Abstract
We study expression learning problems with syntactic restrictions and introduce the class of finite-aspect checkable languages to characterize symbolic languages that admit decidable learning. The semantics of such languages can be defined using a bounded amount of auxiliary information that is independent of expression size but depends on a fixed structure over which evaluation occurs. We introduce a generic programming language for writing programs that evaluate expression syntax trees, and we give a meta-theorem that connects such programs for finite-aspect checkable languages to finite tree automata, which allows us to derive new decidable learning results and decision procedures for several expression learning problems by writing programs in the programming language.
- Rajeev Alur, Rastislav Bodík, Eric Dallal, Dana Fisman, Pranav Garg, Garvit Juniwal, Hadas Kress-Gazit, P. Madhusudan, Milo M. K. Martin, Mukund Raghothaman, Shambwaditya Saha, Sanjit A. Seshia, Rishabh Singh, Armando Solar-Lezama, Emina Torlak, and Abhishek Udupa. 2015. Syntax-Guided Synthesis. In Dependable Software Systems Engineering (NATO Science for Peace and Security Series, D: Information and Communication Security, Vol. 40). IOS Press, 1–25.
Google Scholar
- Angello Astorga, P. Madhusudan, Shambwaditya Saha, Shiyu Wang, and Tao Xie. 2019. Learning Stateful Preconditions modulo a Test Generator. In Proceedings of the 40th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2019). Association for Computing Machinery, New York, NY, USA. 775–787. isbn:9781450367127 https://doi.org/10.1145/3314221.3314641
Google Scholar
Digital Library
- Angello Astorga, Shambwaditya Saha, Ahmad Dinkins, Felicia Wang, P. Madhusudan, and Tao Xie. 2021. Synthesizing Contracts Correct modulo a Test Generator. Proc. ACM Program. Lang., 5, OOPSLA (2021), Article 104, oct, 27 pages. https://doi.org/10.1145/3485481
Google Scholar
Digital Library
- Patrick Blackburn, Maarten de Rijke, and Yde Venema. 2001. Modal Logic. Cambridge University Press. https://doi.org/10.1017/CBO9781107050884
Google Scholar
Cross Ref
- Manuel Bodirsky. 2021. Complexity of Infinite-Domain Constraint Satisfaction. Cambridge University Press. https://doi.org/10.1017/9781107337534
Google Scholar
Cross Ref
- J. Richard Büchi. 1990. On a Decision Method in Restricted Second Order Arithmetic. Springer New York, New York, NY. 425–435. isbn:978-1-4613-8928-6 https://doi.org/10.1007/978-1-4613-8928-6_23
Google Scholar
Cross Ref
- J. Richard Buchi and Lawrence H. Landweber. 1969. Solving Sequential Conditions by Finite-State Strategies. Trans. Amer. Math. Soc., 138 (1969), 295–311. issn:00029947 http://www.jstor.org/stable/1994916
Google Scholar
Cross Ref
- J. Richard Büchi. 1960. Weak Second-Order Arithmetic and Finite Automata. Mathematical Logic Quarterly, 6, 1-6 (1960), 66–92. https://doi.org/10.1002/malq.19600060105 arxiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/malq.19600060105.
Google Scholar
Cross Ref
- Thierry Cachat. 2002. Two-Way Tree Automata Solving Pushdown Games. Springer-Verlag, Berlin, Heidelberg. 303–317. isbn:3540003886
Google Scholar
- José Cambronero, Sumit Gulwani, Vu Le, Daniel Perelman, Arjun Radhakrishna, Clint Simon, and Ashish Tiwari. 2023. FlashFill++: Scaling Programming by Example by Cutting to the Chase. In Principles of Programming Languages. ACM. https://www.microsoft.com/en-us/research/publication/flashfill-scaling-programming-by-example-by-cutting-to-the-chase/
Google Scholar
- Alonzo Church. 1963. Application of Recursive Arithmetic to the Problem of Circuit Synthesis. Journal of Symbolic Logic, 28, 4 (1963), 289–290. https://doi.org/10.2307/2271310
Google Scholar
Cross Ref
- H. Comon, M. Dauchet, R. Gilleron, C. Löding, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. 2007. Tree Automata Techniques and Applications. Available on: http://www.grappa.univ-lille3.fr/tata. release October, 12th 2007.
