some examples of this type of algorithms are Ant Colony Optimization (ACO) [4,7,11,12]; Particle Swarm Optimization (PSO) [20]; Bee Colony Optimization (BCO) [18]; Bird Flocking [23] or Bacterial Foraging [9]. In these algorithms, the population travels through the solution space in order to obtain the best solution to the problem. Each solution is evaluated by a quality function and the resulting value is used to guide the whole population, or swarm, to the optimal solution. This quality function is usually designed as part of the meta-heuristic used by this kind of algorithms. From the different available methods related to Swarm Intelligence, the selection of ant colonies (ACO) algorithms has been made taking into account two main characteristics. On the one hand, ACO algorithms work with a population that allow us to make a simple analogy between the concept of a swarm of creatures, and the ants used to model and solve the problem. On the other hand, the video game selected (Lemmings Game) has several characteristics that makes particularly interesting the application of ACO algorithms, such as the possibility to include some physics in the context-based information, to have a set of finite skills which can be used to model an optimization problem, or the necessity to find the optimum path taking into account previous features, among others (see Section 2 for a detailed description). The increasing interest in the utilization of different techniques in video games [22] from areas like Artificial Intelligence (AI), Computational Intelligence (CI) or Machine Learning (ML), has originated a wide number of gamebased software platforms. In these platforms different classical video games, such as Pac-Man (Ms Pac-Man [21]), Tetris [5], Mario Bros (Platformer AI [24,25]), Mastermind [2], Asteroids (Physical Traveller Salesman Problem [6]) or Starcraft (StarCraft [3]) among others, have been adapted as a new benchmark environment for testing classical and new methods from previous areas. The Lemmings Game is a popular proven NP-hard puzzle game [8] that can be used as a benchmark for CI algorithms. In spite of the popularity that this game obtained in the 1990s, few research has been applied to it. This paper presents a simple ACO algorithm that uses a specifically designed heuristic to be applied in the Lemmings video game, where a set of creatures (Lemmings) must reach the exit point of each level using a subset of finite number of skills. The heuristic designed, named common-sense, takes into account the ”contextual information” that must be used in this game to solve a level. In order to do that, each level is represented as a contextual graph where the edges store the allowed movements inside the world. The goal of the algorithm is to assign the best skills in each position on a particular level, to guide the Lemmings to reach the exit. The common-sense heuristic allows to select the best skill to be applied by the Lemming in a particular level, using the current state of the level represented by this contextual graph. The paper describes both, the contextual graph model designed and the common-sense heuristic. On the other hand, two different kind of simulations have been carried out to analyse the behaviour of the ACO algorithms. It has been designed a micro
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simulation, where each ant is used to model a Lemming, and a macro simulation where a complete swarm of Lemmings is represented using only one ant. In the micro simulations two different approaches have been used: the first one only allows one action from each Lemming per step, therefore any Lemming must take into account the previous modifications made by the rest of the Lemmings. In the second kind of micro simulations, a set of parallel modifications (one per Lemming available) is made in the same step, so these actions ignore the context information from the environment. In the macro simulations, only the first Lemming of the swarm can execute an action in a step, whereas the rest of the Lemmings must follow the ”leader”. Finally, the paper provides a complete experimental evaluation between these three different kinds of simulations, based on the number and quality of solutions found, and the number and complexity of the levels solved. The main goal of these experiments will be to analyse the behaviour of the meta-heuristic designed, when different modifications are applied in the contextual graph that have been designed to model a level game. The rest of the paper is structured as follows: Section 2 provides a detailed description of the Lemmnings video game; Section 3 presents the basics on both, the model design to implement the contextual graph, and the main features of the ACO algorithm proposed; Section 4 analyses the two different simulations approaches (micro and macro) considered in this work; Section 6 provides a complete description of the experimental settings, and the results obtained from the simulations; finally, Section 5 summarizes the conclusions and introduces some futures lines of work.
