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In four dimensions, we consider a generalized scalar-tensor theory where thecoupling functions only depend on the kinetic term of the scalar field. Forthis model, we obtain a set of hairy Anti-de-Sitter black hole solutions,allowing us to calculate the computational complexity, according to theComplexity equals Action conjecture. To perform this, the system contains aparticle moving on the boundary, corresponding to the insertion of afundamental string in the bulk. The effect string is given by the Nambu-Gototerm, analyzing the time development of this system. Together with the above,we calculate the shear viscosity, where the viscosity/entropy density ratio canviolate the Kovtun-Son-Starinets bound for a suitable choice of couplingfunctions.

Complexity of four-dimensional hairy Anti-de-Sitter black holes with a rotating string and shear viscosity in generalized scalar-tensor theories