# Decomposition theorems and model-checking for the modal μ-calculus

@article{Bojanczyk2014DecompositionTA, title={Decomposition theorems and model-checking for the modal $\mu$-calculus}, author={Mikolaj Bojanczyk and Christoph Dittmann and Stephan Kreutzer}, journal={Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, year={2014} }

We prove a general decomposition theorem for the modal μ-calculus Lμ in the spirit of Feferman and Vaught's theorem for disjoint unions. In particular, we show that if a structure (i.e., transition system) is composed of two substructures M1 and M2 plus edges from M1 to M2, then the formulas true at a node in M only depend on the formulas true in the respective substructures in a sense made precise below. As a consequence we show that the model-checking problem for Lμ is fixed-parameter…

## 7 Citations

### DAG-width of Control Flow Graphs with Applications to Model Checking

- Mathematics, Computer ScienceArXiv
- 2015

One consequence of this result is that parity games (and hence the $\mu$-calculus model checking problem), which are known to be tractable on graphs of bounded DAG-width, can be solved efficiently in practice on control flow graphs.

### Graph-theoretic Properties of Control Flow Graphs and Applications

- Mathematics, Computer Science
- 2015

This thesis shows that the DAG-width of control flow graphs arising from structured programs is at most three and gives a linear-time algorithm to compute the corresponding DAG decomposition and shows that control flow graph can have unbounded entanglement.

### Clique-Width and Directed Width Measures for Answer-Set Programming

- Computer ScienceECAI
- 2016

This paper first gives some negative observations showing that directed width measures on the dependency graph of a program do not lead to FPT results, then considers the graph parameter of signed clique-width and presents a novel dynamic programming algorithm that is FPT w.r.t. this parameter.

### א1 and the Modal μ-Calculus

- MathematicsLog. Methods Comput. Sci.
- 2019

It is proved that any formal expression built from 0, 1, $\omega$, $\omego_1$ by using the binary operator symbol + gives rise to a closure ordinal and that $\omegas_1$, the least uncountable ordinal, is such aclosure ordinal.

### Aleph1 and the Modal mu-Calculus

- MathematicsCSL
- 2017

It is proved that any formal expression built from 0, 1, $\omega$, $\omego_1$ by using the binary operator symbol + gives rise to a closure ordinal and that $\omegas_1$, the least uncountable ordinal, is such aclosure ordinal.

### A I ] 3 0 D ec 2 01 6 Clique-Width and Directed Width Measures for Answer-Set Programming

- Computer Science
- 2018

This paper considers the graph parameter of signed clique-width and presents a novel dynamic programm ing algorithm that is FPT w.r.t. this parameter, and is the first FPT algorithm for bounded cliques-width for reasoning probl ems beyond SAT.

### Parameterized Algorithms for Parity Games

- Computer ScienceMFCS
- 2015

Determining the winner of a Parity Game is a major problem in computational complexity with a number of applications in verification, and the problem has often been considered with parameters such as treewidth or clique-width, by applying dynamic programming techniques.

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