2
The Lemmings Video Game
The Lemmings are creatures that need to be saved. In each level, Lemmings start in a specific point of the stage and must be guided to the exit point by the player. They live in a two-dimensional space and are affected by gravity. They start walking in a specific direction until they find an obstacle. In this case the Lemming will change the direction and walk back. In the case where the Lemming encounters a hole, it will fall down. The only two ways, considered in this work, by which a Lemming can die is by falling beyond a certain distance, or by falling from the bottom of the level. In order to make Lemmings to reach the exit point, players have a set of skills that must be assigned (not necessarily all of them) to the Lemmings. Using these skills, Lemmings can modify the environment creating tunnels, or bridges, and thus creating a new way to reach the exit. Following, the basic skills that can be used by any lemming, and the basic features to build the game levels are described. On the one hand, there are eight different skills, with different features, that are shown in Table 1. Some of these skills have No Restrictions (NR). This means that although the number of times that these skills can be assigned is limited, once it is assigned to the Lemming, it does not have any restriction to use it (i.e. Climber or Floater) several times in the same level. Other are Restricted (RE)
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Table 1. Lemmings skills and basic features (NR:No Restrictions, NE: No Exit, and RE: Restricted)
Skill Description Features Climber A Lemming given the climber skill can scale vertical walls NR Floater This skill allow the Lemming to open an umbrella if it NR falls beyond a high distance, avoiding its dead. Exploder The Lemming will explode after a short delay NE Blocker Using this skill, a Lemming will halt and the rest of LemNE ming will turn around Builder The Lemming with this skill will build a bridge of a speRE cific length Basher To create horizontal tunnels if the environment allows it RE Miner This skill is similar to the previous one, but in this case RE the tunnel is dug in diagonal direction Digger The Lemming will dig vertically downwards until it found RE air or a solid material
skills, so the Lemming only can use it a maximum number of times (i.e. Builder, Miner or Digger). For example, if the a Lemming has to dig in two separated locations this lemming must be assigned the Digger skill twice. Finally, there are some skills that do Not allow to reach the Exit (NE) to the Lemming, because the Lemming will die (i.e Exploder), or because it will not be able to make more movements (i.e. Blocker). On the other hand, in the Lemmings’ world there are a huge number of materials, but all of them can be grouped into two different classes: the ones that can be modified (i.e. it can be dug) and the ones that cannot be altered. In the former type, skills like Basher, Miner and Digger are allowed. In the case that a Lemming is digging and finds a material that cannot be dug, the Lemming will stop digging and start walking. Furthermore, each game level has its own skill configuration, where each skill can be used (i.e. assigned) a maximum number of times. It is not necessary to use all of the skills in the levels. Based on both kind of materials, editable and non editable, three different kinds of levels have been designed: – Easy. These levels use both kind of materials, and the human-likes solution is a short path (few lemmings actions) with few skills are required to reach the exit. When non editable material is used, the lemmings colonies are ”guided” to the exit because those skills related to ”digging” abilities cannot be used (therefore the search space is reduced). – Medium. In these kind of levels, both materials can be used and the solutions can be a mixture of actions. In the level, it is possible to find parts with a high level of freedom for the lemmings (they can use all of the available skills), and some other parts where the number of skills that can be used are reduced.
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– Hard. These type of levels only use editable materials, and the solution to reach the exit needs from a large number of skills and actions (large solution paths) to be taken. The Lemmings’ game can be considered an interesting research video game problem specially for optimization algorithms. Three main objectives are necessary to optimize in this game: to save the maximum number of Lemmings in each level, to minimize the use of skills needed to reach the exit of the level, and finally to find the best path that allows to save as many Lemmings as possible using the less number of skills. The Lemmings’ game have been studied in [19]. In this work, authors apply a genetic algorithm to solve the different levels and the goal is the study of how the individuals initialization can affect to the performance of the GA. Summarizing, the Lemmings video game provides (at least) two new interesting features. On the one hand, the video game provides different kind of terrains, that the algorithm must take into account to avoid a premature dead of the lemming, or to decide an adequate selection from the available skills. This characteristic provides an interesting ”context” that should be handled by the algorithm (for instance, by using a constraint-based modelling of the environment or a meta-heuristic to select the best skill). On the other hand, the game itself needs from the management and control of a colony of Lemmings. It is necessary to coordinate those lemmings to look for the best solution (which is based on a mixture of different goals).
