\newpage
\section{State of the Art and Limits of Current Practice}

The principal aim of this research is to create autonomous reactor control
systems that are tractably safe. But, to understand what exactly is being
automated, it is important to understand how nuclear reactors are operated
today. First, the reactor operator themselves is discussed. Then, operating
procedures that we aim to leverage later are examined. Next, limitations of
human-based operation are investigated, while finally we discuss current formal
methods based approaches to building reactor control systems.

\subsection{Current Reactor Procedures and Operation}

Nuclear plant procedures exist in a hierarchy: normal operating procedures for
routine operations, abnormal operating procedures for off-normal conditions,
Emergency Operating Procedures (EOPs) for design-basis accidents, Severe
Accident Management Guidelines (SAMGs) for beyond-design-basis events, and
Extensive Damage Mitigation Guidelines (EDMGs) for catastrophic damage
scenarios. These procedures must comply with 10 CFR 50.34(b)(6)(ii) and are
developed using guidance from NUREG-0900~\cite{NUREG-0899, 10CFR50.34}, but their
development process relies fundamentally on expert judgment and simulator
validation rather than formal verification. Procedures undergo technical
evaluation, simulator validation testing, and biennial review as part of
operator requalification under 10 CFR 55.59~\cite{10CFR55.59}. Despite these
rigorous development processes, procedures fundamentally lack formal
verification of key safety properties. There is no mathematical proof that
procedures cover all possible plant states, that required actions can be
completed within available timeframes under all scenarios, or that transitions
between procedure sets maintain safety invariants. 

\textbf{LIMITATION:} \textit{Procedures lack formal verification of correctness
and completeness.} Current procedure development relies on expert judgment and
simulator validation. No mathematical proof exists that procedures cover all
possible plant states, that required actions can be completed within available
timeframes, or that transitions between procedure sets maintain safety
invariants. Paper-based procedures cannot ensure correct application, and even
computer-based procedure systems lack the formal guarantees that automated
reasoning could provide.

Nuclear plants operate with multiple control modes: automatic control where the
reactor control system maintains target parameters through continuous reactivity
adjustment, manual control where operators directly manipulate the reactor, and
various intermediate modes. In typical pressurized water reactor operation, the
reactor control system automatically maintains a floating average temperature
and compensates for changes in power demand with reactivity feedback loops
alone. Safety systems instead operate with implemented automation. Reactor
Protection Systems trip automatically on safety signals with millisecond
response times, and engineered safety features actuate automatically on accident
signals without operator action required.

% \textbf{LIMITATION:} \textit{Current practice treats continuous plant
% dynamics and discrete control logic separately.} No application of
% hybrid control theory exists that could provide mathematical guarantees
% across mode transitions, verify timing properties formally, or optimize
% the automation-human interaction trade-off with provable safety bounds.
%
The current division between automated and human-controlled functions reveals
the fundamental challenge of hybrid control. Highly automated systems handle
reactor protection like automatic trips on safety parameters, emergency core
cooling actuation, containment isolation, and basic process
control~\cite{WRPS.Description, gentillon_westinghouse_1999}. Human operators,
however, retain control of strategic decision-making such as power level
changes, startup/shutdown sequences, mode transitions, and procedure
implementation. %%%NEED MORE


\subsection{Human Factors in Nuclear Accidents}
The persistent role of human error in nuclear safety incidents, despite
decades of improvements in training and procedures, provides perhaps the
most compelling motivation for formal automated control with
mathematical safety guarantees.

Current generation nuclear power plants employ 3,600+ active NRC-licensed
reactor operators in the United States~\cite{operator_statistics}. These
operators are divided into Reactor Operators (ROs) who manipulate reactor
controls and Senior Reactor Operators (SROs) who direct plant operations and
serve as shift supervisors~\cite{10CFR55}. Staffing typically requires 2+ ROs
with at least one SRO for current generation units~\cite{10CFR50.54}. To become
a reactor operator, an individual spends several years to pass completed
training.

The role of these human operators is paradoxically both critical and
problematic. Operators hold legal authority under 10 CFR Part 55 to make
critical decisions including departing from normal regulations during
emergencies. The Three Mile Island (TMI) accident demonstrated how combination
of personnel error, design deficiencies, and component failures led to partial
meltdown when operators misread confusing and contradictory readings and shut
off the emergency water system~\cite{Kemeny1979}. The President's Commission on
TMI identified a fundamental ambiguity: placing responsibility and
accountability for safe power plant operations on the licensee in all
circumstances without formal verification that operators can fulfill this
responsibility under all conditions does not guarantee safety. This tension
between operational flexibility and safety assurance remains unresolved in
current practice as the person responsible for reactor safety simultaneously is
usually the root cause of a failure.

Multiple independent analyses converge on a striking statistic: 70--80\% of all
nuclear power plant events are attributed to human error versus approximately
20\% to equipment failures~\cite{WNA2020}. More significantly, the root cause of
all severe accidents at nuclear power plants such as those at Three Mile Island,
Chernobyl, and Fukushima Daiichi, has been identified as poor safety management
and poor safety culture--primarily human factors~\cite{hogberg_root_2013}. A
detailed analysis of 190 events at Chinese nuclear power plants from
2007--2020~\cite{zhang_analysis_2025} found that 53\% of events involved active
errors while 92\% were associated with latent errors (organizational and
systemic weaknesses that create conditions for failure). 

