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Automata-Based Quantitative Verification RICE UNIVERSITY Automata-Based Quantitative Reasoning by Suguman Bansal A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved, Thesis Committee: Rajeev Alur Zisman Family Professor of Computer and Information Science, University of Pennsylvania Konstantinos Mamouras Assistant Professor of Computer Science, Rice University Moshe Y. Vardi, Chair University Professor, Karen Ostrum arXiv:2010.02055v1 [cs.FL] 5 Oct 2020 George Distinguished Service Professor in Computational Engineering, Rice University Peter J. Varman Professor of Electrical and Computer Engineering, Rice University Houston, Texas June, 2020 Dedicated to my family, My mother, Anita, who now resides among the stars My father, Rajeev My brother, Sanchit ABSTRACT Automata-Based Quantitative Reasoning by Suguman Bansal The analysis of quantitative properties of computing systems, or quantitative anal- ysis in short, is an emerging area in automated formal analysis. Such properties address aspects such as costs and rewards, quality measures, resource consumption, distance metrics, and the like. So far, several applications of quantitative analysis have been identified, including formal guarantees for reinforcement learning, plan- ning under resource constraints, and verification of (multi-agent) on-line economic protocols. Existing solution approaches for problems in quantitative analysis suffer from two challenges that adversely impact the theoretical understanding of quantitative anal- ysis, and large-scale applicability due to limitations on scalability. These are the lack of generalizability, and separation-of-techniques. Lack of generalizability refers to the issue that solution approaches are often specialized to the underlying cost model that evaluates the quantitative property. Different cost models deploy such disparate algorithms that there is no transfer of knowledge from one cost model to another. Separation-of-techniques refers to the inherent dichotomy in solving prob- lems in quantitative analysis. Most algorithms comprise of two phases: A structural phase, which reasons about the structure of the quantitative system(s) using tech- niques from automata or graphs; and a numerical phase, which reasons about the quantitative dimension/cost model using numerical methods. The techniques used in both phases are so unlike each other that they are difficult to combine, forcing the phases to be performed sequentially, thereby impacting scalability. This thesis contributes towards a novel framework that addresses these chal- lenges. The introduced framework, called comparator automata or comparators in short, builds on automata-theoretic foundations to generalize across a variety of cost models. The crux of comparators is that they enable automata-based methods in the numerical phase, hence eradicating the dependence on numerical methods. In doing so, comparators are able to integrate the structural and numerical phases. On the theoretical front, we demonstrate that comparator-based solutions have the advantage of generalizable results, and yield complexity-theoretic improvements over a range of problems in quantitative analysis. On the practical front, we demonstrate through empirical analysis that comparator-based solutions render more efficient, scalable, and robust performance, and hold the ability to integrate quantitative with qualitative objectives. List of Publications This thesis is based on the following publications. Publications1-2 are yet to appear in an official proceedings at the time of thesis submission. 1. On the analysis of quantitative games Suguman Bansal, Krishnendu Chatterjee, and Moshe Y. Vardi 2. Anytime discounted-sum inclusion Suguman Bansal and Moshe Y. Vardi 3. Safety and co-safety comparator automata for discounted-sum inclu- sion Suguman Bansal and Moshe Y. Vardi In Proceedings of International Conference on Computer-Aided Verification (CAV) 2019 4. Automata vs linear-programming discounted-sum inclusion Suguman Bansal, Swarat Chaudhuri, and Moshe Y. Vardi In Proceedings of International Conference on Computer-Aided Verification (CAV) 2018 5. Comparator automata in quantitative verification Suguman Bansal, Swarat Chaudhuri, and Moshe Y. Vardi In Proceedings of International Conference on Foundations of Software Science and Computation Structures (FoSSaCS) 2018 (Extended version with additional results on Arxiv) Acknowledgements Foremost, I would like to thank my advisor and mentor Moshe Vardi for his unwa- vering support, constant guidance, and encouragement to pursue my ideas. He took me under his wings during the most difficult times I have faced, both academic and personal. Needless to say, there has been no looking back since. Thank you for giving me the freedom for ample exploration while also nudging me towards the right path. Most importantly, thank you for believing in me. I hope to be the advisor you have been to me to another dreamer someday. I am fortunate to have worked with an excellent thesis committee: Rajeev Alur, Konstantinos