Thesis/5-risks-and-contingencies/v1.tex

\section{Risks and Contingencies}

This research relies on several critical assumptions that, if invalidated, require
scope adjustment or methodological revision. Four primary risks could prevent
successful completion: 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 early warning indicators and
contingency plans that preserve research value even if core assumptions prove
false. The staged project structure ensures that partial success yields
publishable results while clearly identifying 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, creating the 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 FRET
or Strix 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.

If computational tractability becomes the limiting factor, we reduce scope to a
minimal viable startup sequence covering only cold shutdown to criticality to low-power hold, omitting power ascension and other operational phases. This reduced sequence still demonstrates the complete methodology—procedure formalization, discrete synthesis, continuous verification, and hardware implementation—while reducing computational burden. The research contribution remains valid: we prove that formal hybrid control synthesis is achievable for safety-critical nuclear applications and clearly identify which operational complexities exceed current computational capabilities. We document the limitation as a scaling constraint requiring future work, not a methodological failure.

\subsection{Discrete-Continuous Interface Formalization}

Computational tractability represents one dimension of risk. A more fundamental challenge represents the second critical assumption: mapping boolean guard
conditions in temporal logic to 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. Guard conditions
requiring complex nonlinear predicates may resist boolean abstraction, 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 previous two risks concern verification infrastructure—computational scaling and mathematical formalization. The third assumption addresses the source material itself: 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.


These risks and contingencies demonstrate that while the research faces real challenges, each has identifiable early indicators and viable mitigation strategies. The staged approach ensures valuable contributions even if assumptions prove invalid. With risks addressed, the next section examines broader impacts: who cares about this work and why it matters now.