Term-modal logic and quantifier-free dynamic assignment logic

Thalmann, Lars 2000: Term-Modal Logic and Quantifier-free Dynamic Assignment Logic. Uppsala Theses in Computing Science 34. 140 pp. Uppsala. ISSN 0283–359X, ISBN 91–506–1443–6. In this dissertation, we present two new sorts of computer science logics. Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way. To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce in the first part of the dissertation a family of logics called term-modal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators. This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. We give examples of the expressiveness of the languages and provide sequent-style and tableau-based proof systems for the logics. Furthermore, we give proofs of soundness and completeness with respect to the possible world semantics. In the second part of the dissertation, we treat another problem in reasoning about multi-agent systems, namely the problem of information updating. We develop a dynamic logic of assignments with a scoping operator instead of quantifiers. Function, relation symbols and logic variables are all rigidly interpreted in our semantics, while program variables are non-rigid. The scoping operator is used to distinguish between the value of a program variable before and after the execution of a program. We provide a tableau proof system for the logic. First, the system is proved complete without the star operator, and then with the star operator using an omega rule. The full logic is shown to be undecidable, while some interesting fragments are decidable. Lars Thalmann, Computing Science Department, Information Technology, Uppsala University, Box 337, SE–751 05 Uppsala, Sweden. c © Lars Thalmann 2000 www.LarsThalmann.com ISSN 0283–359X ISBN 91–506–1443–6 Printed by Nina Tryckeri HB, Uppsala 2000 To my parents and my sister Acknowledgments I would first of all like to thank my advisor, Prof. Andrei Voronkov, for his advise and guidance, first during the years at Uppsala University, and later during my visits to Manchester University. His suggestions and comments have been of utmost importance for completing the research made in this dissertation and for getting to know what characterize good computer science. From the summer of 1998, until late 1999, I spent a year visiting Prof. Melvin Fitting at the City University of New York. His suggestions and encouragement were key ingredients in formulating and carrying out the work especially behind the second part of the thesis. The many discussions we have had, have been both fun and interesting, much due to Mel’s ability to discuss complicated things in a simple way. During the work at Uppsala University, Faron Moller has shown me ways to prove things rigorously as well as structured. Our many discussions about computer science, as well as teaching, has taught me a lot. My appreciation also goes to the present and past members of the Computing Science Department and other departments at Uppsala University: Anatoli, Anders, Arne, Cons, Evgeny, Greger, Göran, Happi, H̊akan, Helena, Hessmo, Jan, Joel, Kostis, Margus, Marianne, Marko, Mikael, Monika, Sergei, Sven-Olof, Thomas, Peder, Per, Pierangelo, Plopp, Rafal, Richard, Roland, and all other I have forgotten to mention. Finally, I would especially like to thank my family and all my friends, for all the great times, at work as well as off work. Special thanks also to Andrei, Carin, Maria and Rafal for proofreading the dissertation.