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On the Limits of Decision: the Adjacent Fragment of First-Order Logic Bednarczyk, Bartosz Kojelis, Daumantas Pratt-Hartmann, Ian decidability satisfiability variable-ordered logics complexity We define the adjacent fragment AF of first-order logic, obtained by restricting the sequences of variables occurring as arguments in atomic formulas. The adjacent fragment generalizes (after a routine renaming) two-variable logic as well as the fluted fragment. We show that the adjacent fragment has the finite model property, and that its satisfiability problem is no harder than for the fluted fragment (and hence is Tower-complete). We further show that any relaxation of the adjacency condition on the allowed order of variables in argument sequences yields a logic whose satisfiability and finite satisfiability problems are undecidable. Finally, we study the effect of the adjacency requirement on the well-known guarded fragment (GF) of first-order logic. We show that the satisfiability problem for the guarded adjacent fragment (GA) remains 2ExpTime-hard, thus strengthening the known lower bound for GF. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Bartosz Bednarczyk and Daumantas Kojelis and Ian Pratt-Hartmann 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.111 urn:nbn:de:0030-drops-181632 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.111 eng https://creativecommons.org/licenses/by/4.0/legalcode