concept called affine computation was introduced [@DCY16A] and its finite automata versions (AfAs) have been examined [@DCY16A; @VilY16A; @BMY16A; @HMY17]. Some underlying results are as follows: (i) they are more powerful than their probabilistic and quantum counterparts (PFAs and QFAs) with bounded and unbounded error; (ii) one-sided bounded-error AfAs and nondeterministic QFAs define the same class when using rational number transitions; and, (iii) AfAs can distinguish any given pair of strings by using two states with zero-error. Very recently, affine OBDD was introduced in [@IKPVY17A] and it was shown that they can be exponentially narrower than bounded-error quantum and classical OBDDs.

In this paper, we introduce (realtime) AfA augmented with a counter (AfCAs), and analyze their ability as well as Las Vegas AfAs. It is already known that AfAs can simulate QFAs exactly by a quadratic increase in the number of states [@VilY16A]. However, this simulation cannot be extended to the simulation of QFAs with a counter (QCAs). Therefore, the quantum interference used by QCAs cannot be trivially used by AfCAs. Besides, the well-formed conditions for QCAs can be complicated, but as seen soon, they are easy to check for AfCAs. Thus, we believe that AfCAs may present classical and 