%%%%% This seems like a bad paragraph. Doesn't really connect with the idea of 
%%%%% autonomy. Seems more like a design issue for the control room. With more
%%%%% time it could be explained how doing things this way with requirements to 
%%%%% logic would catch that no indicator of actual valve position is a problem.
% The Three Mile Island Unit 2 accident on March 28, 1979 remains the definitive
% case study in human factors failures in nuclear operations. The accident began
% at 4:00 AM with a routine feedwater pump trip, escalating when a
% pressure-operated relief valve (PORV) stuck open---draining reactor
% coolant---but control room instrumentation showed only whether the valve had
% been commanded to close, not whether it actually closed. When Emergency Core
% Cooling System pumps automatically activated as designed, operators made the
% fateful decision to shut them down based on their incorrect assessment of plant
% conditions. The result was a massive loss of coolant accident and the core
% quickly began to overheat. During the emergency, operators faced more than 100
% simultaneous alarms, overwhelming their cognitive capacity~\cite{Kemeny1979}.
% The core suffered partial meltdown with 44\% of the fuel melting before the
% situation was stabilized. Quantitative risk analysis revealed the magnitude of
% failure in existing safety assessment methods: the actual core damage
% probability was approximately 5\% per year while Probabilistic Risk Assessment
% had predicted 0.01\% per year---a 500-fold underestimation. This
% dramatic failure demonstrated that human reliability could not be adequately
% assessed through expert judgment and historical data alone. 
% % how does autonomy fix these issues exactly? This seems like

\textbf{LIMITATION:} \textit{Human factors impose fundamental reliability limits
that cannot be overcome through training alone.} The persistent human
error contribution despite four decades of improvements demonstrates that these
limitations are fundamental rather than remediable part of human-driven control.

\subsection{HARDENS and Formal Methods}

The High Assurance Rigorous Digital Engineering for Nuclear Safety (HARDENS)
project represents the most advanced application of formal methods to nuclear
reactor control systems to date~\cite{Kiniry2024}. HARDENS aimed to address the nuclear industry's
fundamental dilemma: existing U.S. nuclear control rooms rely on analog
technologies from the 1950s--60s. This technology is woefully out of date
compared to modern control technologies, and incurs significant risk and cost to
plant operation. The NRC contracted Galois, a company of formal methods experts,
to demonstrate that Model-Based Systems Engineering and formal methods could
design, verify, and implement a complex protection system meeting regulatory
criteria at a fraction of typical cost. The project delivered a Reactor Trip
System (RTS) implementation with full traceability from NRC Request for
Proposals and IEEE standards through formal architecture specifications to
formally verified software.
%%% did it actually do an FPGA demonstration? Dubious.

HARDENS employed an array of formal methods tools and techniques across the
verification hierarchy. High-level specifications used Lando, SysMLv2, and FRET
(NASA Formal Requirements Elicitation Tool) to capture stakeholder
requirements, domain engineering, certification requirements, and safety
requirements. % this sentence is long af
Requirements were formally analyzed for consistency, completeness,
and realizability using SAT and SMT solvers. Executable formal models employed
Cryptol to create an executable behavioral model of the entire RTS including all
subsystems, components, and limited digital twin models of sensors, actuators,
and compute infrastructure. Automatic code synthesis generated formally
verifiable C implementations and System Verilog hardware implementations
directly from Cryptol models---eliminating the traditional gap between
specification and implementation where errors commonly arise.

Despite its accomplishments, HARDENS has a fundamental limitation directly
relevant to hybrid control synthesis: the project addressed only discrete
digital control logic without modeling or verifying continuous reactor dynamics.
The Reactor Trip System specification and formal verification covered discrete
state transitions such as trip/no-trip decisions, digital sensor input
processing through discrete logic, and discrete actuation outputs such as
reactor trip commands. However, the project did not address continuous dynamics
of nuclear reactor physics. Real reactor safety depends on the interaction
between continuous processes such as temperature, pressure, neutron flux
evolving due to discrete control decisions. HARDENS verified the discrete
controller in isolation but not the closed-loop hybrid system behavior.

\textbf{LIMITATION:} \textit{HARDENS addressed discrete control logic without
continuous dynamics or hybrid system verification.} Verifying discrete control
logic alone provides no guarantee that the closed-loop system exhibits desired
continuous behavior such as stability, convergence to setpoints, or maintained
safety margins.

HARDENS produced a demonstrator system at Technology Readiness Level 2--3
(analytical proof of concept with laboratory breadboard validation) rather than
a deployment-ready system validated through extended operational testing. The
NRC Final Report explicitly notes~\cite{Kiniry2024} that all material is
considered in development and not a finalized product and ``The demonstration
of its technical soundness was to be at a level consistent with satisfaction of
the current regulatory criteria, although with no explicit demonstration of how
regulatory requirements are met.'' The project did not include deployment in
actual nuclear facilities, testing with real reactor systems under operational
conditions, side-by-side validation with operational analog RTS systems,
systematic failure mode testing (radiation effects, electromagnetic
interference, temperature extremes), actual NRC licensing review, or human
factors validation with licensed nuclear operators in realistic control room
scenarios.

\textbf{LIMITATION:} \textit{HARDENS achieved TRL 2--3 without experimental
validation.} While formal verification provides mathematical correctness
guarantees for the implemented discrete logic, the gap between formal
verification and actual system deployment involves myriad practical
considerations: integration with legacy systems, long-term reliability
under harsh environments, human-system interaction in realistic
operational contexts, and regulatory acceptance of formal methods as
primary assurance evidence.
