  We introduce a new notion of C-simple problems for a class C of decision
problems (i.e. languages), w.r.t. a particular reduction. A problem is C-simple
if it can be reduced to each problem in C. This can be viewed as a conceptual
counterpart to C-hard problems to which all problems in C reduce. Our concrete
example is the class of non-regular deterministic context-free languages
(DCFL'), with a truth-table reduction by Mealy machines (which proves to be a
preorder). The main technical result is a proof that the DCFL' language
$L=\{0^n1^n; n\geq 1\}$ is DCFL'-simple, which can thus be viewed as the
simplest problem in the class DCFL'. This result has already provided an
application, to the computational model of neural networks 1ANN at the first
level of analog neuron hierarchy. This model was proven not to recognize $L$,
by using a specialized technical argument that can hardly be generalized to
other languages in DCFL'. By the result that $L$ is DCFL'-simple, w.r.t. the
reduction that can be implemented by 1ANN, we immediately obtain that 1ANN
cannot accept any language in DCFL'. It thus seems worthwhile to explore if
looking for C-simple problems in other classes C under suitable reductions
could provide effective tools for expanding the lower-bound results known for
single problems to the whole classes of problems.
