# Sequential Relational Decomposition

## TL;DR

This paper formalizes the concept of sequential decomposition in computational tasks, demonstrating its computational complexity varies across different specification methods, and exploring how human input can simplify the process.

## Contribution

It introduces a formal framework for sequential decomposition, analyzes its computational complexity in various settings, and investigates how human hints can reduce complexity.

## Key findings

- Decomposition is NP-complete when tasks are explicitly specified.
- Decomposition is NEXPTIME-complete with Boolean circuit specifications.
- Evidence suggests decomposition may be undecidable with automatic relations.

## Abstract

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple and natural formalization of sequential decomposition, in which a task is decomposed into two sequential sub-tasks, with the first sub-task to be executed before the second sub-task is executed. These tasks are specified by means of input/output relations. We define and study decomposition problems, which is to decide whether a given specification can be sequentially decomposed. Our main result is that decomposition itself is a difficult computational problem. More specifically, we study decomposition problems in three settings: where the input task is specified explicitly, by means of Boolean circuits, and by means of automatic relations. We show that in the first setting decomposition is NP-complete, in the second setting it is NEXPTIME-complete, and in the third setting there is evidence to suggest that it is undecidable. Our results indicate that the intuitive idea of decomposition as a system-design approach requires further investigation. In particular, we show that adding a human to the loop by asking for a decomposition hint lowers the complexity of decomposition problems considerably.