Simplest Non-Regular Deterministic Context-Free Language

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. The main technical result is a proof that the DCFL′ language L# = {01 | n ≥ 1} is DCFL′-simple, and can be thus viewed as one of the simplest languages in the class DCFL′, in a precise sense. The notion of DCFL′-simple languages is nontrivial: e.g., the language LR = {wcw | w ∈ {a, b}∗} is not DCFL′-simple. By describing an application in the area of neural networks (elaborated in another paper), we demonstrate that C-simple problems under suitable reductions can provide a tool for expanding the lower-bound results known for single problems to the whole classes of problems. 2012 ACM Subject Classification Theory of computation → Grammars and context-free languages; Theory of computation → Problems, reductions and completeness; Theory of computation → Transducers