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Quantitative Graph Games: Theory and Applications

Final Report Summary - GRAPH GAMES (Quantitative Graph Games: Theory and Applications)

Games played on graphs are central in many problems in computer science, especially in the analysis of reactive systems. Formal methods aim at rigorous techniques to improve quality in software and hardware designs by preventing design errors. In the core of formal methods is the logical notion of correctness of systems, and the basic mathematical foundation to analyze such logical frameworks for dynamical systems (that evolve over time, such as hardware and software systems that interact with environment and their behavior depends on the input) is graph games. While traditional graph games have been studied for Boolean (correct vs incorrect) properties we study quantitative aspects of such graph games that are required to model robustness, timeliness of responses, amount of resource consumption, and many such important properties.

Our project was organized along four inter-connected yet complementary directions, and significant progress was achieved in each one of them. First, we considered graph games with quantitative objectives. Second, we studied compositional techniques to analyze large-scale graph games. Third, we considered the applications of graph games in diverse domains, such as analysis of security protocols, as well as analysis of evolutionary game theory. Finally, we considered some fundamental theoretical questions in analysis of infinite-state games with quantitative objectives. Along with the above four directions, we made unexpected progress in improving the basic algorithmic techniques for several graph games improving the long-standing best-known bounds for the problems.

The main discoveries made in this project include the following new and novel results: analysis of multi-dimensional quantitative objectives in graph games (a technique that can be used to quantitatively estimate quality of a design); compositional techniques to analyze games (that allows large systems to be decomposed into smaller systems and combine the analysis to enable the analysis techniques to scale to large-scale systems); apply graph game techniques for correct design of security protocols (that are correct by construction and attack-free) and evolutionary game theory (for modeling problems related to dynamics of population or model growth of cancer); analysis of infinite-state systems (such as programs with recursion) with quantitative objectives; and developing new algorithmic techniques that improve the complexity of long-standing open theoretical questions (such as concurrent ergodic mean-payoff games and Buechi games that are central for formal methods and we improved the existing bounds known for decades).