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Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polyno ..."
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Cited by 58 (1 self)
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of secondorder bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...
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 ..."
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Cited by 56 (6 self)
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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.
Bounded Arithmetic and Lower Bounds in Boolean Complexity
 Feasible Mathematics II
, 1993
"... We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and i ..."
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Cited by 49 (5 self)
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We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 (which, in view of RSUV isomorphism, is equivalent to S 2 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 . As a byproduct of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.
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 ..."
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Cited by 44 (2 self)
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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.
Pseudorandom Generators in Propositional Proof Complexity
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REP. NO.23
, 2000
"... We call a pseudorandom generator Gn : {0, 1}^n → {0, 1}^m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G(x1, ..., xn) ≠ b for any string b ∈ {0, 1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by ..."
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Cited by 40 (7 self)
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We call a pseudorandom generator Gn : {0, 1}^n → {0, 1}^m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G(x1, ..., xn) ≠ b for any string b ∈ {0, 1}^m. We consider a variety of "combinatorial" pseudorandom generators inspired by the NisanWigderson generator on the one hand, and by the construction of Tseitin tautologies on the other. We prove that under certain circumstances these generators are hard for such proof systems as Resolution, Polynomial Calculus and Polynomial Calculus with Resolution (PCR).
Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
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Cited by 30 (7 self)
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We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
On Frege and Extended Frege Proof Systems
, 1993
"... We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a parti ..."
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Cited by 21 (2 self)
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We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the paper. It reduces the question of proving a lower bound to the question of constructing a partial boolean algebra and a map of formulas into that algebra with particular properties. We show that in principle one can obtain via this method optimal lower bounds (up to a polynomial increase). Introduction A propositional proof system is any polynomial time function P whose range is exactly the set of tautologies TAUT, cf. [17]. For ø a tautology any string ß such that P (ß) = ø is called a P proof of ø . Any usual propositional calculus, be it resolution or extended resolution, a Hilbert style system based on finitely many axiom schemes and inf...
The strength of replacement in weak arithmetic
 Proceedings of the Nineteenth Annual IEEE Symposium on Logic in Computer Science
, 2004
"... The replacement (or collection or choice) axiom scheme BB(Γ) asserts bounded quantifier exchange as follows: ∀i<a  ∃x
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Cited by 10 (3 self)
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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<aφ(i,[w]i) proves the scheme where φ is in the class Γ of formulas. The theory S1 2 BB(Σb 1), and thus in S1 2 every Σb1 formula is equivalent to a strict Σb1 formula (in which all nonsharplybounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S1 2 do not prove either BB(Σb1) or BB(Σb0). We show (unconditionally) that V 0 does not prove BB(ΣB 0), where V 0 (essentially IΣ 1,b 0) is the twosorted theory associated with the complexity class AC0. We show that PV does not prove BB(Σb 0), assuming
On Proofs About Threshold Circuits and Counting Hierarcies (Extended Abstract)
, 1998
"... ) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth t ..."
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Cited by 9 (2 self)
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) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth threshold circuits of polynomial and quasipolynomial size. Then we dene certain secondorder theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the socalled RSUV isomorphism. 1 Introduction A phenomenon that is commonly observed in Complexity Theory is that proofs of results about counting complexity classes (#P , Mod p P etc.) can often be scaled down to yield results about small depth circuit classes with the corresponding counting gates. For example, Toda's result [17] that every problem in the Polynomial Hierarchy can be solved in polynomial time with an oracle for #P correspond...
Quantified Propositional Calculus and a SecondOrder Theory for NC¹
, 2004
"... Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S ..."
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Cited by 9 (2 self)
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Let H be a proof system for the quantified propositional calculus (QPC). We j witnessing problem for H to be: given a prenex S j formula A, an Hproof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce