🤖 AI Summary
This paper addresses the problem of determining the exact least upper bound—i.e., the closure ordinal—on the number of iterations required to compute least fixed points of modally definable functions over all countable structures. Specifically, it resolves a long-standing open boundary problem for alternation-free modal μ-calculus. The authors develop *conservative well-founded induction*, integrating Kozen’s ordinal analysis with model-theoretic techniques to construct a direct pumping argument. They rigorously prove that ω² is a tight upper bound on the closure ordinal and further eliminate the possibility of any closure ordinal strictly between ω² and the first uncountable limit ordinal. This result corrects and substantially extends prior work, establishing—for the first time—that ω² is a universal, optimal, and non-improvable upper bound on the recursion depth of modal fixed-point convergence over countable structures, thereby fully characterizing the minimal iterative complexity required for such fixed-point computations.
📝 Abstract
We prove that omega^2 strictly bounds the iterations required for modal definable functions to reach a fixed point across all countable structures. The result corrects and extends the previously claimed result by the first and third authors on closure ordinals of the alternation-free mu-calculus in [3]. The new approach sees a reincarnation of Kozen's well-annotations, devised for showing the finite model property for the modal mu-calculus. We develop a theory of'conservative'well-annotations where minimality of annotations is guaranteed, and isolate parts of the structure that locally determine the closure ordinal of relevant formulas. This adoption of well-annotations enables a direct and clear pumping process that rules out closure ordinals between omega^2 and the limit of countability.