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Intuitionistic Compression and Optimal Verification Trees

Primarily human-written textHuman understanding: all partsmath.LO — Logic

Contributed by Zhenyu Sun ↗, Jilin Wang ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

In strict finitistic generation semantics, a formula is assertible at a generation if some stage in its verification tree forces it. We ask when assertibility agrees with intuitionistic truth on the poset of generations. The leaves of the verification trees, ordered by generation extension, form an intuitionistic Kripke model that preserves truth at every leaf, even without atomic prevalence. For a finite bundle, compression to the generations is exact if and only if each fiber of leaves has a greatest intuitionistic theory and the map selecting these theories has the back property. Compression remains exact under every restriction of the generation poset if and only if atomic prevalence holds. Conversely, consider coherent tree representations of a finite Kripke model whose atomic valuation reflects its order. Among representations with atomic prevalence, the minimum numbers of leaves and nodes over uu are h(u)h(u) and δC+∑x⪯uh(x)\delta_C+\sum_{x\preceq u}h(x), where h(x)h(x) counts saturated paths from minimal worlds to xx, and δC\delta_C is 11 if the comparability component of uu has no least element and 00 otherwise. Saturated-path trees attain both minima simultaneously. Among representations attaining the node minimum at every world, the common root labels in components without a least element determine the representation up to isomorphism over the outer frame. Allowing rooted posets in place of trees can reduce the number of nodes exponentially.

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Version history

  1. v1 · 21 pagesSubmitted by Zhenyu SunInitial depositCurrentNov 03, 2026