On the Bit-Comprehension Rule

Johannsen, Jan; Pollett, Chris

doi:10.1017/9781316756140.019

Cited by 13 publications

(28 citation statements)

References 10 publications

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“…was introduced by Johannsen and Pollett [1998] (where they call it C 0 2 ), and proved to be ∀ b 1 conservative over b 1 − CR in Johannsen and Pollett [2000]. From these conservativity results it follows that V 0 +BB( B 0 ) does not prove the pigeonhole principle and b 1 − CR + BB( b 0 ) does not prove the circuit value principle (unless P equals uniform TC 0 ), which gives us the separations between the three theories with replacement.…”

Section: Introductionmentioning

confidence: 99%

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The strength of replacement in weak arithmetic

Cook

Thapen

2006

ACM Trans. Comput. Logic

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The replacement (or collection or choice) axiom scheme BB( ) asserts bounded quantifier exchange as follows:for φ in the class of formulas. The theory S 1 2 proves the scheme BB( b 1 ), and thus in S 1 2 every b 1 formula is equivalent to a strict b 1 formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S 1 2 do not prove either BB( b 1 ) or BB( b 0 ). We show (unconditionally) that V 0 does not prove BB( B 0 ), where V 0 (essentially I 1,b 0 ) is the twosorted theory associated with the complexity class AC 0 . We show that PV does not prove BB( b 0 ), assuming that integer factoring is not possible in probabilistic polynomial time. Johannsen and Pollett introduced the theory C 0 2 associated with the complexity class TC 0 , and later introduced an apparently weaker theory b 1 − CR for the same class. We use our methods to show that b 1 − CR is indeed weaker than C 0 2 , assuming that RSA is secure against probabilistic polynomial time attack. Our main tool is the KPT witnessing theorem.

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Section: Introductionmentioning

confidence: 99%

“…The question is raised in Johannsen and Pollett [2000], whether this theory is strictly stronger than b 1 − CR. We show that it is, under a cryptographic assumption.…”

Section: Introductionmentioning

confidence: 99%

The strength of replacement in weak arithmetic

Cook

Thapen

2006

ACM Trans. Comput. Logic

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“…It follows that there is a fixed upper bound of the nesting depth of applications of the ∆ b 1 bit-comprehension rule required for proofs in ∆ b 1 -CR, which answers an open question in [JP00]. Section 5 summarizes our main contributions.…”

Section: Theories For Tc 0 and Other Small Complexity Classesmentioning

confidence: 94%

Theories for TC0 and Other Small Complexity Classes

Nguyen¹,

Cook²

2006

Logical Methods in Computer Science

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Abstract. We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC 0 , whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC 0 in which the provably-total functions are those associated with TC 0 , and then prove that VTC 0 is "isomorphic" to a differentlooking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.

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“…In the case of Ì ¼ , a number of theories have been proposed. Perhaps the most remarkable among them is the first-order theory ¡ ½ -Ê by Johannsen and Pollett [19], who have shown (among other results) that: if ¡ ½ Ê Ë ½ ¾ , then AEÈ is contained in non-uniform Ì ¼ .…”

Section: Introductionmentioning

confidence: 99%

VTC/sup O/: a second-order theory for TC/sup 0/

Nguyen¹,

Cook²

2004

Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.

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We introduce a finitely axiomatizable second-order theory ÎÌ ¼ , which is associated with the class Ç-uniform Ì ¼ . It consists of the base theory Î ¼ for ¼ reasoning together with the axiom AEÍÅÇAE Ë, which states the existence of a "counting array" for any string : the th row of contains only the number of 1 bits up to (excluding) bit of . We introduce the notion of "strong ¡ ½ -definability" for relations in a theory, and use a recursive characterization of the Ì ¼ relations (rather than functions) to show that the Ì ¼ relations are strongly ¡ ½ -definable. It follows that the Ì ¼ functions are ¦ ½ -definable in ÎÌ ¼ .We prove a general witnessing theorem for second-order theories and conclude that the ¦ ½ theorems of ÎÌ ¼ are witnessed by Ì ¼ functions. We prove that ÎÌ ¼ is RSUV isomorphic to the first order theory ¡ ½ -Ê of Johannsen and Pollett (the "minimal theory for Ì ¼ "). ¡ ½ -Ê includes the ¡ ½ comprehension rule, and J and P ask whether there is an upper bound to the nesting depth required for this rule. We answer "yes", because ÎÌ ¼ , and therefore ¡ ½ -Ê, are finitely axiomatizable. Finally, we show that ¦ ¼ theorems of ÎÌ ¼ translate to families of tautologies which have polynomial-size constant-depth Ì ¼ -Frege proofs. We also show that ÈÀÈ is a ¦ ¼ theorem of ÎÌ ¼ . These together imply that the family of propositional tautologies associated with ÈÀÈ has polynomialsize constant-depth Ì ¼ -Frege proofs.

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On the Bit-Comprehension Rule

Cited by 13 publications

References 10 publications

The strength of replacement in weak arithmetic

The strength of replacement in weak arithmetic

Theories for TC0 and Other Small Complexity Classes

VTC/sup O/: a second-order theory for TC/sup 0/

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