Google Scholar
- Bruno Courcelle. 1990. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85, 1 (1990), 12–75. issn:0890-5401 https://doi.org/10.1016/0890-5401(90)90043-H
Google Scholar
Digital Library
- Professor Bruno Courcelle and Dr Joost Engelfriet. 2012. Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach (1st ed.). Cambridge University Press, New York, NY, USA. isbn:0521898331, 9780521898331
Google Scholar
- John Doner. 1970. Tree acceptors and some of their applications. J. Comput. System Sci., 4, 5 (1970), 406–451. issn:0022-0000 https://doi.org/10.1016/S0022-0000(70)80041-1
Google Scholar
Digital Library
- Calvin C. Elgot. 1961. Decision Problems of Finite Automata Design and Related Arithmetics. Trans. Amer. Math. Soc., 98, 1 (1961), 21–51. issn:00029947 http://www.jstor.org/stable/1993511
Google Scholar
Cross Ref
- Javier Esparza, Orna Kupferman, and Moshe Y. Vardi. 2021. Verification. In Handbook of Automata Theory, Jean-Éric Pin (Ed.). European Mathematical Society Publishing House, Zürich, Switzerland, 1415–1456.
Google Scholar
- Richard Evans and Edward Grefenstette. 2018. Learning Explanatory Rules from Noisy Data. J. Artif. Int. Res., 61, 1 (2018), Jan., 1–64. issn:1076-9757
Google Scholar
Digital Library
- Azadeh Farzan, Danya Lette, and Victor Nicolet. 2022. Recursion Synthesis with Unrealizability Witnesses. In Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation (PLDI 2022). Association for Computing Machinery, New York, NY, USA. 244–259. isbn:9781450392655 https://doi.org/10.1145/3519939.3523726
Google Scholar
Digital Library
- Henning Fernau. 2009. Algorithms for learning regular expressions from positive data. Information and Computation, 207, 4 (2009), 521–541. issn:0890-5401 https://doi.org/10.1016/j.ic.2008.12.008
Google Scholar
Digital Library
- J. Flum and M. Grohe. 2006. Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series). Springer-Verlag, Berlin, Heidelberg. isbn:3540299521 https://doi.org/10.1007/3-540-29953-X
Google Scholar
Cross Ref
- Maurice Funk, Jean Christoph Jung, Carsten Lutz, Hadrien Pulcini, and Frank Wolter. 2019. Learning Description Logic Concepts: When can Positive and Negative Examples be Separated? In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19. International Joint Conferences on Artificial Intelligence Organization, 1682–1688. https://doi.org/10.24963/ijcai.2019/233
Google Scholar
Cross Ref
- Pranav Garg, Christof Löding, P. Madhusudan, and Daniel Neider. 2014. ICE: A Robust Framework for Learning Invariants. In Computer Aided Verification, Armin Biere and Roderick Bloem (Eds.). Springer International Publishing, Cham. 69–87. isbn:978-3-319-08867-9
Google Scholar
- Pranav Garg, Christof Löding, P. Madhusudan, and Daniel Neider. 2015. Quantified data automata for linear data structures: a register automaton model with applications to learning invariants of programs manipulating arrays and lists. Formal Methods in System Design, 47, 1 (2015), 01 Aug, 120–157. issn:1572-8102 https://doi.org/10.1007/s10703-015-0231-6
Google Scholar
Digital Library
- 2002. Automata Logics, and Infinite Games: A Guide to Current Research, Erich Grädel, Wolfgang Thomas, and Thomas Wilke (Eds.). Springer-Verlag, Berlin, Heidelberg. isbn:3540003886
Google Scholar
- Sumit Gulwani. 2011. Automating String Processing in Spreadsheets Using Input-Output Examples. In Proceedings of the 38th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL ’11). Association for Computing Machinery, New York, NY, USA. 317–330. isbn:9781450304900 https://doi.org/10.1145/1926385.1926423
Google Scholar
Digital Library
- Annegret Habel. 1992. Graph-theoretic aspects of HRL’s. Springer Berlin Heidelberg, Berlin, Heidelberg. 117–144. isbn:978-3-540-47340-4 https://doi.org/10.1007/BFb0013882
