Results 1 
6 of
6
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 in Izvestiya of the Russian Academy of Science, mathematics
, 1995
"... To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators ..."
Abstract

Cited by 54 (6 self)
 Add to MetaCart
To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constantdepth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolationlike theorems for certain “split versions ” of classical systems of Bounded Arithmetic introduced in this paper.
On provably disjoint NPpairs
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 1994
"... In this paper we study the pairs (U; V ) of disjoint NPsets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NPpair which is complete for the class of disjoint NPpairs representable in T . Th ..."
Abstract

Cited by 42 (2 self)
 Add to MetaCart
In this paper we study the pairs (U; V ) of disjoint NPsets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NPpair which is complete for the class of disjoint NPpairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [11]. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NPpair by a (quasi)polytime computable set. We remark that such a separation is obvious for the theory S(S 2 ) + S \Sigma 2 \Gamma PIND considered in [11], and this gives an alternative proof of the main result from that paper.
Tennenbaum’s Theorem for Models of Arithmetic
, 2006
"... This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
This paper discusses Tennenbaum’s Theorem in its original context of models of arithmetic, which states that there are no recursive nonstandard models of Peano Arithmetic. We focus on three separate areas: the historical background to the theorem; an understanding of the theorem and its relationship with the Gödel–Rosser Theorem; and extensions of Tennenbaum’s theorem to diophantine problems in models of arthmetic, especially problems concerning which diophantine equations have roots in some model of a given theory of arithmetic.
Translating I∆0(exp) proofs into weaker systems
 Mathematical Logic Quarterly
, 2000
"... The purpose of this paper is to explore the relationship between I∆0+exp and its weaker subtheories. We give a method of translating certain classes of I∆0+exp proofs into weaker systems of arithmetic such as Buss ’ systems S2. We show if IEi(exp) ⊢ A with a proof P of expindrank(P) ≤ n + 1 where ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The purpose of this paper is to explore the relationship between I∆0+exp and its weaker subtheories. We give a method of translating certain classes of I∆0+exp proofs into weaker systems of arithmetic such as Buss ’ systems S2. We show if IEi(exp) ⊢ A with a proof P of expindrank(P) ≤ n + 1 where all ( ∀ ≤: right) or ( ∃ ≤: left) have bounding terms not containing function symbols then S i 2 ⊇ IEi,2 ⊢ A n. Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) � I∆0(exp). Here I∆0(exp) is a conservative extension of I∆0+exp obtained by adding to I∆0 a symbol for 2 x to the language as well as defining axioms for it.
The SkolemBang Theorems in Ordered Fields with an IP
, 705
"... This paper is concerned with the extent to which the SkolemBang theorems in Diophantine approximations generalise from the standard setting of 〈R, Z 〉 to structures of the form 〈F, I〉, where F is an ordered field and I is an integer part of F. We show that some of these theorems are hold unconditio ..."
Abstract
 Add to MetaCart
This paper is concerned with the extent to which the SkolemBang theorems in Diophantine approximations generalise from the standard setting of 〈R, Z 〉 to structures of the form 〈F, I〉, where F is an ordered field and I is an integer part of F. We show that some of these theorems are hold unconditionally in general case (ordered fields with an integer part). The remainder results are based on Dirichlet’s and Kronecker’s theorems. Finally we extend Dirichlet’s theorem to ordered fields with IE1 integer part.