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The capacity region of the Finite-State Multiple Access Channel (FS-MAC) withfeedback that may be an arbitrary time-invariant function of the channel outputsamples is considered. We characterize both an inner and an outer bound forthis region, using Masseys's directed information. These bounds are shown tocoincide, and hence yield the capacity region, of FS-MACs where the stateprocess is stationary and ergodic and not affected by the inputs.  Though `multi-letter' in general, our results yield explicit conclusions whenapplied to specific scenarios of interest. E.g., our results allow us to:  - Identify a large class of FS-MACs, that includes the additive mod-2 noiseMAC where the noise may have memory, for which feedback does not enlarge thecapacity region.  - Deduce that, for a general FS-MAC with states that are not affected by theinput, if the capacity (region) without feedback is zero, then so is thecapacity (region) with feedback.  - Deduce that the capacity region of a MAC that can be decomposed into a`multiplexer' concatenated by a point-to-point channel (with, without, or withpartial feedback), the capacity region is given by $\sum_{m} R_m \leq C$, whereC is the capacity of the point to point channel and m indexes the encoders.Moreover, we show that for this family of channels source-channel codingseparation holds.

Capacity Region of the Finite-State Multiple Access Channel with and without Feedback