Agent Interval Temporal Logic Agent Interval Temporal Logic
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Agent Interval Temporal Logic Agent Interval Temporal Logic by Johannes Francois Oberholzer A dissertation submitted in fulfilment of the requirements for the degree Masters in Philosophy in the Department of Philosophy at the UNIVERSITY OF PRETORIA FACULTY OF HUMANITIES SUPERVISOR: Prof Emma Ruttkamp-Bloem CO-SUPERVISOR: Prof Stefan Gruner January 2020 Acknowledgements I would like to thank my supervisor Professor Emma Ruttkamp-Bloem of the Phi- losophy Department at the University of Pretoria. Without her endless patience and expert feedback, this work would not have been possible. Her endurance is legendary, and her caring attitude makes her a beloved mentor. I would like to thank my co-supervisor Professor Stefan Gruner of the Computer Science Department at the University of Pretoria. He consistently allowed this paper to be my own work, but steered me in the right the direction whenever he thought I needed it. His unique insights were valuable. I would like to thank Professor Valentin Goranko of the Philosophy Department at Stockholm University, who provided crucial advice at various stages of writing this dissertation. I am also grateful for his assistance in helping me attend the Nordic Logic Summer School 2017 in Stockholm, where I gained many valuable perspectives. I would like to thank Dr. Nils Timm of the Computer Science Department at the University of Pretoria. His assistance with revising multiple key chapters is greatly appreciated. I would like to thank Ken Halland of the School of Computing at the Univer- sity of South Africa, as an additional reader of this dissertation. I am gratefully indebted to him for his very valuable comments on the entire dissertation in its final stages, comments without which this work might never have gotten done. I would like to thank the CSIR and the Centre for Artificial Intelligence Re- search for funding this dissertation. Finally, I would like to thank my parents, family and friends, for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this dissertation. This accomplishment would not have been possible without them. Thank you. Francois i Abstract Alternating-Time Temporal Logic (ATL), introduced by Alur, Henzinger and Kupferman, is a logic involving coalitions of agents performing actions which cause a state change in a turn-based time system. There have been game theoretic ex- tensions on ATL, and they are very good at specifying systems of multiple agents cooperating or competing in a game-like situation. Unfortunately neither ATL nor its extensions are able to capture the idea of gradual change, or duration of actions or events. The concurrent game model of ATL operates like a turn based game, with sets of agents taking their turn, and then the environment changing based on their actions, before they take their next turn. The fact that some actions take longer than others, or that sometimes a state changes gradually, rather than immediately, is not representable in ATL. As an example, take a train entering a tunnel. Before the train enters the tunnel, it is outside the tunnel, after it has entered the tunnel, it is inside the tunnel, but for the few seconds it takes the train to enter the tunnel, it is neither inside nor outside the tunnel. ATL cannot represent this basic intuitive truth. A family of logics called Interval Logic (IL) use finite state sequences called \intervals", which allow it to describe a more continuous model of time, rather than a discrete state based one such as ATL. This allows it to capture the idea of gradual change, of a train entering a tunnel, and the fact that actions and events have various durations. Most of the IL formulations do however not have any way of distinguishing multiple agents acting at the same time. Both of these logics - ATL and IL - are useful for specific things, but combining them might produce new applications which are not possible when only using the one or the other. In this dissertation we present one such possible combination, called Agent Interval Temporal Logic (AITL). AITL combines the notion of agents, coalitions and strategies from ATL with the interval based model of time from IL, thus creating a new logic which might have some powerful applications in a wide range of areas in which gradual change and multiple agents acting at the same time can both be accommodated. ii Contents 1 Introduction vi 2 Background 1 2.1 Historical Development . 2 2.1.1 Classical Logic . 2 2.1.2 Modal Logic . 4 2.1.3 Temporal Logic . 6 2.1.4 LTL, CTL, and CTL* . 