Abstract
We prove that the observational equivalence problem for a finitary fragment of the programming langauge ML is recursively equivalent to the reachability problem for extended branching vector addition systems with states (EBVASS). This result has two natural and independent parts. We first prove that the observational equivalence problem is equivalent to the emptiness problem for a new class of class memory automata equipped with a visibly pushdown stack, called Visibly Pushdown Class Memory Automata (VPCMA). Our proof uses the fully abstract game semantics of the language. We then prove that the VPCMA emptiness problem is equivalent to the reachability problem for EBVASS. The results of this article complete our programme to give an automata classification of the ML types with respect to the observational equivalence problem for closed terms.
- Samson Abramsky, Radha Jagadeesan, and Pasquale Malacaria. 2000. Full abstraction for PCF. Inf. Comput. 163, 2 (2000), 409--470. Google Scholar
Digital Library
- Samson Abramsky and Guy McCusker. 1996. Linearity, sharing and state: A fully abstract game semantics for Idealized Algol with active expressions. Electron. Not. Theor. Comput. Sci. 3 (1996), 2--14.Google Scholar
Cross Ref
- Samson Abramsky and Guy McCusker. 1997. Call-by-value games. In Proceedings of the Annual Conference on Computer Science Logic (CSL’97). Lecture Notes in Computer Science, Vol. 1414. Springer, 1--17. Google Scholar
Digital Library
- Rajeev Alur and P. Madhusudan. 2004. Visibly pushdown languages. In Proceedings of the Annual Symposium on Theory of Computing Conference (STOC’04). ACM, 202--211. Google Scholar
Digital Library
- Toshiro Araki and Tadao Kasami. 1976. Some decision problems related to the reachability problem for Petri nets. Theor. Comput. Sci. 3, 1 (1976), 85--104.Google Scholar
Cross Ref
- Henrik Björklund and Thomas Schwentick. 2007. On notions of regularity for data languages. In Proceedings of the Symposium on Fundamentals of Computation Theory (FCT’07). Lecture Notes in Computer Science, Vol. 4639. Springer, 88--99. Google Scholar
Digital Library
- Henrik Björklund and Thomas Schwentick. 2010. On notions of regularity for data languages. Theor. Comput. Sci. 411, 4--5 (2010), 702--715. Google Scholar
Digital Library
- Conrad Cotton-Barratt. 2017. Using Class Memory Automata in Algorithmic Game Semantics. Ph.D. Dissertation. University of Oxford.Google Scholar
- Conrad Cotton-Barratt, David Hopkins, Andrzej S. Murawski, and C.-H. Luke Ong. 2015a. Fragments of ML decidable by nested data class memory automata. In Proceedings of the International Conference on Foundations of Software Science and Computation Structures (FOSSACS’15). Lecture Notes in Computer Science, Vol. 9034. Springer, 249--263.Google Scholar
Cross Ref
- Conrad Cotton-Barratt, Andrzej S. Murawski, and C.-H. Luke Ong. 2015b. Weak and nested class memory automata. In Proceedings of the International Conference on Language and Automata Theory and Applications (LATA’15). Lecture Notes in Computer Science, Vol. 8977. Springer, 188--199.Google Scholar
Cross Ref
- Conrad Cotton-Barratt, Andrzej S. Murawski, and C.-H. Luke Ong. 2017. ML and extended branching VASS. In Proceedings of the European Symposium on Programming (ESOP’17). Lecture Notes in Computer Science, Vol. 8977. Springer, 188--199.Google Scholar
Cross Ref
- Philippe de Groote, Bruno Guillaume, and Sylvain Salvati. 2004. Vector addition tree automata. In Proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS’04). IEEE Computer Society, 64--73. Google Scholar
Digital Library
- Dan R. Ghica and Guy McCusker. 2000. Reasoning about Idealized ALGOL using regular languages. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’00). Lecture Notes in Computer Science, Vol. 1853. Springer, 103--115. Google Scholar
Digital Library
- Benny Godlin and Ofer Strichman. 2009. Regression verification. In Proceedings of the Design Automation Conference (DAC’09). ACM, 466--471. Google Scholar
Digital Library
- Kohei Honda and Nobuko Yoshida. 1999. Game-theoretic analysis of call-by-value computation. Theor. Comput. Sci. 221, 1--2 (1999), 393--456. Google Scholar
Digital Library
- David Hopkins. 2012. Game Semantics Based Equivalence Checking of Higher-Order Programs. Ph.D. Dissertation. Department of Computer Science, University of Oxford.Google Scholar
