\section{Risks and Contingencies}

This research relies on several critical assumptions that, if invalidated,
would require scope adjustment or methodological
revision. The primary risks to successful
completion fall into four categories: computational tractability of synthesis
and verification, complexity of the discrete-continuous interface,
completeness of procedure formalization, and hardware-in-the-loop integration
challenges. Each risk has associated indicators for early detection and
contingency plans that preserve research value even if core assumptions prove
false. The staged project structure ensures that partial success yields
publishable results and clear identification of remaining barriers to
deployment.

\subsection{Computational Tractability of Synthesis}

The first major assumption is that formalized startup procedures will yield
automata small enough for efficient synthesis and verification. Reactive synthesis scales exponentially with specification
complexity,\footnote{``Scales exponentially'' means the computation time
doesn't just grow---it \textit{explodes}. If doubling the problem size
multiplied the time by 4, then tripling it would multiply it by 8, and so on.
A problem that takes 1 second with 10 requirements might take days with 50.
This is a fundamental challenge in computer science, not a limitation of
Dane's approach specifically.} creating risk that temporal logic
specifications derived from complete startup procedures may produce automata
with thousands of states. Such large automata would
require synthesis times exceeding days or weeks, preventing demonstration of
the complete methodology within project timelines. Reachability analysis for
continuous modes with high-dimensional state spaces may similarly prove
computationally intractable. Either barrier would constitute a fundamental
obstacle to achieving the research objectives.

Several indicators would provide early warning of computational tractability
problems. Synthesis times exceeding 24 hours for simplified procedure subsets
would suggest complete procedures are intractable. Generated automata
containing more than 1,000 discrete states would indicate the discrete state
space is too large for efficient verification. Specifications flagged as
unrealizable by realizability checking
tools would reveal fundamental conflicts in the formalized procedures.
Reachability analysis failing to converge within reasonable time bounds would
show that continuous mode verification cannot be completed with available
computational resources.

The contingency plan for computational intractability is to reduce scope to a
minimal viable startup sequence. This reduced sequence would cover only cold
shutdown to criticality to low-power hold, omitting power ascension and other
operational phases. The subset would still demonstrate the complete
methodology while reducing computational burden. The research contribution
would remain valid even with reduced scope, proving that formal hybrid
control synthesis is achievable for safety-critical nuclear applications. The
limitation to simplified operational sequences would be explicitly documented
as a constraint rather than a failure.

\subsection{Discrete-Continuous Interface Formalization}

The second critical assumption concerns the mapping between boolean guard
conditions in temporal logic and continuous state boundaries required for mode
transitions. This interface represents the fundamental challenge of hybrid
systems: relating discrete switching logic to continuous dynamics. Temporal
logic operates on boolean predicates, while continuous control requires
reasoning about differential equations and reachable sets. Some guard conditions
may require complex nonlinear predicates that cannot be cleanly expressed as
boolean combinations of simple threshold checks, making synthesis intractable.
Continuous safety regions that cannot be expressed as conjunctions of verifiable
constraints would similarly create insurmountable verification challenges. The
risk extends beyond static interface definition to dynamic behavior across
transitions: barrier certificates may fail to exist for proposed transitions, or
continuous modes may be unable to guarantee convergence to discrete transition
boundaries.

Early indicators of interface formalization problems would appear during both
synthesis and verification phases. Guard conditions requiring complex
nonlinear predicates that resist boolean abstraction would suggest
fundamental misalignment between discrete specifications and continuous
realities. Continuous safety regions that cannot be expressed as conjunctions
of half-spaces or polynomial inequalities would indicate the interface
between discrete guards and continuous invariants is too complex. Failure to
construct barrier certificates proving safety across mode transitions would
reveal that continuous dynamics cannot be formally related to discrete
switching logic. Reachability analysis showing that continuous modes cannot
reach intended transition boundaries from all possible initial conditions
would demonstrate the synthesized discrete controller is incompatible with
achievable continuous behavior.