8 2.1.5 Agents and Strategies . 12 2.2 Characteristics of Temporal Logics . 14 2.2.1 Propositional or Predicate . 14 2.2.2 Points or Intervals . 15 2.2.3 Duration . 16 2.2.4 Properties of Time . 16 2.3 Problems with Prediction . 18 2.4 Conclusion . 19 3 Problems Illustrating Practical Application of Temporal Logics 20 3.1 The Banker's Algorithm . 21 3.2 The Sleeping Barber Problem . 23 3.3 The Trains Problem . 26 4 Alternating-time Temporal Logic 29 4.1 Syntax and Semantics . 29 4.1.1 Concurrent Game Models . 29 4.1.2 Syntax . 34 4.1.3 Semantics . 34 4.1.4 ATL* . 35 4.2 Problems . 35 4.2.1 The Banker's Algorithm . 35 4.2.2 The Sleeping Barber . 37 iii CONTENTS 4.2.3 The Trains Problem . 40 4.3 Extensions of ATL . 42 4.3.1 Strategic Commitment . 43 4.3.2 Explicit Strategies . 44 4.3.3 Incomplete Information . 46 4.4 Conclusion . 47 5 The Logic of Intervals 48 5.1 Preliminaries . 48 5.1.1 Intervals . 49 5.1.2 Relations . 50 5.2 Interval Logics . 52 5.2.1 Propositional Modal Logic of Time Intervals . 52 5.2.2 Interval Temporal Logic . 54 5.2.3 Neighbourhood Logic . 56 5.2.4 Duration Calculus . 57 5.3 Allen's Logic of Actions and Events . 57 5.3.1 Explanation Closure Axioms . 61 5.4 Problems - The Banker's Algorithm . 62 5.5 Conclusion . 67 6 Agent Interval Temporal Logic 69 6.1 Domain . 69 6.1.1 Scenario . 70 6.1.2 States and Events . 70 6.1.3 Agents . 71 6.1.4 Actions and Tasks . 71 6.1.5 Intervals . 72 6.1.6 Relations . 73 6.1.7 Restrictions . 75 6.1.8 Strategies and Goals . 76 6.2 Model . 77 6.2.1 Scenario Model . 77 6.2.2 Strategies . 78 6.3 Syntax . 79 6.4 Semantics . 80 6.5 Example . 81 6.6 Problems . 89 6.6.1 The Banker's Algorithm . 89 6.6.2 The Sleeping Barber . 92 6.6.3 The Trains Problem . 94 iv CONTENTS 6.7 Conclusion . 95 7 Conclusion 97 7.1 Summary and Contribution . 97 7.2 Applications . 98 7.3 Future Work . 99 Appendix A 101 Bibliography 110 v Chapter 1 Introduction This dissertation aims to contribute to the field of formal logic and knowledge representation and reasoning by presenting a logic for reasoning about the strate- gic abilities of agents. Specifically we consider agents in a multi-agent system while incorporating aspects of interval time models. This new logic is an exten- sion of Alternating-Time Temporal Logic (ATL) and will be called Agent Interval Temporal Logic (AITL). Before any agent can act, it must decide on what action to take. Before it can make a decision on the best action to take, it must consider the possible outcomes of the action, and try to predict what effect the action will have. While in ordinary circumstances humans do not necessarily consider the process by which they make decisions, in the context of Artificial Intelligence for instance, it becomes necessary to have the ability to formally represent the decision making process for the purposes of programming. One of the functions of logic is to enable us to represent knowledge of a system and predict the outcomes of various events in that system. This will be our main focus in this dissertation: to develop a formal mechanism that allows correct prediction of some future state based on the current state and events which occur to change it. McCarthy and Hayes (1981) state that: \A computer program capable of acting intelligently in the world must have a general representation of the world in terms of which its inputs are interpreted... More specifically, we want a computer program that decides what to do by inferring in a formal language that a certain strategy will achieve its assigned goal. This requires formalising concepts of causality, ability, and knowledge". While the field of AI has since moved from knowledge-based to data-driven platforms, the need for formal mechanisms to accurately represent the essence of these concepts remains. A very important aspect of acting in the world is time. Shoham (1987) states that: \It is hard to think of a research area within Artificial Intelligence (AI) that does not involve reasoning about time in one way or another". Temporal vi logic (the logic of time) has been very successful at allowing reasoning about time and has had \numerous important applications not only to philosophy, but also to computer science, artificial intelligence, and linguistics" (Goranko and Galton, 2015). Alternating-Time Temporal Logic (ATL), introduced by Alur et al. (2002), is a very popular temporal logic for reasoning about the strategic abilities of agents Goranko et al. (2018), and will be fundamental to this dissertation. ATL is however restricted to a turn-based time model, where the agents and the system are in a back and forth game with each other.