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Published Article / Planning problems are frequently encountered in everyday situations. The brute force approach of evaluating every possible solution for any medium size planning problem is just not feasible. Drools Planner is an open source Java library developed to help solve planning problems by using meta-heuristic algorithms. Drools Planner uses the Drools Expert (rule engine) for score calculation to greatly reduce the complexity and effort required to write scalable constraints in a declarative manner. This paper presents an introduction to Drools Planner, how it can be used to solve problems and concludes with an example scenario.

We study games where a team of agents needs to collaborate against an adversary to achieve a common goal. The agents make their moves simultaneously, and they have different perceptions about the system state after each move, due to different sensing capabilities. Each agent can only act based on its own experiences, since no communication is assumed during the game. However, before the game begins, the agents can agree on some strategy. A strategy is winning if it guarantees that the agents achieve their goal regardless of how the opponent acts. Identifying a winning strategy, or determining that none exists, is known as the strategy synthesis problem. In this thesis, we only consider a simple objective where the agents must force the game into a given state. Much of the literature is focused on strategies that either rely on that the agents (a) can remember everything that they have perceived or (b) can only remember the last thing that they have perceived. The strategy synthesis problem is (in the general case) undecidable in (a) and has exponential running time in (b). We are interested in the middle, where agents can have finite memory. Specifically, they should be able to keep a finite-state machine, which they update when they make new observations. In our case, the internal state of each agent represents its knowledge about the state of affairs. In other words, an agent is able to update its knowledge, and act based on it. We propose an algorithm for constructing the finite-state machine for each agent, and assigning actions to the internal states before the game begins. Not every winning strategy can be found by the algorithm, but we are convinced that the ones found are valid ones. An important building block for the algorithm is the knowledge-based subset construction (KBSC) used in the literature, which we generalise to games with multiple agents. With our construction, the game can be reduced to another game, still with uncertain state information, but with less or equal uncertainty. The construction can be applied arbitrarily many times, but it appears as if it stabilises (so that no new knowledge is gained) after only a few steps. We discuss this and other interesting properties of our algorithm in the final chapters of this thesis. / Vi studerar spel där ett lag agenter behöver samarbeta mot en motståndare för att uppnå ett mål. Agenterna agerar samtidigt, och vid varje steg av spelet så har de olika uppfattning om spelets tillstånd. De antas inte kunna kommunicera under spelets gång, så agenterna kan bara agera utifrån sina egna erfarenheter. Innan spelet börjar kan agenterna dock komma överrens om en strategi. En sådan strategi är vinnande om den garanterar att agenterna når sitt mål oavsett hur motståndaren beter sig. Att hitta en vinnande strategi är känt som syntesproblemet. I den här avhandlingen behandlar vi endast ett enkelt mål där agenterna måste tvinga in spelet i ett givet tillstånd. Mycket av litteraturen handlar om strategier där agenterna antingen antas (a) kunna minnas allt som de upplevt eller (b) bara kunna minnas det senaste de upplevt. Syntesproblemet är (i det generella fallet) oavgörbart i (a) och tar exponentiell tid i (b). Vi är intressede av fallet där agenter kan ha ändligt minne. De ska kunna ha en ändlig automat, som de kan uppdatera när de får nya observationer. I vårt fall så representerar det interna tillståndet agentens kunskap om spelets tillstånd. En agent kan då uppdatera sin kunskap och agera utifrån den. Vi föreslår en algoritm som konstruerar en ändlig automat åt varje agent, samt instruktioner för vad agenten ska göra i varje internt tillstånd. Varje vinnande strategi kan inte hittas av algoritmen, men vi är övertygade om att de som hittas är giltiga. En viktig byggsten är den kunskapsbaserade delmängskonstruktionen (KBSC), som vi generaliserar till spel med flera agenter. Med vår konstruktion kan spelet reduceras till ett annat spel som har mindre eller lika mycket osäkerhet. Detta kan göras godtyckligt många gånger, men det verkar som om att ingen ny kunskap tillkommer efter bara några gånger. Vi diskuterar detta vidare tillsammans med andra intressanta egenskaper hos algoritmen i de sista kapitlen i avhandlingen.