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The SAT problem is based upon Boolean expressions, but is there a generalized SAT problem based upon higher order logics?

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Yes, you may be interested in the paper "Higher-Order Quantified Boolean Satisfiability" by Chistikov, Haase, Hadizadeh, and Mansutti (https://doi.org/10.4230/LIPIcs.MFCS.2022.33)

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