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On Bounded Set Theory
"... We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation ..."
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Cited by 9 (1 self)
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We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation between BST, possibly with AntiFoundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory.
On Feasible Numbers
 Logic and Computational Complexity, LNCS Vol. 960
, 1995
"... . A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The "vague" set F of feasible numbers intuitively satisfies the axioms 0 2 F , F + 1 ` F and 2 1000 62 F , where the latter is ..."
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. A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The "vague" set F of feasible numbers intuitively satisfies the axioms 0 2 F , F + 1 ` F and 2 1000 62 F , where the latter is stronger than a condition considered by Parikh, and seems to be treated rigorously here for the first time. Our technical considerations, though quite simple, have some unusual consequences. A discussion of methodological questions and of relevance to the foundations of mathematics and of computer science is an essential part of the paper. 1 Introduction How to formalize the intuitive notion of feasible numbers? To see what feasible numbers are, let us start by counting: 0,1,2,3, and so on. At this point, A.S. YeseninVolpin (in his "Analysis of potential feasibility", 1959) asks: "What does this `and so on' mean?" "Up to what extent `and so on'?" And he answers: "Up to exhaustion!" Note that by cos...