Diagonalizing Through the $ω$-Chain: Iterated Self-Certification on Bounded Turing Machines and its Least Fixed Point
This provides a novel perspective on the halting problem for theoretical computer science, though it is incremental in framing the transition from finite observability to fixed points.
The paper tackled the problem of bounded self-certification in Turing machines by translating operational constraints into a domain-theoretic framework, showing that iterative self-simulation forms an ascending ω-chain whose Scott limit resolves to the least fixed point, representing an unbounded computation that fully captures halting behavior.
Bounded self-certification in Turing machines fails because self-simulation necessarily incurs a strictly positive temporal overhead. We translate this operational constraint into a domain-theoretic framework, defining an operator that advances a finite halting observation from time bound $i$ to $i+1$. While no bounded machine can achieve a fixed point under this operator, the iterative process forms an ascending $ω$-chain. The Scott limit of this chain resolves to the least fixed point of the operator, representing an unbounded computation that fully captures the machine's halting behavior. Our construction provides a novel perspective on the halting problem, framing the transition from finite observability to the least fixed point as the continuous deferral of the diagonal.