3
The ACO Approach for the Lemmings Video Game
The Lemmings Game can be seen as a Constraint Satisfaction Problem (CSP ), where the variables (denoted as X) represent the different positions of the levels, and the possible values (D) represent are the skills that Lemmings can execute in each position. The set of constraints, C, is composed by the number of lemmings that must be saved, the maximum number of skills that can be applied in each level, or the different destination from a given position taking into account the applied skill (i.e. given a position the set of possible destination nodes is different whether the skill is Builder or Digger). In order to execute an ACO algorithm to solve a CSP, traditionally authors model the CSP as a graph where the nodes represent the variable/value pairs (< variable, value >) and the edges connect those nodes whose variable X are different. The problem with this representation is the size of the resulting graph. In this work, the model used to represent CSP as a graph is the one described in [15]. If the Lemmings level is mapped into a graph using the classical approach for each position, the resulting graph would have eight nodes (each of them represents the action that can be applied in the corresponding position). With the approach used in this work, each node only represents a position and the ants are in charge of selecting a specific skill to be applied in this position. The adaptation of a Lemmings level into the simplified approach is performed in two different phases. First of all, the level is represented in a two-dimensional
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representation that contains information about the starting point, the exit point and the terrain information of the level. In Figure 1 there are shown an original lemmings level (Figure 1(a)) and the simplification of this level into a twodimensional representation (Figure 1(b)). This representation is mapped into a constraint-based graph as Figure 1(c) shows. The constraint-based graph, or contextual graph, contains as many nodes as squares are contained in the two dimensional representation and the edges represent the default movement that ants can performed. It is important to note that the application of different skills in the graph will produce the creation of new edges in the graph, thus ants deal with a dynamic graph.
(a)
(b)
(c)
Fig. 1. An easy Lemmings level. The Figure a) shows one of the Lemmings level designed for the experiments carried out in this paper, the Figure b) shows a two dimensional representation of this level where only the starting and exit point, and the walls are represented. Finally, the Figure c) shows the constraint-based graph model for this level.
This work uses a classical ACO approach to search for the best paths of the levels. In this case, the nest of the colony is located in the node that represents the level starting point (marked as a ”S” node in Figure 1(c)), and the food is located in the node that represents the level exit point (marked as a ”G” node in Figure 1(c)). From the nest, ants start building their own local solution while they travel through the graph. In order to do that, each ant executes the behaviour shown in Algorithm 1. The first step in the algorithm corresponds to the heuristic information retrieval (line 2). In this work, a heuristic called Common-Sense has been used. Using this heuristic, ants can perceive the environment (i.e. ants know the type of terrain of the surrounding nodes) and filter the skills that they can apply depending on this environment. For example, given an ant if the type of the node where the ant is placed and their surrounding are Air, the ant knows that the Lemming is falling and a possible skill to apply is Floater but not Builder. Once the ants have the values for the different skills, corresponding to the heuristic function and the pheromones, the decision of selecting one of them is computed using the classical proportional selection.
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Algorithm 1. ACO algorithm for the Lemmings game
1 2 3 4 5 6 7 8 9 10 11 12 13 14
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Parameter: A contextual graph. A Swarm Sw composed by L Lemmings A set of available skills Sk Result: A path plan P to reach the Exit G, from the Start S. foreach li ∈ L do HeuristicSkillList ← getSkillsUsingHeuristic P heromoneV alues ← getPheromoneValues newAction ← selectAction(HeuristicSkillList, P heromoneV alues) if newAction = currenAction then if newAction canBeExecuted then putPheromone updateRemainingActions currentAction ← newAction add currentAction to P end end goToNextNodeAccordingTo(currentAction) end
The Micro and Macro Lemmings Simulations
