# Proof Theory

Proof theory serves as one of the central pillars of mathematical logic and concerns the study and application of formal mathematical proofs. Typically, proofs are defined as syntactic objects inductively constructible through applications of inference rules to a given set of assumptions, axioms, or previously constructed proofs. Since proofs are built by means of inference rules, which manipulate formulae and symbols, proof theory is syntactic in nature, making proof systems well-suited for logical reasoning in a computational environment. As such, proof theory has proven to be an effective tool in automated reasoning, allowing for the design of complexity-optimal decision procedures providing verifiable witnesses for (un)satisfiable logical statements. On the theoretical side, techniques in proof theory can be used to establish the consistency, decidability, or interpolability of logics, among other significant properties.
Common questions that arise in the domain of proof theory are: What constitutes a mathematical proof? How can proof systems be constructed so that they are suitable in automated reasoning? What are the relationships between proofs in differing formalisms? How does one construct a proof system for a logic whereby all proofs are analytic (i.e. all information used in the proof is contained in the conclusion of the proof)?

#### Journal Articles

Bartosz Bednarczyk, Sebastian Rudolph
How to Tell Easy from Hard: Complexities of Conjunctive Query Entailment in Extensions of ALC
Journal of Artificial Intelligence Research, 78:385–458, November 2023
Tim Lyon
Nested Sequents for Intermediate Logics: The Case of Gödel-Dummett Logics
Journal of Applied Non-Classical Logics, 33(2):121-164, 2023
Details
Agata Ciabattoni, Tim Lyon, Revantha Ramanayake, Alwen Tiu
Display to Labeled Proofs and Back Again for Tense Logics
ACM Transactions on Computational Logic, 22(3):1-31, August 2021
Tim Lyon, Christian Ittner, Timo Eckhardt, Norbert Gratzl
The Basics of Display Calculi
Kriterion -- Journal of Philosophy, 31(2):55-100, 2017

#### Proceedings Articles

Tim Lyon, Jonas Karge
Constructive Interpolation and Concept-Based Beth Definability for Description Logics via Sequents
Proceedings of the 33rd International Joint Conference on Artificial Intelligence (IJCAI 2024), 2024. ijcai.org
Tim Lyon, Piotr Ostropolski-Nalewaja
Connecting Proof Theory and Knowledge Representation: Sequent Calculi and the Chase with Existential Rules
Proceedings of the 20th International Conference on Principles of Knowledge Representation and Reasoning, 769-773, 2023. IJCAI Inc
Bartosz Bednarczyk, Reijo Jaakkola
Towards Model Theory of Ordered Logics: Expressivity and Interpolation
In Stefan Szeider, Robert Ganian, Alexandra Silva, eds., Proceedings of the 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics, 15:1--15:14, August 2022. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik
Tim Lyon, Jonas Karge
Uniform and Modular Sequent Systems for Description Logics
In Ofer Arieli, Martin Homola, Jean Christoph Jung, Marie-Laure Mugnier, eds., Proceedings of the 35th International Workshop on Description Logics (DL 2022), 2022
Tim Lyon
A Framework for Intuitionistic Grammar Logics
In Pietro Baroni, Christoph Benzmüller, Yὶ N. Wang, eds., Logic and Argumentation, volume 13040, 495-503, October 2021. Springer International Publishing
Tim Lyon
Nested Sequents for Intuitionistic Modal Logics via Structural Refinement
In Anupam Das, Sara Negri, eds., Automated Reasoning with Analytic Tableaux and Related Methods, 409-427, August 2021. Springer International Publishing
Kees van Berkel, Tim Lyon
The Varieties of Ought-implies-Can and Deontic STIT Logic
In Fenrong Liu, Alessandra Marra, Paul Portner and Frederik Van De Putte, eds., Deontic Logic and Normative Systems: 15th International Conference, DEON 2020/2021, 57-76, July 2021. College Publications
Tim Lyon, Alwen Tiu, Rajeev Goré, Ranald Clouston
Syntactic Interpolation for Tense Logics and Bi-Intuitionistic Logic via Nested Sequents
In Maribel Fernández and Anca Muscholl, eds., 28th EACSL Annual Conference on Computer Science Logic (CSL 2020), volume 152, 28:1--28:16, 2020. Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Tim Lyon, Kees van Berkel
Automating Agential Reasoning: Proof-Calculi and Syntactic Decidability for STIT Logics
In Baldoni, Matteo and Dastani, Mehdi and Liao, Beishui and Sakurai, Yuko and Zalila Wenkstern, Rym, eds., PRIMA 2019: Principles and Practice of Multi-Agent Systems, volume 11873, 202-218, 2019. Springer
Kees van Berkel, Tim Lyon
Cut-Free Calculi and Relational Semantics for Temporal STIT Logics
In Calimeri, Francesco and Leone, Nicola and Manna, Marco, eds., Logics in Artificial Intelligence, volume 11468, 803-819, 2019. Springer