Complexity in Interaction
Lena Kurzen
Abstract:
This dissertation presents a formal analysis of the computational complexity aspects of interaction of agents.
As many interactive scenarios can be represented as relational structures, we use modal logic for formalizing them. This leads us to an investigation of the complexity of modal logics for reasoning about interaction. Capturing the complexity of interaction in terms of the complexity of satisfiability or model checking however might not be very accurate as these problems are concerned with arbitrary formulas of the associated language and not only with those expressing relevant concepts for interaction.
In order to capture more interaction-specific complexities in game-like interactive processes, we can investigate the complexity of deciding if a player has a winning strategy in a given game. Testing the robustness of this complexity with respect to small changes in the game rules or changing objectives of the players then helps to determine the sources of the complexity.
The complexity notions discussed above capture the complexity of interaction from an external perspective by focussing on how difficult it is to reason about interaction. Taking a more agent-oriented perspective leads us to the following question. What is the complexity of comparing agents' information structures and in how far is it influenced by underlying simplifying assumptions on the agents?
The final task arising for a formal analysis of the complexity of interaction is then to determine to what extent the analysis has implications for real-life interactive processes.
In this dissertation, it is shown how strategic ability of groups and individuals can be made explicit in modal logic in terms of actions and a representation of agents' preferences (Chapter 2). We discuss the conceptual benefits of such an approach and also determine the computational consequences for the satisfiability problem of the resulting logic, which turns out to be decidable in NEXPTIME and EXPTIME-hard.
More generally, considering simple coalition-labeled transition systems and action and power based normal modal logics for reasoning about cooperative ability, Chapter 3 shows that the choice of primitive influences whether weak or strong stability notions are easier to express.
Focussing on more specific game-like interaction and the complexity of determining if a player has a wining strategy, Chapter 4 analyzes different versions of Sabotage Games, two player games played on a graph with one player moving through the graph trying to reach a goal vertex and the other player obstructing him by removing edges. It is shown that cooperative versions of this game are easiest (NL-complete), while the non-cooperative versions with reachability and safety objectives for the first player are both PSPACE- complete. The complexities are robust with respect to small changes in the procedural rules of the games.
Zooming in onto the agents involved in interaction, Chapter 5 analyzes the complexity of determining the relationship between different information structures of agents. It is shown that when assuming knowledge to be truthful and fully introspective a complexity jump occurs with the introduction of a second agent, while in the general case more agents do not increase the complexity. For problems related to the manipulation of agents' information it is shown that the assumptions on agents' knowledge can make intractable problems tractable.
In Chapter 6, the computational complexity of actually playing a recreational game is investigated for the inductive inference game Eleusis. Using Post's Correspondence Problem, it is shown that players can be forced to face undecidable problems during the game, which makes the game in principle unplayable. Recommendations are given for adjusting the rules of the game.