On nondeterminism in combinatorial filters

TL;DR

This paper broadens the concept of combinatorial filter minimization to include nondeterministic filters, demonstrating that such problems are PSPACE-hard and can significantly reduce filter size, posing new challenges.

Contribution

It introduces a new, broader definition of filter minimization allowing nondeterminism, and proves the problem is PSPACE-hard, unlike the deterministic case.

Findings

Nondeterministic filters can reuse states for more behavior.

Size reduction in nondeterministic filters can be larger than polynomial.

Producing nondeterministic minimizers is PSPACE-hard.

Abstract

The problem of combinatorial filter reduction arises from questions of resource optimization in robots; it is one specific way in which automation can help to achieve minimalism, to build better, simpler robots. This paper contributes a new definition of filter minimization that is broader than its antecedents, allowing filters (input, output, or both) to be nondeterministic. This changes the problem considerably. Nondeterministic filters are able to re-use states to obtain, essentially, more 'behavior' per vertex. We show that the gap in size can be significant (larger than polynomial), suggesting such cases will generally be more challenging than deterministic problems. Indeed, this is supported by the core computational complexity result established in this paper: producing nondeterministic minimizers is PSPACE-hard. The hardness separation for minimization which exists between deterministic filter and deterministic automata, thus, does not hold for the nondeterministic case.