  We study the communication complexity of computing functions
$F:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}$ in the memoryless
communication model. Here, Alice is given $x\in \{0,1\}^n$, Bob is given $y\in
\{0,1\}^n$ and their goal is to compute F(x,y) subject to the following
constraint: at every round, Alice receives a message from Bob and her reply to
Bob solely depends on the message received and her input x; the same applies to
Bob. The cost of computing F in this model is the maximum number of bits
exchanged in any round between Alice and Bob (on the worst case input x,y). In
this paper, we also consider variants of our memoryless model wherein one party
is allowed to have memory, the parties are allowed to communicate quantum bits,
only one player is allowed to send messages. We show that our memoryless
communication model capture the garden-hose model of computation by Buhrman et
al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13)
and the overlay communication complexity by Papakonstantinou et al. (CCC'14).
Thus the memoryless communication complexity model provides a unified framework
to study space-bounded communication models. We establish the following: (1) We
show that the memoryless communication complexity of F equals the logarithm of
the size of the smallest bipartite branching program computing F (up to a
factor 2); (2) We show that memoryless communication complexity equals
garden-hose complexity; (3) We exhibit various exponential separations between
these memoryless communication models.
  We end with an intriguing open question: can we find an explicit function F
and universal constant c>1 for which the memoryless communication complexity is
at least $c \log n$? Note that $c\geq 2+\varepsilon$ would imply a
$\Omega(n^{2+\varepsilon})$ lower bound for general formula size, improving
upon the best lower bound by Ne\v{c}iporuk in 1966.
