Abstract:We study finite Kripke semantics as an explicit search and certification problem for modal formulas. Sets of worlds are encoded as integer bitmasks, so Boolean connectives, $\Box$, and $\Diamond$ reduce to word-level containment and intersection tests. This gives a deterministic evaluator with an independent certificate checker, then scales it through a fused CUDA kernel for exhaustive small-frame scans. Over $K,T,S4,S5$, a corpus of 5,624 formulas is evaluated on all frames through five worlds, performing $1.63\times 10^{14}$ formula evaluations in 45 minutes on one H100. All 20,990 emitted countermodel certificates verify. In this bounded corpus, every $K$-refutable formula has a countermodel on at most two worlds, far below the standard filtration bound $2^{|\mathrm{Sub}(\varphi)|}$. We then turn pairwise formula equivalence into a minimal-countermodel problem for biconditionals and synthesize semantic mirages: formulas that agree on every model up to a finite size and split only later. In particular, $\alpha_2=(\Box\Diamond)^2\top$ and $\alpha_3=(\Box\Diamond)^3\top$ agree on all frames of at most five worlds but are separated by a checked six-world path. Finally, we build a density-aggregated semantic atlas for representation-guided candidate retrieval and compare raw features, PCA, UMAP, spectral layouts, and random layouts under a common million-pair verifier budget. The result is a reproducible bridge between modal finite-model theory, GPU enumeration, certificate checking, and graphics-supported semantic exploration.
From: Baris Basaran [view email]
[v1]
Sat, 13 Jun 2026 19:01:41 UTC (4,793 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。