Dynamic Complexity of Formal Languages

TL;DR

This paper explores the dynamic complexity of formal languages, establishing the boundaries of what can be maintained efficiently in various dynamic classes and demonstrating that regular languages are exactly those maintainable in DynPROP.

Contribution

It characterizes the classes of formal languages maintainable in DynPROP, DynQF, and DynFO, providing new lower bounds and separation results among these classes.

Findings

Regular languages are exactly those maintainable in DynPROP.

All context-free languages can be maintained in DynFO.

Some properties are not maintainable in DynPROP, but are in DynQF with precomputation.

Abstract

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.