In [16], some initial experiments were made using the common-sense heuristic and a simplified contextual graph.In our initial experiments, the model and the heuristic were compared (using some few levels and a simple configuration) against a Genetic Algorithm approach. No modifications were allowed in the contextual graph during the simulation process and the experimental results were used to demonstrate the feasibility of the approach. In this new work, the contextual graph will be modified by the ants (Lemmings) inside a simulation step, considering (or not) the context of the level. Therefore, two different kinds of simulations have been carried out to analyse the behaviour of the ACO algorithm. On the one hand, a micro simulation, where each ant is used to model a Lemming, and a macro simulation where a complete swarm of Lemmings is represented using only one ant. The main characteristics of both simulations can be summarized as follows: 1. In the micro simulations two different approaches have been used: ”one to one sequential” (1to1S ) and ”one to one parallel” (1to1P ). In the 1to1S simulations, only one action (the application of one skill) from each Lemming is allowed per step. Therefore, any Lemming must take into account the modifications than the rest of the Lemmings previously have made in the environment, so this kind of simulation can be considered as contextualbased, because the actions previously made by others Lemmings will affect to the current (scheduled) Lemming decision. In the 1to1P simulations, in each step all of the available Lemmings can make one action ignoring the
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contextual information from the environment. The main difference between both kind of algorithms is related to the contextual graph modification, the first one will provide a smooth modification of the graph, increasing the importance of the common-sense heuristic. The second approach will allow a fast modification of the graph, so the relevance of the meta-heuristic will be lower in the solution process. 2. In the macro simulations, denoted as 1toN (one to N ), only the first Lemming from the swarm can execute an action (skill) in a particular step, whereas the rest of the Lemmings must follow the ”leader” [16]. This kind of simulations provides a semi-static modification of the contextual graph. The graph is slowly modified, so the relevance of the meta-heuristic and the pheromone values will be increased. The parallel simulation is similar to the 1to1S ones, but the latter allows to explore better the solution space (any Lemming has an opportunity to apply an skill), whereas the first reduce the solution space by following a particular Lemming leader. Previous simulations allows to analyse the behaviour of our approach by modifying three essential features: how fast the graph could be modified, how affects the contextual information to the searching process, and finally the importance of the pheromone concentration in the searching process. Table 2 shows a summary of both, the simulations designed and their basic features. Table 2. Lemmings simulations and their related basic features. Simulation Graph modification Contextual inf. Num. Pheromones 1to1S medium high high 1to1P high low low 1toN low very high very high
5
Experimental Results
Fifteen different levels1 have been designed, by hand, to measure the efficiency our approach under different simulations configurations. The complexity of the levels is based on the size of the level, the different blocks contained into each level, the distance from the entry point to the exit point, the number of skills needed to solve the level, the type of terrains contained in the levels, etc. In this work, three different complexity levels are considered: easy, medium and hard, and 5 different levels have been designed per category. All the experiments have been repeated 50 times, using the described contextual graph and the commonsense heuristic. In each experiment, the ant colony is composed by 100 ants that execute during 500 steps. The evaporation rate of the system is 1% and α and β parameters (needed to measure the influence of the heuristic and the pheromone values) are fixed to 1. The number, and quality of the different found paths (solutions), have been used to compare the performance of our approach. 1
http://aida.ii.uam.es/researchers/facultystaff/gonzalez-pardo-antonio/
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The quality of any solution is composed by the number of lemmings that reach the exit (Eq. 1), the time needed to solve the problem (2) and the number of skills used (Eq. 3). The goal is to maximize the number of lemmings saved while the time needed to solve the problem and the number of skills used are minimized, but instead of facing this multi-objective problem, Eq. 4 is used and the goal is to maximize it. S(p) = T otalLemm − Blockers − ExplodedLemmings
(1)
T (p) = M axT ime − ExpendedT ime A(p) = T otalActGiv − ActionU sed(p) T (p) + A(p) + S(p) Q(p) = M axT ime + T otalActGiv + T otalLemm
(2) (3) (4)
Table 3 and Figure 2, shows the results of our approach with the three different simulations carried out. Figure 2 shows the number of different paths (solutions) found by each algorithm, whereas Table 3 shows the average and standard deviation of the solutions quality. Table 3. Average and standard deviation of the best solutions found by the ACO algorithm under different simulation configurations. These results have been obtained executing the different algorithms 50 different times.