Mamouras, and Peter Varman. Their suggestions, feedback, and thor- ough evaluations of this thesis have led to numerous improvements and pursuits for future work. Thanks go to Krishnendu Chatterjee and Swarat Chaudhuri, collabo- rators on parts of this thesis. Their feedback on my doctoral research from the very beginning has been crucial in shaping the course. I am grateful to have found mentors in Swarat and Kedar Namjoshi. Right from building my foundations in Computer Science, Swarat has helped me navigate through the bigger career decisions. Incidentally, it was upon his advice that I took up the internship offer from Bell Labs to work with Kedar, which has turned out to be one of my best collaborative experiences. The daily, intense brainstorming sessions with Kedar were tiring yet immensely fulfilling. That said, a respite from the sweltering Houston weather takes the cherry on the cake for both of my summers at Bell Labs. During the Ph.D., I had the opportunity to collaborate with several brilliant researchers from around the globe: Rajeev Alur, Shaull Almagor, Swarat Chaudhuri, Krishnendu Chatterjee, Dror Fried, Yong Li, Kuldeep Meel, Kedar Namjoshi, Yaniv Sa'ar, Lucas Tabajara, Moshe Vardi, Andrew Wells. I am, perhaps, most partial to collaborations with my peers at Rice - Yong Li and Lucas Tabajara - where discussions would begin in our offices but end in Valhalla (Viva Valhalla!). The fact that there is a large overlap between my friends and (extended) research group LAPIS is a testament to how instrumental they have been in this journey. Dror Fried has been so much more than a friend to me. I am eternally grateful to his wife Sagit, him, and their three little kids (not so little anymore) for giving me an abode in their hearts. In a discipline that suffers from a dearth of women, I have been very lucky to have Afsaneh Rahbar and Shufang Zhu by my side. These women have truly made my successes and failures their own, as we continue to inspire each other with our \We can do it" chant. Because Aditya and I share a history of borrowing each other's sentences, I'll paraphrase him here - Thank you to Yong Li and Aditya Shrotri for keeping me company during the late nights and weekends in the office, the car rides home, coffee breaks, lunches, and dinners. I am humbled to share a special siblings-like bond with Vu Phan. I have relied on Jeffery Dudek, Antonio Di Stasio, my officemate Lucas Tabajara, Kevin Smith, Abhinav Verma, and Zhiwei Zhang for cerebral discussions and simple reasons for laughter. My life outside the colorful walls of Duncan Hall would have been so lack-luster had it not been for the friendships of Priyadarsini (Priya) Dasari, Vaideesh Logan, Rakesh Malladi, Sushma Sri Pamulapati, and the folks@IMT. In addition to keeping our apartment in pristine condition, my flatmate Priya has made me a better cook, and counseled me through personal and work-related turmoils; all of this over a steaming cup of chai. The folks@IMT have been a source of strength in the times of social-distancing/isolation during the ongoing Covid-19 pandemic. In these uncertain times, their small gestures have gone a long way to ease the anxiety around the global upheaval. Thank you all for making Houston my home. My friends from undergraduate and school have been available in moments of despair and self-doubt. I cannot thank Saheli, Siddharth, Siddhesh, and Visu enough for having been just a phone call away for almost a decade. Of all the things I had imagined graduate school would entail, meeting the love of my life was not on the agenda. Yet, here we are. Whether we are ten centimeters or ten thousand miles apart, Kuldeep can always make me smile. Thank you, ghano saaro, for brightening my day, every day. Finally but most importantly, I owe my deepest gratitude to my parents, Anita Bansal and Rajeev Kumar, and brother Sanchit Bansal. Right since I can remember, they filled me with curiosity, creativity, and imagination, and have shunned societal norms so that I could run, stumble, and chase after my dreams. Their love and support have been unconditional. It is to them that I dedicate this thesis. Yet there is a pit in my stomach. We lost my mother, Anita, four years ago. She was my loudest champion and strictest critic. Never had I imagined crossing this milestone without her cheering from the stands. But I am not upset, because I know that wherever she is, she is very proud of her guriya. We miss you every day, Mummy! Contents Abstract List of Illustrations 1 Introduction1 1.1 Quantitative analysis . .1 1.2 Challenges in quantitative analysis . .5 1.3 Thesis contributions . .8 1.4 Outline . 10 2 Background 12 2.1 Automata and formal languages . 12 2.2 Games over graphs . 14 2.3 Aggregate functions . 17 2.4 Quantitative inclusion . 20 2.5 Solving quantitative games . 21 I Theoretical framework 23 3 Comparator automata 25 3.1 Comparison language and comparator automata . 28 3.1.1 !-regular comparator . 29 3.1.2 !-pushdown comparator . 34 3.2 Generalizability with !-regular comparators .
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