Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems

Higher-order pushdown systems (PDSs) generalise pushdown systems through the use of higher-order stacks, that is, a nested "stack of stacks" structure. These systems may be used to model higher-order programs and are closely related to the Caucal hierarchy of infinite graphs and safe higher-order recursion schemes. We consider the backwards-reachability problem over higher-order Alternating PDSs (APDSs), a generalisation of higher-order PDSs. This builds on and extends previous work on pushdown systems and context-free higher-order processes in a non-trivial manner. In particular, we show that the set of configurations from which a regular set of higher-order APDS configurations is reachable is regular and computable in n-EXPTIME. In fact, the problem is n-EXPTIME-complete. We show that this work has several applications in the verification of higher-order PDSs, such as linear-time model-checking, alternation-free mu-calculus model-checking and the computation of winning regions of reachability games

Saturation methods for global model-checking pushdown systems

Pushdown systems equip a finite state system with an unbounded stack memory, and are thus infinite state. By recording the call history on the stack, these systems provide a natural model for recursive procedure calls. Model-checking for pushdown systems has been well-studied. Tools implementing pushdown model-checking (e.g. Moped) are an essential back-end component of high-profile software model checkers such as SLAM, Blast and Terminator.Higher-order pushdown systems define a more complex memory structure: a higher-order stack is a stack of lower-order stacks. These systems form a robust hierarchy closely related to the Caucal hierarchy and higher-order recursion schemes. This latter connection demonstrates their importance as models for programs with higher-order functions.We study the global model-checking problem for (higher-order) pushdown systems. In particular, we present a new algorithm for computing the winning regions of a parity game played over an order-1 pushdown system. We then show how to compute the winning regions of two-player reachability games over order-n pushdown systems. These algorithms extend the saturation methods of Bouajjani, Esparza and Maler for order-1 pushdown systems, and Bouajjani and Meyer for higher-order pushdown systems with a single control state. These techniques begin with an automaton recognising (higher-order) stacks, and iteratively add new transitions until the automaton becomes saturated.The reachability result, presented at FoSSaCS 2007 and in the LMCS journal, is the main contribution of the thesis. We break the saturation paradigm by adding new states to the automaton during the iteration. We identify the fixed points required for termination by tracking the updates that are applied, rather than by observing the transition structure. We give a number of applications of this result to LTL model-checking, branching-time model-checking, non-emptiness of higher-order pushdown automata and Büchi games.Our second major contribution is the first application of the saturation technique to parity games. We begin with a mu-calculus characterisation of the winning region. This formula alternates greatest and least fixed point operators over a kind of reachability formula. Hence, we can use a version of our reachability algorithm, and modifications of the Büchi techniques, to compute the required result. The main advantages of this approach compared to existing techniques due to Cachat, Serre and Vardi et al. are that it is direct and that it is not immediately exponential in the number of control states, although the worst-case complexity remains the same.</p

Saturation Methods for Global Model−Checking Pushdown Systems

Pushdown systems equip a finite state system with an unbounded stack memory, and are thus infinite state. By recording the call history on the stack, these systems provide a natural model for recursive procedure calls. Model-checking for pushdown systems has been well-studied. Tools implementing pushdown model-checking (e.g. Moped) are an essential back-end component of high-profile software model checkers such as SLAM, Blast and Terminator.Higher-order pushdown systems define a more complex memory structure: a higher-order stack is a stack of lower-order stacks. These systems form a robust hierarchy closely related to the Caucal hierarchy and higher-order recursion schemes. This latter connection demonstrates their importance as models for programs with higher-order functions.We study the global model-checking problem for (higher-order) pushdown systems. In particular, we present a new algorithm for computing the winning regions of a parity game played over an order-1 pushdown system. We then show how to compute the winning regions of two-player reachability games over order-n pushdown systems. These algorithms extend the saturation methods of Bouajjani, Esparza and Maler for order-1 pushdown systems, and Bouajjani and Meyer for higher-order pushdown systems with a single control state. These techniques begin with an automaton recognising (higher-order) stacks, and iteratively add new transitions until the automaton becomes saturated.The reachability result, presented at FoSSaCS 2007 and in the LMCS journal, is the main contribution of the thesis. We break the saturation paradigm by adding new states to the automaton during the iteration. We identify the fixed points required for termination by tracking the updates that are applied, rather than by observing the transition structure. We give a number of applications of this result to LTL model-checking, branching-time model-checking, non-emptiness of higher-order pushdown automata and Büchi games.Our second major contribution is the first application of the saturation technique to parity games. We begin with a mu-calculus characterisation of the winning region. This formula alternates greatest and least fixed point operators over a kind of reachability formula. Hence, we can use a version of our reachability algorithm, and modifications of the Büchi techniques, to compute the required result. The main advantages of this approach compared to existing techniques due to Cachat, Serre and Vardi et al. are that it is direct and that it is not immediately exponential in the number of control states, although the worst-case complexity remains the same.</p

Saturation Methods for Global Model-Checking Pushdown Systems

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