Level Complexity 1to1S 1to1P 1toN 1 Easy 0,86 ± 0,009 0,69 ± 0,084 0,65 ± 0,089 2 Easy 0,88 ± 0,021 0,85 ± 0,032 0,83 ± 0,040 3 Easy 0,94 ± 0,038 0,94 ± 0,012 0,90 ± 0,024 4 Easy 0,82 ± 0,021 0,78 ± 0,030 0,77 ± 0,030 5 Easy 0,96 ± 0,015 0,95 ± 0,008 0,94 ± 0,019 6 Medium 0,91 ± 0,005 0,88 ± 0,031 0,85 ± 0,040 7 Medium 0,94 ± 0,000 0,89 ± 0,027 0,86 ± 0,037 8 Medium 0 ± 0,000 0,90 ± 0,020 0,91 ± 0,044 9 Medium 0 ± 0,000 0,93 ± 0,006 0,63 ± 0,018 10 Medium 0,69 ± 0,034 0,69 ± 0,093 0,78 ± 0,110 11 Hard 0 ± 0,000 0 ± 0,000 0,67 ± 0,052 12 Hard 0 ± 0,000 0 ± 0,000 0,88 ± 0,050 13 Hard 0,72 ± 0,061 0,91 ± 0,013 0,75 ± 0,034 14 Hard 0,78 ± 0,002 0,82 ± 0,029 0,90 ± 0,052 15 Hard ± 0,000 0,95 ± 0,003 0,94 ± 0,020
Analyzing the quality of the solutions (Table 3) the 1to1S approach obtains better solutions that 1to1P and 1toN in easy and medium levels. This is produced by two different reasons. On the one hand, with easy and medium levels the solution approach is not as bigger as the one in hard levels. So a sequential depth-first search is able to find good solutions, in the maximum number of simulation steps allowed. In this kind of levels is better to strongly use the
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contextual information than to make a wide parallel search. On the other hand, 1to1S approach performs a depth-first search in the solution space, while 1to1P and 1toN algorithms make a breadth-first search.
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Fig. 2. The figure shows the number of different solutions found by the algorithms. The Y axis represent the log2 of the different solutions found by the algorithms.
The Figure 2 shows the number of different paths that each algorithm is able to identify. In general, 1toN and 1to1P finds more paths than 1to1S . This effect is produced because the solution space is more explored by the 1toN and 1to1P than 1to1S . In a single execution of 1to1P and 1toN , the number of parallel searches are equals to the number of ants that compose the colony, while in 1to1S al the ants compose a single search. Comparing 1to1P and 1toN can be seen that 1toN , in general, finds more different paths that 1to1P . This is an expected results because although both algorithms make a breadth-first search, 1toN makes more parallel searches than 1to1P .
6
Conclusions
This paper analyses the behaviour of three different ACO-based approaches related to the automatic solving level problem in games. The application domain of this work is the well-known Lemmings game, where a set of Lemmings need to apply different skills in order to reach the exit. Three different categories of levels have been designed, with five levels per each category. The complexity of each level is defined by the size of the level, the number of available skills that can be applied, and the different types of terrains that compose the level. The three different approaches considered in this work are: macro simulations, denoted as 1toN (one to N ) where a swarm of lemmings is represented using only one ant, and two micro simulations denoted as 1to1S and 1to1P . In the 1to1S simulations, only one action (the application of one skill) to each Lemming is allowed per step. Also, the Lemmings share the context, this means that
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any modification performed by one Lemming is visible to the rest of them, so the contextual information will be used by the rest of the Lemmings. However, in the 1to1P simulations, in each step all of the available Lemmings can make one action, therefore they can ignore the contextual information from the environment. One of the main differences between previous approaches (macro and micro) is related to the global behaviour of the searching algorithm for those kinds of simulations. In 1toN and 1to1P the algorithm makes a parallel search because they allow the application of different skills at the same time with different lemmings. On the other hand, the 1to1S approach is likely a sequential search because the Lemmings must apply their skills in order taking into account the contextual information. From the experimental results shown in Table 3 and Figure 2, two main conclusions can be summarized. On the one hand, for easy and some (few) medium levels, the 1to1S approach obtains the highest quality solutions because in those simple levels the ants are able to find short paths (solutions) from the exit using the contextual information. However, once the complexity of the level is increased, this approach has problems to find good quality solutions, or even a solution. In these levels the parallel approaches, 1toN and 1to1P , find the best solutions. From both approaches, the 1toN simulation, is able to find solutions for all the levels considered. This means that the contextual information, used through the common-sense heuristic, guides efficiently the algorithm enabling it to solve the hardest designed levels. Acknowledgments. This work has been partly supported by Spanish Ministry of Science and Education under grant TIN2010-19872 (ABANT) and Savier project (Airbus Defense & Space project, FUAM-076914).
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