Monotonic and Non-Monotonic Solution Concepts for Generalized Circuits

TL;DR

This paper identifies a non-monotonic flaw in the solution concept of generalized circuits and proposes a way to simulate stronger, monotonic Boolean gates to resolve this issue, simplifying future research.

Contribution

It demonstrates that Boolean gates in generalized circuits are redundant and can be replaced with monotonic versions, fixing the non-monotonicity problem.

Findings

Identified non-monotonicity in generalized circuits' solution concept.

Proposed simulation of monotonic Boolean gates to fix the flaw.

Simplified the theoretical framework for generalized circuits.

Abstract

Generalized circuits are an important tool in the study of the computational complexity of equilibrium approximation problems. However, in this paper, we reveal that they have a conceptual flaw, namely that the solution concept is not monotonic. By this we mean that if $\varepsilon < \varepsilon'$, then an $\varepsilon$-approximate solution for a certain generalized circuit is not necessarily also an $\varepsilon'$-approximate solution. The reason for this non-monotonicity is the way Boolean operations are modeled. We illustrate that non-monotonicity creates subtle technical issues in prior work that require intricate additional arguments to circumvent. To eliminate this problem, we show that the Boolean gates are a redundant feature: one can simulate stronger, monotonic versions of the Boolean gates using the other gate types. Arguing at the level of these stronger Boolean gates eliminates all of the aforementioned issues in a natural way. We hope that our results will enable new studies of sub-classes of generalized circuits and enabler simpler and more natural reductions from generalized circuits to other equilibrium search problems.