### Abstract

We formalize pushdown systems and the correctness of the pushdown reachability algorithms post* (forward search), pre* (backward search) and dual* (bi-directional search). For pre* we refine the algorithm to an executable version for which one can generate code using Isabelle's code generator. For pre* and post* we follow Stefan Schwoon's PhD thesis [Sch02a]. The dual* algorithm is from a paper by Jensen et. al presented at ATVA2021 [JSS+21]. The formalization is described in our FMCAD2022 paper [SSST22] in which we also document how we have used it to do differential testing against a C++ implementation of pushdown reachability called PDAAAL. Lammich et al. [Lam09, LMW09] formalized the pre* algorithm for dynamic pushdown networks (DPN) which is a generalization of pushdown systems. Our work is independent from that because the post* of DPNs is not regular and additionally the DPN formalization does not support epsilon transitions which we use for post* and dual*.

[JSS+21] Peter Gjøl Jensen, Stefan Schmid, Morten Konggaard Schou, Jirí Srba, Juan Vanerio, and Ingo van Duijn. Faster pushdown reachability analysis with applications in network verification. In Zhe Hou and Vijay Ganesh, editors, Automated Technology for Verification and Analysis - 19th International Symposium, ATVA 2021, Gold Coast, QLD, Australia, October 18-22, 2021, Proceedings, volume 12971 of Lecture Notes in Computer Science, pages 170-186. Springer, 2021.
[Lam09] Peter Lammich. Formalization of dynamic pushdown networks in Isabelle/HOL, 2009. https://www21.in.tum.de/~lammich/isabelle/dpn-document.pdf.
[LMW09] Peter Lammich, Markus Müller-Olm, and Alexander Wenner. Predecessor sets of dynamic pushdown networks with tree-regular constraints. In Ahmed Bouajjani and Oded Maler, editors, CAV2009, volume 5643 of LNCS, pages 525-539. Springer, 2009.
[Sch02a] Stefan Schwoon. Model checking pushdown systems. PhD thesis, Technical University Munich, Germany, 2002. https://d-nb.info/96638976X/34.
[SSST22] Anders Schlichtkrull, Morten Konggaard Schou, Jirí Srba, and Dmitriy Traytel. Differential testing of pushdown reachability with a formally verified oracle. In Alberto Griggio and Neha Rungta, editors, 22nd Formal Methods in Computer-Aided Design, FMCAD 2022, Trento, Italy, October 17-21, 2022, pages 369-379. IEEE, 2022.

### License

BSD License

### Topics

- Computer science/Automata and formal languages
- Computer science

### Related publications

- Schlichtkrull, A., Konggaard Schou, M., Srba, J., & Traytel, D. (2022). Differential Testing of Pushdown Reachability with a Formally Verified Oracle. TU Wien. https://doi.org/10.34727/2022/ISBN.978-3-85448-053-2_44

### Session Pushdown_Systems

- P_Automata
- PDS
- PDS_Code
- Ex

### Depends on

- Labeled Transition Systems

## Cite

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Pushdown_Systems-AFP

    @article{Pushdown_Systems-AFP,
      author  = {Anders Schlichtkrull and Morten Konggaard Schou and Jiří Srba and Dmitriy Traytel},
      title   = {Pushdown Systems},
      journal = {Archive of Formal Proofs},
      month   = {October},
      year    = {2023},
      note    = {\url{https://isa-afp.org/entries/Pushdown_Systems.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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