Computational Complexity of Propositional Dynamic Logics Summary of Dissertation

In 1979, Fischer and Ladner introduced Propositional Dynamic Logic (PDL) as a logical formalism for reasoning about programs [9]. Since then, PDL has become a classic of logic in computer science [11], and many extensions and variations have been proposed. Several of these extensions are inspired by the original application of reasoning about programs, while others aim at the numerous novel applications that PDL has found since its invention. Notable examples of such applications include agent-based systems [14], regular path constraints for querying semi-structured data [2], and XML-querying [1, 17, 16]. In artifical intelligence, PDL received attention due to its close relationship to description logics [10] and epistemic logic [18, 19]. The models of PDL formulas and programs are transition systems (also called Kripke structures) whose transitions are labeled with atomic programs and whose states are labeled with atomic propositions. Formulas are defined from atomic propositions, are closed by the boolean operations, and finally for each program π and each formula φ, 〈π〉φ is a again a formula – with the semantics that there is some state both satisfying φ and reachable by executing π. Hence, formulas define subsets of the state set of Kripke structures. Programs are built up from atomic ones, are closed under the operations union, composition, and Kleene star, and finally whenever φ is a formula, then φ? is a program defining loops in those states that satisfy φ. Hence programs define binary relations on the state set of Kripke structures. The satisfiability problems asks, given a formula φ, whether there exists a Kripke structure K and a state of K that satisfies φ. The model checking problem asks, given a Kripke structure K, a state x of K, and a formula φ, whether x satisfies φ.