The primary contingency for interface complexity is restricting continuous
modes to operate within polytopic invariants. Polytopes are state regions
defined as intersections of linear half-spaces, which map directly to boolean
predicates through linear inequality checks. This restriction ensures
tractable synthesis while maintaining theoretical rigor, though at the cost
of limiting expressiveness compared to arbitrary nonlinear regions. The
discrete-continuous interface remains well-defined and verifiable with
polytopic restrictions, providing a clear fallback position that preserves
the core methodology. Conservative over-approximations offer an alternative
approach: a nonlinear safe region can be inner-approximated by a polytope,
sacrificing operational flexibility to maintain formal guarantees. The
three-mode classification already structures the problem to minimize complex
transitions, with critical safety properties concentrated in expulsory modes
that can receive additional design attention.

Mitigation strategies focus on designing continuous controllers with discrete
transitions as primary objectives from the outset. Rather than designing
continuous control laws independently and verifying transitions post-hoc, the
approach uses transition requirements as design constraints. Control barrier
functions provide a systematic method to synthesize controllers that
guarantee forward invariance of safe sets and convergence to transition
boundaries. This design-for-verification approach reduces the likelihood that
interface complexity becomes insurmountable. Focusing verification effort on
expulsory modes---where safety is most critical---allows more complex
analysis to be applied selectively rather than uniformly across all modes,
concentrating computational resources where they matter most for safety
assurance.

\subsection{Procedure Formalization Completeness}

The third assumption is that existing startup procedures contain sufficient
detail and clarity for translation into temporal logic specifications.
Nuclear operating procedures, while extensively detailed, were written for
human operators who bring contextual understanding and adaptive reasoning to
their interpretation. Procedures may contain implicit knowledge, ambiguous
directives, or references to operator judgment that resist formalization in
current specification languages. Underspecified timing constraints, ambiguous
condition definitions, or gaps in operational coverage would cause synthesis
to fail or produce incorrect automata. The risk is not merely that
formalization is difficult, but that current procedures fundamentally lack
the precision required for autonomous control, revealing a gap between
human-oriented documentation and machine-executable specifications.

Several indicators would reveal formalization completeness problems early in
the project. FRET realizability checks failing due to underspecified
behaviors or conflicting requirements would indicate procedures do not form a
complete specification. Multiple valid interpretations of procedural steps
with no clear resolution would demonstrate procedure language is
insufficiently precise for automated synthesis. Procedures referencing
``operator judgment,'' ``as appropriate,'' or similar discretionary language
for critical decisions would explicitly identify points where human reasoning
cannot be directly formalized. Domain experts unable to provide crisp answers
to specification questions about edge cases would suggest the procedures
themselves do not fully define system behavior, relying instead on operator
training and experience to fill gaps.

The contingency plan treats inadequate specification as itself a research
contribution rather than a project failure. Documenting specific ambiguities
encountered would create a taxonomy of formalization barriers: timing
underspecification, missing preconditions, discretionary actions, and
undefined failure modes. Each category would be analyzed to understand why
current procedure-writing practices produce these gaps and what specification
languages would need to address them. Proposed extensions to FRETish or
similar specification languages would demonstrate how to bridge the gap
between current procedures and the precision needed for autonomous control.
The research output would shift from ``here is a complete autonomous
controller'' to ``here is what formal autonomous control requires that
current procedures do not provide, and here are language extensions to bridge
that gap.'' This contribution remains valuable to both the nuclear industry
and formal methods community, establishing clear requirements for
next-generation procedure development and autonomous control specification
languages.

Early-stage procedure analysis with domain experts provides the primary
mitigation strategy. Collaboration through the University of Pittsburgh Cyber
Energy Center enables identification and resolution of ambiguities before
synthesis attempts, rather than discovering them during failed synthesis
runs. Iterative refinement with reactor operators and control engineers can
clarify procedural intent before formalization begins, reducing the risk of
discovering insurmountable specification gaps late in the project. Comparison
with procedures from multiple reactor designs---pressurized water reactors,
boiling water reactors, and advanced designs---may reveal common patterns and
standard ambiguities amenable to systematic resolution. This cross-design
analysis would strengthen the generalizability of any proposed specification
language extensions, ensuring they address industry-wide practices rather
than specific quirks.
