\chapter{Related Work}
\label{ch:related-work}

The field of neural network verification is a highly active research area, driven by the need to
provide formal safety for artificial intelligence applications in safety-critical systems.
Existing verification techniques can be classified into complete and incomplete methods.

\paragraph{Complete solvers}
Complete verification methods guarantee to either find a counterexample violating the property or
prove that the property holds. Katz et al.~\cite{katz2017reluplex} presented \textit{Reluplex}, a
pioneering SMT solver designed specifically for networks with ReLU activation functions. It extends
the standard simplex method, originally created by Dantzig (as described by
Nash~\cite{dantzig1947simplex}), to handle ReLU constraints directly. Katz et
al.~\cite{katz2019marabou} also presented \textit{Marabou}, the successor of \textit{Reluplex}, a
more modular and highly optimized framework that supports a wider range of activation functions.

\paragraph{Incomplete solvers}
Incomplete verification methods use abstract interpretation to soundly approximate neural networks.
Wang et al.~\cite{wang2018efficientformalsafetyanalysis} presented \textit{Neurify}, a tool that
utilizes symbolic interval analysis to compute tight bounds on neurons and prune ReLUs that are
guaranteed to be stably active or inactive.

\paragraph{Hybrid solvers}
\textit{$\alpha$-$\beta$-CROWN} is a neural network verifier that combines a series of complete
verification methods (based on branch-and-bound) and incomplete verification methods (based on
bound-propagation): \textit{CROWN} (Zhang et al.~\cite{zhang2018efficient}), \textit{auto\_LiRPA}
(Xu et al.~\cite{xu2020automatic}), \textit{$\alpha$-CROWN} (Xu et al.~\cite{xu2021fast}),
\textit{$\beta$-CROWN} (Wang et al.~\cite{wang2021beta}), \textit{GenBaB} (Shi et
al.~\cite{shi2024genbab}), \textit{GCP-CROWN} (Zhang et al.~\cite{zhang2022general}) and
\textit{BICCOS} (Zhou et al.~\cite{zhou2024scalable}). Thanks to this hybrid approach, 
\textit{$\alpha$-$\beta$-CROWN} achieved state-of-the-art performance and won VNNCOMP 2021-2025.

\paragraph{Equivalence Checking}
Closest to the scope of this work, Eleftheriadis et al.~\cite{eleftheriadis2022equivalence} study
the problem of neural network equivalence checking using SMT solvers, and define the notions of
strict, epsilon, and argmax equivalence adopted in \textbf{\Cref{sec:benchmarks}}. They approach the
equivalence problem by directly encoding an SMT formula and relying on internal heuristics of the
solver to handle the search space. Instead, \textbf{VEIN} introduces an explicit, formally verified
preprocessing stage that reduces the networks to a normal form before encoding them into an SMT
formula.