Google Scholar
Cross Ref
- Travis Hance, Marijn Heule, Ruben Martins, and Bryan Parno. 2021. Finding Invariants of Distributed Systems: It’ s a Small (Enough) World After All. In 18th USENIX Symposium on Networked Systems Design and Implementation (NSDI 21). USENIX Association, 115–131. isbn:978-1-939133-21-2 https://www.usenix.org/conference/nsdi21/presentation/hance
Google Scholar
- Shivam Handa and Martin C. Rinard. 2020. Inductive Program Synthesis over Noisy Data. In Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering (ESEC/FSE 2020). Association for Computing Machinery, New York, NY, USA. 87–98. isbn:9781450370431 https://doi.org/10.1145/3368089.3409732
Google Scholar
Digital Library
- Wilfrid Hodges. 1993. The countable case. Cambridge University Press, 323–359. https://doi.org/10.1017/CBO9780511551574.009
Google Scholar
Cross Ref
- Qinheping Hu, John Cyphert, Loris D’Antoni, and Thomas Reps. 2020. Exact and Approximate Methods for Proving Unrealizability of Syntax-Guided Synthesis Problems. In Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2020). Association for Computing Machinery, New York, NY, USA. 1128–1142. isbn:9781450376136 https://doi.org/10.1145/3385412.3385979
Google Scholar
Digital Library
- Radoslav Ivanov, Kishor Jothimurugan, Steve Hsu, Shaan Vaidya, Rajeev Alur, and Osbert Bastani. 2021. Compositional Learning and Verification of Neural Network Controllers. ACM Trans. Embed. Comput. Syst., 20, 5s (2021), Article 92, sep, 26 pages. issn:1539-9087 https://doi.org/10.1145/3477023
Google Scholar
Digital Library
- Michael J. Kearns and Umesh Vazirani. 1994. An Introduction to Computational Learning Theory. The MIT Press. isbn:9780262276863 https://doi.org/10.7551/mitpress/3897.001.0001
Google Scholar
Cross Ref
- Jason R. Koenig, Oded Padon, Neil Immerman, and Alex Aiken. 2020. First-Order Quantified Separators. In Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2020). Association for Computing Machinery, New York, NY, USA. 703–717. isbn:9781450376136 https://doi.org/10.1145/3385412.3386018
Google Scholar
Digital Library
- Jason R. Koenig, Oded Padon, Sharon Shoham, and Alex Aiken. 2022. Inferring Invariants with Quantifier Alternations: Taming the Search Space Explosion. In Tools and Algorithms for the Construction and Analysis of Systems, Dana Fisman and Grigore Rosu (Eds.). Springer International Publishing, Cham. 338–356. isbn:978-3-030-99524-9
Google Scholar
- James Koppel. 2021. Version Space Algebras are Acyclic Tree Automata. https://doi.org/10.48550/ARXIV.2107.12568
Google Scholar
- James Koppel, Zheng Guo, Edsko de Vries, Armando Solar-Lezama, and Nadia Polikarpova. 2022. Searching Entangled Program Spaces. Proc. ACM Program. Lang., 6, ICFP (2022), Article 91, aug, 29 pages. https://doi.org/10.1145/3547622
Google Scholar
Digital Library
- Paul Krogmeier and P. Madhusudan. 2022. Learning Formulas in Finite Variable Logics. Proc. ACM Program. Lang., 6, POPL (2022), Article 10, jan, 28 pages. https://doi.org/10.1145/3498671
Google Scholar
Digital Library
- Paul Krogmeier and P. Madhusudan. 2023. Languages With Decidable Learning: A Meta-Theorem. https://doi.org/10.48550/ARXIV.2302.05741
Google Scholar
- Paul Krogmeier, Umang Mathur, Adithya Murali, P. Madhusudan, and Mahesh Viswanathan. 2020. Decidable Synthesis of Programs with Uninterpreted Functions. In Computer Aided Verification, Shuvendu K. Lahiri and Chao Wang (Eds.). Springer International Publishing, Cham. 634–657. isbn:978-3-030-53291-8
Google Scholar
- Orna Kupferman, P. Madhusudan, P. S. Thiagarajan, and Moshe Y. Vardi. 2000. Open Systems in Reactive Environments: Control and Synthesis. In CONCUR (Lecture Notes in Computer Science, Vol. 1877). Springer, 92–107.
