# A Strategy for Dynamic Programs: Start over and Muddle through

## TL;DR

This paper presents a new strategy for dynamic programs that combines periodic recomputation with limited change maintenance, enabling the maintenance of complex queries in DynFO under certain conditions.

## Contribution

It introduces a novel approach for constructing dynamic programs using periodic full recomputations and limited change steps, extending the capabilities of DynFO.

## Key findings

- MSO-definable problems are in DynFO for bounded treewidth graphs
- A Feferman-Vaught-type composition theorem for MSO is established
- Dynamic maintenance of queries with limited change steps is feasible

## Abstract

In the setting of DynFO, dynamic programs update the stored result of a query whenever the underlying data changes. This update is expressed in terms of first-order logic. We introduce a strategy for constructing dynamic programs that utilises periodic computation of auxiliary data from scratch and the ability to maintain a query for a limited number of change steps. We show that if some program can maintain a query for log n change steps after an AC$^1$-computable initialisation, it can be maintained by a first-order dynamic program as well, i.e., in DynFO. As an application, it is shown that decision and optimisation problems defined by monadic second-order (MSO) formulas are in DynFO, if only change sequences that produce graphs of bounded treewidth are allowed. To establish this result, a Feferman-Vaught-type composition theorem for MSO is established that might be useful in its own right.