Simplest Non-Regular Deterministic Context-Free Language

TL;DR

This paper introduces the concept of C-simple problems within decision problem classes, demonstrating that the language {0^n1^n} is the simplest in the class of non-regular deterministic context-free languages, with implications for neural network recognition limits.

Contribution

The paper defines C-simple problems, proves {0^n1^n} is the simplest in DCFL', and applies this to show neural networks cannot recognize any language in DCFL'.

Findings

{0^n1^n} is DCFL'-simple under a specific reduction.

Neural networks 1ANN cannot recognize any language in DCFL'.

The concept of C-simple problems aids in understanding computational lower bounds.

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 $L=\{0^n1^n; n\geq 1\}$ 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.