Google Scholar
- Orna Kupferman, Nir Piterman, and Moshe Y. Vardi. 2010. An Automata-Theoretic Approach to Infinite-State Systems. Springer Berlin Heidelberg, Berlin, Heidelberg. 202–259. isbn:978-3-642-13754-9 https://doi.org/10.1007/978-3-642-13754-9_11
Google Scholar
Cross Ref
- Pat Langley and Sean Stromsten. 2000. Learning Context-Free Grammars with a Simplicity Bias. In Machine Learning: ECML 2000, Ramon López de Mántaras and Enric Plaza (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg. 220–228. isbn:978-3-540-45164-8
Google Scholar
- Yunyao Li, Rajasekar Krishnamurthy, Sriram Raghavan, Shivakumar Vaithyanathan, and H. V. Jagadish. 2008. Regular Expression Learning for Information Extraction. In EMNLP.
Google Scholar
- P. Madhusudan. 2011. Synthesizing Reactive Programs. In Computer Science Logic (CSL’11) - 25th International Workshop/20th Annual Conference of the EACSL, Marc Bezem (Ed.) (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 12). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany. 428–442. isbn:978-3-939897-32-3 issn:1868-8969 https://doi.org/10.4230/LIPIcs.CSL.2011.428
Google Scholar
Cross Ref
- Kenneth L. McMillan. 1992. Symbolic Model Checking: an approach to the state explosion problem. Ph.D. Dissertation. Carnegie Mellon. thesis.pdf CMU Tech Rpt. CMU-CS-92-131.
Google Scholar
- Anders Miltner, Adrian Trejo Nuñez, Ana Brendel, Swarat Chaudhuri, and Isil Dillig. 2022. Bottom-up Synthesis of Recursive Functional Programs Using Angelic Execution. Proc. ACM Program. Lang., 6, POPL (2022), Article 21, jan, 29 pages. https://doi.org/10.1145/3498682
Google Scholar
Digital Library
- Anders Miltner, Saswat Padhi, Todd Millstein, and David Walker. 2020. Data-Driven Inference of Representation Invariants. In Proceedings of the 41st ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2020). Association for Computing Machinery, New York, NY, USA. 1–15. isbn:9781450376136 https://doi.org/10.1145/3385412.3385967
Google Scholar
Digital Library
- Tom M. Mitchell. 1982. Generalization as search. Artificial Intelligence, 18, 2 (1982), 203–226. issn:0004-3702 https://doi.org/10.1016/0004-3702(82)90040-6
Google Scholar
Cross Ref
- Thomas M. Mitchell. 1997. Machine Learning (1 ed.). McGraw-Hill, Inc., USA. isbn:0070428077
Google Scholar
Digital Library
- Ulrich Möncke and Reinhard Wilhelm. 1991. Grammar flow analysis. In Attribute Grammars, Applications and Systems, Henk Alblas and Bořivoj Melichar (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg. 151–186. isbn:978-3-540-38490-8
Google Scholar
- Stephen H. Muggleton, Dianhuan Lin, Niels Pahlavi, and Alireza Tamaddoni-Nezhad. 2014. Meta-interpretive learning: application to grammatical inference. Machine Learning, 94, 1 (2014), 01 Jan, 25–49. issn:1573-0565 https://doi.org/10.1007/s10994-013-5358-3
Google Scholar
Digital Library
- Daniel Neider and Ivan Gavran. 2018. Learning Linear Temporal Properties. In 2018 Formal Methods in Computer Aided Design (FMCAD). 1–10. https://doi.org/10.23919/FMCAD.2018.8603016
Google Scholar
Cross Ref
