A categorical account of composition methods in logic

Marsden, Dan; Shah, Nihil; Jakl, Tomáš

Abstract

We present a categorical theory of the composition methods in finite model theory – a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the composition methods are Feferman–Vaught–Mostowski (FVM) type theorems, which characterize how logical equivalence be-haves under composition and transformation of models.Our results are developed by extending the recently introduced game comonad semantics for model comparison games. This level of abstraction allow us to give conditions yielding FVM type results in a uniform way. Our theorems are parametric in the classes of models, logics and operations involved. Furthermore, they naturally account for the positive existential fragment, and extensions with counting quantifiers of these logics. We also reveal surprising connections between FVM type theorems, and classical concepts in the theory of monads.We illustrate our methods by recovering many classical theorems of practical interest, including a refinement of a previous result by Dawar, Severini, and Zapata concerning the 3-variable counting logic and cospectrality. To highlight the importance of our techniques being parametric in the logic of interest, we prove a family of FVM theorems for products of structures, uniformly in the logic in question, which cannot be done using specific game arguments.

Citation

Marsden, D., Shah, N., & Jakl, T. (2023, June). A categorical account of composition methods in logic. Presented at Thirty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Boston, USA

• Presentation Conference Type: Edited Proceedings
• Conference Name: Thirty-Eighth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
• Start Date: Jun 26, 2023
• End Date: Jun 29, 2023
• Acceptance Date: Jun 26, 2023
• Online Publication Date: Jun 26, 2023
• Publication Date: Jun 26, 2023
• Deposit Date: May 15, 2023
• Publicly Available Date: Jun 28, 2023
• Publisher: Institute of Electrical and Electronics Engineers
• Peer Reviewed: Peer Reviewed
• Pages: 126-139
• Book Title: 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
• ISBN: 979-8-3503-3588-0
• DOI: https://doi.org/10.1109/LICS56636.2023.10175751
• Public URL: https://nottingham-repository.worktribe.com/output/20811994
• Publisher URL: https://www.computer.org/csdl/proceedings-article/lics/2023/10175751/1OM4Xl3Z6Hm
• Related Public URLs: https://lics.siglog.org/lics23/