  Gr\"{o}bner bases are nowadays central tools for solving various problems in
commutative algebra and algebraic geometry. A typical use of Gr\"{o}bner bases
is the multivariate polynomial system solving, which enables us to construct
algebraic attacks against post-quantum cryptographic protocols. Therefore, the
determination of the complexity of computing Gr\"{o}bner bases is very
important both in theory and in practice: One of the most important cases is
the case where input polynomials compose an (overdetermined) affine
semi-regular sequence. The first part of this paper aims to present a survey on
Gr\"{o}bner basis computation and its complexity. In the second part, we shall
give an explicit formula on the (truncated) Hilbert-Poincar\'{e} series
associated to the homogenization of an affine semi-regular sequence. Based on
the formula, we also study (reduced) Gr\"{o}bner bases of the ideals generated
by an affine semi-regular sequence and its homogenization. Some of our results
are considered to give mathematically rigorous proofs of the correctness of
methods for computing Gr\"{o}bner bases of the ideal generated by an affine
semi-regular sequence.