- Daniel Neider, P. Madhusudan, Shambwaditya Saha, Pranav Garg, and Daejun Park. 2020. A Learning-Based Approach to Synthesizing Invariants for Incomplete Verification Engines. Journal of Automated Reasoning, 64, 7 (2020), 01 Oct, 1523–1552. issn:1573-0670 https://doi.org/10.1007/s10817-020-09570-z
Google Scholar
Digital Library
- Peter-Michael Osera and Steve Zdancewic. 2015. Type-and-Example-Directed Program Synthesis. In Proceedings of the 36th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI ’15). Association for Computing Machinery, New York, NY, USA. 619–630. isbn:9781450334686 https://doi.org/10.1145/2737924.2738007
Google Scholar
Digital Library
- Amir Pnueli. 1977. The temporal logic of programs. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977). 46–57. https://doi.org/10.1109/SFCS.1977.32
Google Scholar
Digital Library
- Amir Pnueli and Roni Rosner. 1989. On the Synthesis of a Reactive Module. In POPL. ACM Press, 179–190.
Google Scholar
- Amir Pnueli and Roni Rosner. 1990. Distributed Reactive Systems Are Hard to Synthesize. In FOCS. IEEE Computer Society, 746–757.
Google Scholar
- Nadia Polikarpova, Ivan Kuraj, and Armando Solar-Lezama. 2016. Program Synthesis from Polymorphic Refinement Types. In Proceedings of the 37th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI ’16). Association for Computing Machinery, New York, NY, USA. 522–538. isbn:9781450342612 https://doi.org/10.1145/2908080.2908093
Google Scholar
Digital Library
- Oleksandr Polozov and Sumit Gulwani. 2015. FlashMeta: A Framework for Inductive Program Synthesis. In Proceedings of the 2015 ACM SIGPLAN International Conference on Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA 2015). Association for Computing Machinery, New York, NY, USA. 107–126. isbn:9781450336895 https://doi.org/10.1145/2814270.2814310
Google Scholar
Digital Library
- Michael O. Rabin. 1969. Decidability of Second-Order Theories and Automata on Infinite Trees. Trans. Amer. Math. Soc., 141 (1969), 1–35. issn:00029947 http://www.jstor.org/stable/1995086
Google Scholar
- Michael Oser Rabin. 1972. Automata on Infinite Objects and Church’s Problem. American Mathematical Society, Boston, MA, USA. isbn:0821816632
Google Scholar
- Yasubumi Sakakibara. 2005. Learning context-free grammars using tabular representations. Pattern Recognition, 38, 9 (2005), 1372–1383. issn:0031-3203 https://doi.org/10.1016/j.patcog.2004.03.021 Grammatical Inference.
Google Scholar
Digital Library
- Armando Solar-Lezama, Liviu Tancau, Rastislav Bodik, Sanjit Seshia, and Vijay Saraswat. 2006. Combinatorial Sketching for Finite Programs. In Proceedings of the 12th International Conference on Architectural Support for Programming Languages and Operating Systems (ASPLOS XII). Association for Computing Machinery, New York, NY, USA. 404–415. isbn:1595934510 https://doi.org/10.1145/1168857.1168907
Google Scholar
Digital Library
- James W. Thatcher and Jesse B. Wright. 1968. Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical systems theory, 2 (1968), 57–81. https://doi.org/10.1007/BF01691346
Google Scholar
Cross Ref
- Boris A. Trakhtenbrot. 1961. Finite automata and logic of monadic predicates. Doklady Akademii Nauk SSSR, 140, 326-329 (1961), 122–123.