- David Hopkins, Andrzej S. Murawski, and C.-H. Luke Ong. 2011. A fragment of ML decidable by visibly pushdown automata. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’11). Lecture Notes in Computer Science, Vol. 6756. Springer, 149--161. Google Scholar
Digital Library
- J. Martin E. Hyland and C.-H. Luke Ong. 2000. On full abstraction for PCF: I, II, and III. Inf. Comput. 163, 2 (2000), 285--408. Google Scholar
Digital Library
- Florent Jacquemard, Luc Segoufin, and Jerémie Dimino. 2016. FO2(<, +1, ˜) on data trees, data tree automata and branching vector addition systems. Logic. Methods Comput. Sci. 12, 2 (2016).Google Scholar
- Ranko Lazic and Sylvain Schmitz. 2015. Nonelementary complexities for branching VASS, MELL, and extensions. ACM Trans. Comput. Log. 16, 3 (2015), 20:1--20:30. Google Scholar
Digital Library
- Andrzej S. Murawski. 2005. Functions with local state: Regularity and undecidability. Theor. Comput. Sci. 338(1/3) (2005), 315--349. Google Scholar
Digital Library
- Andrzej S. Murawski, C.-H. Luke Ong, and Igor Walukiewicz. 2005. Idealized Algol with ground recursion, and DPDA equivalence. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP). Lecture Notes in Computer Science, Vol. 3580. Springer, 917--929. Google Scholar
Digital Library
- Andrzej S. Murawski, Steven J. Ramsay, and Nikos Tzevelekos. 2015. Game semantic analysis of equivalence in IMJ. In Proceedings of the International Symposium on Automated Technology for Verification and Analysis (ATVA’15). Lecture Notes in Computer Science, Vol. 9364. Springer, 411--428.Google Scholar
Cross Ref
- Andrzej S. Murawski and Nikos Tzevelekos. 2009. Full abstraction for reduced ML. In Proceedings of the International Conference on Foundations of Software Science and Computation Structures (FOSSAC’09). Lecture Notes in Computer Science, Vol. 5504. Springer, 32--47.Google Scholar
- Andrzej S. Murawski and Nikos Tzevelekos. 2011. Algorithmic nominal game semantics. In Proceedings of the European Symposium on Programming (ESOP’11). Lecture Notes in Computer Science, Vol. 6602. Springer, 419--438. Google Scholar
Digital Library
- Andrzej S. Murawski and Nikos Tzevelekos. 2012. Algorithmic games for full ground references. In Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP’12). Lecture Notes in Computer Science, Vol. 7392. Springer, 312--324. Google Scholar
Digital Library
- C.-H. Luke Ong. 2002. Observational equivalence of 3rd-order Idealized Algol is decidable. In Proceedings of the IEEE Symposium on Logic in Computer Science. IEEE Press, 245--256. Google Scholar
Digital Library
- Andrew M. Pitts and Ian D. B. Stark. 1998. Operational reasoning for functions with local state. In Higher Order Operational Techniques in Semantics. Cambridge University Press, 227--273. Google Scholar
Digital Library
- John C. Reynolds. 1981. The essence of algol. In Algorithmic Languages. North Holland, 345--372.Google Scholar
Index Terms
ML, Visibly Pushdown Class Memory Automata, and Extended Branching Vector Addition Systems with States
Recommendations
Timed pushdown automata and branching vector addition systems
LICS '17: Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer ScienceWe prove that non-emptiness of timed register pushdown automata is decidable in doubly exponential time. This is a very expressive class of automata, whose transitions may involve state and top-of-stack clocks with unbounded differences. It strictly ...
Binary Reachability of Timed-register Pushdown Automata and Branching Vector Addition Systems
Timed-register pushdown automata constitute a very expressive class of automata, whose transitions may involve state, input, and top-of-stack timed registers with unbounded differences. They strictly subsume pushdown timed automata of Bouajjani et al., ...
Visibly pushdown automata: from language equivalence to simulation and bisimulation
CSL'06: Proceedings of the 20th international conference on Computer Science LogicWe investigate the possibility of (bi)simulation-like preorder/equivalence checking on the class of visibly pushdown automata and its natural subclasses visibly BPA (Basic Process Algebra) and visibly one-counter automata. We describe generic methods ...






Comments