Combining Combination Properties: An Analysis of Stable Infiniteness, Convexity, and Politeness

Guilherme V. Toledo, Yoni Zohar, Clark Barrett

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We make two contributions to the study of theory combination in satisfiability modulo theories. The first is a table of examples for the combinations of the most common model-theoretic properties in theory combination, namely stable infiniteness, smoothness, convexity, finite witnessability, and strong finite witnessability (and therefore politeness and strong politeness as well). All of our examples are sharp, in the sense that we also offer proofs that no theories are available within simpler signatures. This table significantly progresses the current understanding of the various properties and their interactions. The most remarkable example in this table is of a theory over a single sort that is polite but not strongly polite (the existence of such a theory was only known until now for two-sorted signatures). The second contribution is a new combination theorem showing that in order to apply polite theory combination, it is sufficient for one theory to be stably infinite and strongly finitely witnessable, thus showing that smoothness is not a critical property in this combination method. This result has the potential to greatly simplify the process of showing which theories can be used in polite combination, as showing stable infiniteness is considerably simpler than showing smoothness.

Original languageEnglish
Title of host publicationAutomated Deduction – CADE 29 - 29th International Conference on Automated Deduction, Proceedings
EditorsBrigitte Pientka, Cesare Tinelli
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages20
ISBN (Print)9783031384981
StatePublished - 2023
EventProceedings of the 29th International Conference on Automated Deduction, CADE-29 - Rome, Italy
Duration: 1 Jul 20234 Jul 2023

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14132 LNAI
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


ConferenceProceedings of the 29th International Conference on Automated Deduction, CADE-29

Bibliographical note

Publisher Copyright:
© 2023, The Author(s).


Acknowledgment. I thank Alexander Bentkamp, Stephan Schulz, Mark Summer-field, Petar Vukmirović, and Uwe Waldmann for textual suggestions. This research has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant No. 713999, Matryoshka). This research has also received funding from the Netherlands Organization for Scientific Research (NWO) under the Vidi program (project No. 016.Vidi.189.037, Lean Forward). Partially funded by the Royal Society (International Exchanges, grant number IES\R2\212106) Over the years the work in this talk has been supported by the EPSRC, Leverhulme Trust, ERC, NSF, Cambridge Gates Foundation, Cambridge Trust. Research reported in this paper is partially funded by the EPSRC Fellowship ‘VeTSpec: Verified Trustworthy Software Specification’ (EP/R034567/1). Onderzoek – Vlaanderen (project G070521N) and by the EU ICT-48 2020 project TAILOR (GA 952215). Jakob Nordström was supported by the Swedish Research Council grant 2016-00782 and the Independent Research Fund Denmark grant 9040-00389B. Andy Oertel was supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation. Acknowledgment. Thanks to the reviewers for their extensive constructive feedback and to Diego Olivier Fernandez Pons for introducing us to MIP pre-solving. The research was partially funded by the Austrian Science Fund (FWF) under project No. T-1306. Acknowledgments. We thank the reviewers for their helpful comments. The research reported here was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 465447331. Funded by the European Union (ERC, ExtenDD, project number: 101054714). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. Jeremias Berg was fully supported by the Academy of Finland under grant 342145. Bart Bogaerts and Dieter Vandesande were supported by Fonds Wetenschappelijk Acknowledgements. This work was partially supported by the National Natural Acknowledgments. This work has been supported by the Czech Ministry of Education, Youth and Sports ERC.CZ project LL1908, and the FIT BUT internal project FIT-S-20-6427. Supported by JSPS KAKENHI Grant Number JP22K11900. Acknowledgments. This work was funded in part by NSF-BSF grant 2110397 (NSF) and 2020704 (BSF) and ISF grant number 619/21. Acknowledgements. This work has been partially funded by the Swedish Research Council (VR) under grant 2018-04727, the Swedish Foundation for Strategic Research (SSF) under the project WebSec (Ref. RIT17-0011), the Wallenberg project UPDATE, and the NSTC QC project under Grant no. NSTC 111-2119-M-001-004-and 112-2119-M-001-006-. Acknowledgements. This work is the result of a research internship hosted at TU Wien and defended at the University of Liège. The authors would like to thank Pascal Fontaine for valuable discussions and comments. We acknowledge funding from the ERC Consolidator Grant ARTIST 101002685 and the FWF SFB project SpyCoDe F8504. Acknowledgments. This project was funded in part by the Deutsche Forschungs-gemeinschaft (DFG) – 378803395 (ConVeY), an Alexander von Humboldt Professorship, and by the AFOSR under grant number FA9550-16-1-0288. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -235950644 (Project GI 274/6-2).

FundersFunder number
Cambridge Gates Foundation
Czech Ministry of Education, Youth and Sports ERC.CZFIT-S-20-6427, LL1908
Fonds Wetenschappelijk
NSF-BSF2020704, 2110397
National Science Foundation
Air Force Office of Scientific ResearchFA9550-16-1-0288
Alexander von Humboldt-Stiftung
Horizon 2020 Framework Programme713999
Engineering and Physical Sciences Research CouncilEP/R034567/1
Leverhulme Trust
Royal SocietyIES\R2\212106
European Commission
European Research CouncilARTIST 101002685
Deutsche ForschungsgemeinschaftGI 274/6-2, 378803395, 465447331
Japan Society for the Promotion of ScienceJP22K11900
Stiftelsen för Strategisk ForskningRIT17-0011, NSTC 111-2119-M-001-004-and 112-2119-M-001-006
United States-Israel Binational Science Foundation
National Natural Science Foundation of China62002351, 62032024
Academy of Finland342145
Austrian Science FundF8504, T-1306
Nederlandse Organisatie voor Wetenschappelijk Onderzoek016
Israel Science Foundation619/21
Knut och Alice Wallenbergs Stiftelse
Vetenskapsrådet2018-04727, 2016-00782
Danmarks Frie Forskningsfond9040-00389B


    • Satisfiability modulo theories
    • Theory combination
    • Theory politeness


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