# FO(TC) lower bounding games?

Is anyone aware of any games/algebraic structures that provide lower-bounding methodologies for $$FO(TC)$$ formulae? I am aware of EF games as they apply to first-order and second-order statements, but would like to be able to determine lower bounds for $$FO(TC)$$ queries on finite models. Any help in this direction would be appreciated!

• I am not sure whether your 2nd sentence implies that you are aware of EF games for FO(TC). They were defined in a paper by Erichel Grädel, 1992, doi.org/10.1007/BFb0023764. A proof that uses them can be found in the book "Finite Model Theory" by Ebbinghaus and Flum. – Thomas S Mar 4 at 7:42
• I have actually just taken the Ebbinghaus-Flum out of my school library- I believe I found the section you are referencing. Thank you very much for the help! – Will Asness Mar 5 at 15:08

## 1 Answer

As mentioned above: question answered by reference to FO(TC) games presented in Finite Model Theory by Ebbinghaus and Flum.