Google Scholar
- Steffen van Bergerem. 2019. Learning Concepts Definable in First-Order Logic with Counting. In 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). 1–13. https://doi.org/10.1109/LICS.2019.8785811
Google Scholar
Cross Ref
- Steffen van Bergerem, Martin Grohe, and Martin Ritzert. 2022. On the Parameterized Complexity of Learning First-Order Logic. In Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS ’22). Association for Computing Machinery, New York, NY, USA. 337–346. isbn:9781450392600 https://doi.org/10.1145/3517804.3524151
Google Scholar
Digital Library
- Kurt Vanlehn and William Ball. 1987. A Version Space Approach to Learning Context-free Grammars. Machine Learning, 2, 1 (1987), 01 Mar, 39–74. issn:1573-0565 https://doi.org/10.1023/A:1022812926936
Google Scholar
Digital Library
- Moshe Y. Vardi. 1998. Reasoning about the past with two-way automata. In Automata, Languages and Programming, Kim G. Larsen, Sven Skyum, and Glynn Winskel (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg. 628–641. isbn:978-3-540-68681-1
Google Scholar
- Xinyu Wang, Isil Dillig, and Rishabh Singh. 2017. Program Synthesis Using Abstraction Refinement. Proc. ACM Program. Lang., 2, POPL (2017), Article 63, Dec., 30 pages. issn:2475-1421 https://doi.org/10.1145/3158151
Google Scholar
Digital Library
- Xinyu Wang, Isil Dillig, and Rishabh Singh. 2017. Synthesis of Data Completion Scripts Using Finite Tree Automata. Proc. ACM Program. Lang., 1, OOPSLA (2017), Article 62, Oct., 26 pages. https://doi.org/10.1145/3133886
Google Scholar
Digital Library
- Yuepeng Wang, Xinyu Wang, and Isil Dillig. 2018. Relational Program Synthesis. Proc. ACM Program. Lang., 2, OOPSLA (2018), Article 155, Oct., 27 pages. issn:2475-1421 https://doi.org/10.1145/3276525
Google Scholar
Digital Library
- Jianan Yao, Runzhou Tao, Ronghui Gu, Jason Nieh, Suman Jana, and Gabriel Ryan. 2021. DistAI: Data-Driven Automated Invariant Learning for Distributed Protocols. In 15th USENIX Symposium on Operating Systems Design and Implementation (OSDI 21). USENIX Association, 405–421. isbn:978-1-939133-22-9 https://www.usenix.org/conference/osdi21/presentation/yao
Google Scholar
- He Zhu, Stephen Magill, and Suresh Jagannathan. 2018. A Data-Driven CHC Solver. In Proceedings of the 39th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2018). Association for Computing Machinery, New York, NY, USA. 707–721. isbn:9781450356985 https://doi.org/10.1145/3192366.3192416
Google Scholar
Digital Library
Index Terms
Languages with Decidable Learning: A Meta-theorem
Recommendations
Learning formulas in finite variable logics
We consider grammar-restricted exact learning of formulas and terms in finite variable logics. We propose a novel and versatile automata-theoretic technique for solving such problems. We first show results for learning formulas that classify a set of ...
Bootstrapping domain-specific meta-languages in language workbenches
GPCE 2016: Proceedings of the 2016 ACM SIGPLAN International Conference on Generative Programming: Concepts and ExperiencesIt is common practice to bootstrap compilers of programming languages. By using the compiled language to implement the compiler, compiler developers can code in their own high-level language and gain a large-scale test case. In this paper, we ...
The HOM problem is decidable
We close affirmatively a question that has been open for long time: decidability of the HOM problem. The HOM problem consists in determining, given a tree homomorphism H and a regular tree language L represented by a tree automaton, whether H(L) is ...






Comments