Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
Some consequences of cryptographical conjectures for S12 and EF
Information and Computation - Special issue: logic and computational complexity
On Proofs about Threshold Circuits and Counting Hierarchies
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
Feasibly constructive proofs and the propositional calculus (Preliminary Version)
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION
DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION
The Strength of Replacement in Weak Arithmetic
LICS '04 Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science
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The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i a| ∃x aφ(i,x) → ∃w ∀i a|φ(i,[w]i), for φ in the class Γ of formulas. The theory S12 proves the scheme BB(Σb1), and thus in S12 every Σb1 formula is equivalent to a strict Σb1 formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S12 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(Σb0), where V0 (essentially IΣ1,b0) is the two-sorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb0), assuming that integer factoring is not possible in probabilistic polynomial time. Johannsen and Pollett introduced the theory C02 associated with the complexity class TC0, and later introduced an apparently weaker theory Δb1 − CR for the same class. We use our methods to show that Δb1 − CR is indeed weaker than C02, assuming that RSA is secure against probabilistic polynomial time attack.Our main tool is the KPT witnessing theorem.