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.

中文翻译：

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最简单的非常规确定性上下文无关语言

我们为决策问题（即语言）的C类引入了C简单问题的新概念，其中有一个特别的简化。如果一个问题可以简化为C中的每个问题，则它就是C简单的。这可以看作是C困难问题的概念对应物，C中的所有问题都可以归结为C。我们的具体示例是一类非常规的确定性上下文无关语言（DCFL'），并通过Mealy机器（已证明是预购的）简化了真值表。主要的技术结果是证明DCFL语言$ L = \ {0 ^ n1 ^ n; n \ geq 1 \} $是DCFL'-简单的，因此可以将其视为DCFL'类中最简单的问题。该结果已经为模拟神经元层次的第一级神经网络1ANN的计算模型提供了应用。该模型被证明无法识别$ L $，通过使用专门的技术论点，而该论点几乎不能推广到DCFL中的其他语言。通过$ L $是DCFL'的简单结果，通过1ANN可以实现的简化，我们立即获得1ANN无法接受DCFL'中的任何语言的信息。因此，似乎值得探索的是，在适当的归约下寻找其他C类中的C简单问题是否可以提供有效的工具，以将针对单个问题的下界结果扩展到整个问题类别。