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This paper considers a number of arithmetic theories and shows how the strength of these theories relates to certain open problems in complexity theory concerning the polynomial-time hierarchy. These results are proved quite generally and hold for a wide class of subrecursive hierarchies. They can be used to characterize certain properties of functions provable in these theories.doi:10.1016/s0019-9958(86)80041-9 fatcat:mp344yvwefeitja4jhkenj3n7q