## Sylvain Schmitz and Philippe Schnoebelen

- Area: LoCo
- Level: A
- Week: 2
- Time: 09:00 – 10:30
- Room: D1.01

#### Abstract

Well-quasi-orderings (wqos) are a fundamental tool in logic and computer science. They provide termination arguments in a large number of decidability (or finiteness, regularity, …) results. In constraint solving, automated deduction, program analysis, and many more fields, wqos usually appear under the guise of specific tools, like Dickson’s Lemma (for tuples of integers), Higman’s Lemma (for words and their subwords), Kruskal’s Tree Theorem and its variants (for finite trees with embeddings), and recently the Robertson-Seymour Theorem (for graphs and their minors). What is not very well known is that wqo-based proofs have an algorithmic content.

The purpose of this course is to provide an introduction to the algorithmic aspects of wqos: to present generic algorithms working on large classes of problems, to introduce the techniques used to prove complexity upper bounds and lower bounds, to explain the use of wqo ideals in algorithms, and provide several applications in logics (e.g. data logics, relevance logic), verification (prominently for well-structured transition systems), and formal languages. Because wqos are in such wide use, we believe this topic to be of relevance to a broad community with interests in complexity theory and decision procedures for logical theories.

#### Planned Content

- well-quasi-orders (wqos): examples and characterisations
- applications of wqos: well-structured transition systems (WSTS), termination proofs, relevance logic
- complexity: fast-growing complexity, Hardy computations, length function theorems
- ideals: effective representations and algorithmics
- applications of ideals: complete WSTS, coverability algorithms

#### Resources

- lecture notes (100+ pages)
- slides day 1: Basics of wqo theory
- slides day 2: Algorithmic applications of wqos
- slides day 4: Ideals of wqos and their algorithms
- student survey

#### Additional References

- Blondin, M., Finkel, A., and McKenzie, P., 2014. Handling infinitely branching WSTS.

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*Logical Methods in Computer Science*, 8(1):22. doi:10.2168/LMCS-8(1:22)2012. - Finkel, A. and Schnoebelen, Ph., 2001. Well-structured transition systems everywhere!

*Theoretical Computer Science*, 256(1–2):63–92. doi:10.1016/S0304-3975(00)00102-X. - Lazić, R. and Schmitz, S., 2015. The ideal view on Rackoff’s coverability technique. In

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Computation Theory - Schmitz, S. and Schnoebelen, Ph., 2011. Multiply-recursive upper bounds with Higman’s Lemma. In
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