Editing Open Problems (section)

===Context===
There is a rich history of research on reversible space
bounded Turing machines. Following his seminal results on
time-bounded, logically reversible TMs, Bennett (1973) asked whether
TM-SPACE$(s) =$ reversible-TM-SPACE$(s)$. Fifteen years later he made some
progress towards this, showing that TM-SPACE$(s) \subseteq
$ reversible-TM-SPACE$(s^2)$. Many others worked to improve Bennett's
result; finally Lange, McKenzie and Tapp (2000) answered Bennett's
1973 question in the affirmative.  More recently, Thachuk et
al. described what it means for a DNA strand displacement system (DSD)
to use space (or equivalently volume) efficiently. Roughly, just as
memory-efficient TMs reuse memory, space-efficient DSDs "recycle"
strands by running reactions in both the forward and reverse
directions.  We also introduced restricted CRN models that can be
compiled into DSDs in a space-preserving way.

While our CRN models have plausible physical realizations as DSD's, it
is not clear that logically reversible, space-bounded reversible TMs
have such physical realizations that preserve space. Roughly, the
problem is that such TMs may "run" all of their transitions in the
"forward" direction (rather than in both the forward and backwards
directions, as do our recycling CRNs and DSDs), and so when one tries
a straightforward approach to "compile" these space-bounded TMs into
DSDs, the space (or volume) blows up exponentially. Can this
exponential space blow-up be avoided?