Model Checking Recursive Programs with Numeric Data Types

Pushdown systems (PDS) naturally model sequential recursive programs. Numeric data types also often arise in real-world programs. We study the extension of PDS with unbounded counters, which naturally model numeric data types. Although this extension is Turingpowerful, reachability is known to be decidable when the number of reversals between incrementing and decrementing modes is bounded. In this paper, we (1) pinpoint the decidability/complexity of reachability and linear/branching time model checking over PDS with reversal-bounded counters (PCo), and (2) experimentally demonstrate the effectiveness of our approach in analysing software. We show reachability over PCo is NP-complete, while LTL is coNEXP-complete (coNP-complete for fixed formulas). In contrast, we prove that EF-logic over PCo is undecidable. Our NP upper bounds are by a direct poly-time reduction to satisfaction over existential Presburger formulas, allowing us to tap into highly optimized solvers like Z3. Although reversal-bounded analysis is incomplete for PDS with unbounded counters in general, our experiments suggest that some intricate bugs (e.g. from Linux device drivers) can be discovered with a small number of reversals. We also pinpoint the decidability/complexity of various extensions of PCo, e.g., with discrete clocks.