Probabilistic logical systems play an important role in applications of logic to computer science, where one often has to deal with knowledge of a probabilistic nature. The study of computational and model-theoretic properties of such systems (finitary or infinitary) is a relevant task in logic and theoretical computer science. Furthermore, we may be interested in modal enrichments of probabilistic logic systems, which allow us to reason simultaneously about knowledge (modeled by means of S5-modalities) and probability.

Substructural logics are logical systems which lack all or some of the structural rules. Such logics are used for modelling computations with limited resources (unlike a mathematical statement, a resource cannot be used twice); non-commutative substructural logics have linguistic applications. Infinitary extensions of substructural logics enjoy interesting algorithmic and proof-theoretic properties.

The seminar will be organised in long talks lasting for 2-4 meetings. Each talk will be devoted to a specific probabilistic or substructural logical system and will include detailed proofs of the results related to this system. The speakers will be chosen by the heads of the seminar, mostly from students and PhD students.

As the sources for their talks, the speakers may choose articles and chapters from books, as well as their own new texts. Below we list some examples of sources.

Books
[1] J.Y. Halpern, Reasoning about Uncertainty. 2nd edition. MIT Press, 2017.
[2] R. Moot, C. Retoré, The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics. Springer, 2012.
[3] Z. Ognjanović (ed.), Probabilistic Extensions of Various Logical Systems. Springer, 2020.
[4] G. Restall, An Introduction to Substructural Logics. Routledge, 2000.

Articles
[1] M. Abadi, J.Y. Halpern, Decidability and expressiveness for first-order logics of probability. Information and Computation 112, 1-36, 1994.
[2] R. Fagin, J.Y. Halpern, N. Megiddo, A logic for reasoning about probabilities. Information and Computation 87, 78-128, 1990.
[3] R. Fagin, J.Y. Halpern, Reasoning about knowledge and probability. Journal of the ACM, 41, 340-367, 1994.
[4] J.-Y. Girard, Linear logic. Theoretical Computer Science, 50:1, 1-101, 1987.
[5] H.J. Keisler, Probability quantifiers. In: J. Barwise, S. Feferman (eds.), Model-Theoretic Logics. Springer, 1985, 509-556.
[6] D. Kozen, On the complexity of reasoning in Kleene algebra. Information and Computation. 179:2, 152-162, 2002.
[7] S.L. Kuznetsov, S.O. Speranski, Infinitary action logic with exponentiation. Annals of Pure and Applied Logic 173:2, 103057, 2022.
[8] S.L. Kuznetsov, Complexity of the Lambek calculus and its extensions. In: R. Loukanova et al. (eds.), Logic and Algorithms in Computational Linguistics 2021 (LACompLing2021), Studies in Computational Intelligence 1081, pp. 1-29. Springer, 2023.
[9] J. Lambek, The mathematics of sentence structure. American Mathematical Monthly 65, 154-170, 1958.
[10] H. Ono, Semantics for substructural logics. In: P. Schroeder-Heister, K. Došen (eds.), Substructural Logic, Studies in Logic and Computation 2, pp. 259-291. Clarendon Press, 1993.
[11] M. Pentus, Product-free Lambek calculus and context-free grammars. Journal of Symbolic Logic. 62:2, 648-660, 1997.
[12] S.O. Speranski, Complexity for probability logic with quantifiers over propositions. Journal of Logic and Computation 23, 1035-1055, 2013.
[13] S.O. Speranski, Quantifying over events in probability logic: an introduction. Mathematical Structures in Computer Science 27, 1581-1600, 2017.
[14] S.A. Terwijn, Probabilistic logic and induction. Journal of Logic and Computation 15, 507-515, 2005.

Seminar organizers
Kuznetsov Stepan Lvovich
Speranski Stanislav Olegovich

Financial support
The seminar is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center