Lukas Kebuladze https://calfproject.github.io/kebuladze-2025/ kebuladze-2025 /kebuladze-2025/ Amortized Analysis of Splay Trees via a Lax Homomorphism Reference Technical Report, Carnegie Mellon University https://www.cs.cmu.edu/~runmingl/student/kebuladze-splay.pdf 2025, title = {Amortized Analysis of Splay Trees via a Lax Homomorphism}, author = {Lukas Kebuladze}, year = {2025}, month = {aug}, url = {https://www.cs.cmu.edu/~runmingl/student/kebuladze-splay.pdf}, institution = {Carnegie Mellon University} }]]> CMU undergraduate summer SURA project report by Lukas Kebuladze, supervised by Runming Li. Extended abstract, poster, and presentation: 3rd place winner in the undergraduate category of the 2026 POPL Student Research Competition. References Context Andrew Zhou Harrison Grodin Lukas Kebuladze Robert Harper Runming Li https://calfproject.github.io/bib-calf-cost/ bib-calf-cost /bib-calf-cost/ Cost analysis in Calf Harrison Grodin Robert Harper https://calfproject.github.io/grodin-harper-2023/ grodin-harper-2023 /grodin-harper-2023/ Amortized Analysis via Coinduction Reference CALCO '23: 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023) 10.4230/LIPIcs.CALCO.2023.23 2023, title = {{Amortized Analysis via Coinduction}}, author = {Grodin, Harrison and Harper, Robert}, year = {2023}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, volume = {270}, pages = {23:1--23:6}, doi = {10.4230/LIPIcs.CALCO.2023.23}, isbn = {978-3-95977-287-7}, issn = {1868-8969}, url = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.23}, editor = {Baldan, Paolo and de Paiva, Valeria}, urn = {urn:nbn:de:0030-drops-188201}, annote = {Keywords: amortized analysis, coinduction, data structure, mechanized proof} }]]> Amortized analysis is a program cost analysis technique for data structures in which the cost of operations is specified in aggregate, under the assumption of continued sequential use. Typically, amortized analyses are presented inductively, in terms of finite sequences of operations. We give an alternative coinductive formulation and prove that it is equivalent to the standard inductive definition. We describe a classic amortized data structure, the batched queue, and outline a coinductive proof of its amortized efficiency in Calf, a dependent type theory for cost analysis. Harrison Grodin Robert Harper https://calfproject.github.io/grodin-harper-2024/ grodin-harper-2024 /grodin-harper-2024/ Amortized Analysis via Coalgebra Reference MFPS '24: 40th Conference on Mathematical Foundations of Programming Semantics 10.46298/entics.14797 2024, title = {Amortized Analysis via Coalgebra}, author = {Harrison Grodin and Robert Harper}, year = {2024}, month = {Dec}, journal = {Electronic Notes in Theoretical Informatics and Computer Science}, volume = {Volume 4 - Proceedings of MFPS XL}, doi = {10.46298/entics.14797}, issn = {2969-2431}, url = {https://entics.episciences.org/14797}, eid = {10}, keywords = {Computer Science - Programming Languages, Computer Science - Data Structures and Algorithms, Mathematics - Category Theory} }]]> Amortized analysis is a cost analysis technique for data structures in which cost is studied in aggregate: rather than considering the maximum cost of a single operation, one bounds the total cost encountered throughout a session. Traditionally, amortized analysis has been phrased inductively, quantifying over finite sequences of operations. Connecting to prior work on coalgebraic semantics for data structures, we develop the alternative perspective that amortized analysis is naturally viewed coalgebraically in a category of cost algebras, where a morphism of coalgebras serves as a first-class generalization of potential function suitable for integrating cost and behavior. Using this simple definition, we consider amortization of other sample effects, non-commutative printing and randomization. To support imprecise amortized upper bounds, we adapt our discussion to the bicategorical setting, where a potential function is a colax morphism of coalgebras. We support algebraic and coalgebraic operations simultaneously by using coalgebras for an endoprofunctor instead of an endofunctor, combining potential