Model Checking Linear Properties of Prefix-Recognizable Systems

We develop an automata-theoretic framework for reasoning about linear properties of infinite-state sequential systems. Our framework is based on the observation that states of such systems, which carry a finite but unbounded amount of information, can be viewed as nodes in an infinite tree, and transitions between states can be simulated by finite-state automata. Checking that the system satisfies a temporal property can then be done by an alternating two-way automaton that navigates through the tree. For branching properties, the framework is known and the two-way alternating automaton is a tree automaton. Applying the framework for linear properties results in algorithms that are not optimal. Indeed, the fact that a tree automaton can split to copies and simultaneously read all the paths of the tree has a computational price and is irrelevant for linear properties. We introduce path automata on trees. The input to a path automaton is a tree, but the automaton cannot split to copies and it can read only a single path of the tree. In particular, two-way nondeterministic path automata enable exactly the type of navigation that is required in order to check linear properties of infinite-state sequential systems. As has been the case with finite-state systems, the automata-theoretic framework is quite versatile. We demonstrate it by solving several versions of the model-checking problem for LTL specifications and prefix-recognizable systems. Our algorithm is exponential in both the size of (the description of) the system and the size of the LTL specification, and we prove a matching lower bound. This is the first optimal algorithm for solving the LTL model-checking problem for prefix recognizable systems. Our framework also handles systems with regular labeling, and in fact we show that LTL model checking with respect to pushdown systems with regular labeling is intereducible with LTL model checking with respect to prefix-recognizable systems with simple labeling. ∗Address: School of Computer Science and Engineering, Hebrew University, Jerusalem 91904, Israel. Email: [email protected] †Department of Computer Science and Applied Mathematics, Weizmann institute, Rehovot 76100, Israel. Email: [email protected] ‡Address: Department of Computer Science, Rice University, Houston TX 77005-1892, U.S.A. Email: [email protected]