lin-0129

9.12 Iterated repair and coalgebraic stabilization

Repair can be iterated. Let \(x_n\) denote the obstruction state after the \(n\)-th diagnose–repair cycle, and let

\[ x_{n+1}=F_D(x_n) \]

be the realized INC update. This is first of all a discrete dynamical system. A coalgebraic presentation additionally packages an observable certificate, for example

\[ c:M_D\longrightarrow O\times M_D, \qquad c(x)=\bigl(\operatorname {cert}(x),F_D(x)\bigr), \]

which is a coalgebra for the endofunctor \(O\times -\). This packaging exposes the successive observations of a system that repeatedly unfolds and repairs its infinitesimal behavior [ Rutten , 2000 ] .

Theorem 9.13 Contractive stabilization

Let \((M_D,d_D)\) be complete and suppose \(F_D:M_D\to M_D\) is \(\rho \)-contractive for \(0\le \rho {\lt}1\). Then there is a unique fixed point \(x_\ast \), and

\[ d_D(x_n,x_\ast ) \le \frac{\rho ^n}{1-\rho }d_D(x_1,x_0). \]

If approximate updates satisfy \( d_D(\widetilde x_{n+1},F_D\widetilde x_n)\le \varepsilon , \) then

\[ \limsup _n d_D(\widetilde x_n,x_\ast ) \le \frac{\varepsilon }{1-\rho }. \]

Proof ▼

The first claim is the Banach fixed-point argument. Summing the geometric bound on successive differences gives the displayed estimate. For approximate updates, apply contractivity recursively and sum the accumulated \(\varepsilon \)-terms. This is the metric-coinductive stabilization pattern [ Kozen and Ruozzi , 2009 ] .

Contractivity is an additional hypothesis, not a generic property of the tangent tower. Coalgebraic language organizes repeated diagnosis; it does not guarantee convergence of every repair loop.