using a monoidal structure on the underlying category. Finally, we compose amortization arguments in the indexed category of coalgebras to implement one amortized data structure in terms of others. Runming Li Harrison Grodin Robert Harper https://calfproject.github.io/li-grodin-harper-2023/ li-grodin-harper-2023 /li-grodin-harper-2023/ A Verified Cost Analysis of Joinable Red-Black Trees Reference Manuscript 10.48550/arXiv.2309.11056 2023, title = {A Verified Cost Analysis of Joinable Red-Black Trees}, author = {Runming Li and Harrison Grodin and Robert Harper}, year = {2023}, url = {https://arxiv.org/abs/2309.11056}, eprint = {2309.11056}, archiveprefix = {arXiv}, primaryclass = {cs.PL} }]]> Ordered sequences of data, specified with a join operation to combine sequences, serve as a foundation for the implementation of parallel functional algorithms. This abstract data type can be elegantly and efficiently implemented using balanced binary trees, where a join operation is provided to combine two trees and rebalance as necessary. In this work, we present a verified implementation and cost analysis of joinable red-black trees in Calf, a dependent type theory for cost analysis. We implement red-black trees and auxiliary intermediate data structures in such a way that all correctness invariants are intrinsically maintained. Then, we describe and verify precise cost bounds on the operations, making use of the red-black tree invariants. Finally, we implement standard algorithms on sequences using the simple join-based signature and bound their cost in the case that red-black trees are used as the underlying implementation. All proofs are formally mechanized using the embedding of Calf in the Agda theorem prover. Andrew Zhou https://calfproject.github.io/zhou-2025/ zhou-2025 /zhou-2025/ Formally Verified Cost of the Parallel Prefix Sum Algorithm Reference Technical Report, Carnegie Mellon University https://www.cs.cmu.edu/~runmingl/student/zhou-scan.pdf 2025, title = {Formally Verified Cost of the Parallel Prefix Sum Algorithm}, author = {Andrew Zhou}, year = {2025}, month = {aug}, url = {https://www.cs.cmu.edu/~runmingl/student/zhou-scan.pdf}, institution = {Carnegie Mellon University} }]]> CMU undergraduate summer SURA project report by Andrew Zhou, supervised by Runming Li. Lukas Kebuladze https://calfproject.github.io/kebuladze-2025/ kebuladze-2025 /kebuladze-2025/ Amortized Analysis of Splay Trees via a Lax Homomorphism Reference Technical Report, Carnegie Mellon University https://www.cs.cmu.edu/~runmingl/student/kebuladze-splay.pdf 2025, title = {Amortized Analysis of Splay Trees via a Lax Homomorphism}, author = {Lukas Kebuladze}, year = {2025}, month = {aug}, url = {https://www.cs.cmu.edu/~runmingl/student/kebuladze-splay.pdf}, institution = {Carnegie Mellon University} }]]> CMU undergraduate summer SURA project report by Lukas Kebuladze, supervised by Runming Li. Extended abstract, poster, and presentation: 3rd place winner in the undergraduate category of the 2026 POPL Student Research Competition. Backlinks Lukas Kebuladze Runming Li https://calfproject.github.io/ver-splaytree/ ver-splaytree /ver-splaytree/ splay tree Case Study Amortized Analysis of Splay Trees via a Lax Homomorphism Formalization: splay tree Related https://calfproject.github.io/cmu/ cmu /cmu/ Carnegie Mellon University Institution https://www.cmu.edu/ Pittsburgh, PA, USA https://calfproject.github.io/lkebulad/ lkebulad /lkebulad/ Lukas Kebuladze Person Carnegie Mellon University Undergraduate Student https://calfproject.github.io/runmingl/ runmingl /runmingl/ Runming Li Person https://www.cs.cmu.edu/~runmingl/ Carnegie Mellon University 0000-0001-7600-9069 Ph.D. Student Runming Li is a second-year Ph.D. student in the Computer Science Department at Carnegie Mellon University, advised by Prof. Robert Harper. He studies semantics of programming languages from the perspectives of (dependent) type theory and category theory. More broadly, he is interested in the design and implementation of functional programming languages and proof assistants that facilitate the construction and verification of correct and efficient programs. Contributions