Simplest Non-Regular Deterministic Context-Free Language

Abstract

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 n 1 n | n ≥ 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.