We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provably-recursive operations over sets are characterized in terms of explicit definability and PTIME- or LOGSPACE-computability. We also present some conservativity results and describe a relation between BST, possibly with Anti-Foundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory. 1.
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