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Proof Complexity In Algebraic Systems And Bounded Depth Frege Systems With Modular Counting
"... We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, us ..."
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Cited by 30 (9 self)
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We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constantdepth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss
A Complexity Gap for TreeResolution
 COMPUTATIONAL COMPLEXITY
, 1999
"... It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size treeresolution proofs or is "di#cult" i.e requires exponential size treeresolution proofs. It is shown that the class of tautologies which a ..."
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Cited by 16 (2 self)
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It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size treeresolution proofs or is "di#cult" i.e requires exponential size treeresolution proofs. It is shown that the class of tautologies which are hard (for treeresolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Actually it is
Counting axioms do not polynomially simulate counting gates
 In Proceedings 42nd Annual Symposium on Foundations of Computer Science
, 2001
"... For every prime m ≥ 2, we give a family of tautologies that require superpolynomial size constantdepth Frege proofs from Countm axioms, and whose algebraic translations have constantdegree, polynomial size polynomial calculus refutations over Zm. This shows that constantdepth Frege systems with ..."
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Cited by 8 (2 self)
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For every prime m ≥ 2, we give a family of tautologies that require superpolynomial size constantdepth Frege proofs from Countm axioms, and whose algebraic translations have constantdegree, polynomial size polynomial calculus refutations over Zm. This shows that constantdepth Frege systems with counting axioms modulo m do not polynomially simulate constantdepth Frege systems with counting gates modulo m. Our primary technical tool is a switching lemma that uses random substitutions of one variable for another in addition to random 0/1 restrictions. 1
Constantdepth frege systems with counting axioms polynomially simulate nullstellensatz refutations. August 05 2003
 Comment
, 1998
"... Abstract. We show that constantdepth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previous ..."
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Cited by 4 (0 self)
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Abstract. We show that constantdepth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from propositional formulas to systems of polynomials. Using our definition of reducibility, most previously studied propositional formulas reduce to their polynomial translations. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new upper bounds on refutation sizes for certain CNFs in constantdepth Frege with counting axioms systems. 1
Count(q) versus the PigeonHole Principle
, 1996
"... For each p 2 there exists a model M of I \Delta 0 (ff) which satisfies the Count(p) principle. Furthermore, if p contains all prime factors of q there exist n; r 2 M and a bijective map f 2 dom(M ) mapping f1; 2; :::; ng onto f1; 2; :::; n+ q r g. A corollary is a complete classificati ..."
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Cited by 4 (1 self)
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For each p 2 there exists a model M of I \Delta 0 (ff) which satisfies the Count(p) principle. Furthermore, if p contains all prime factors of q there exist n; r 2 M and a bijective map f 2 dom(M ) mapping f1; 2; :::; ng onto f1; 2; :::; n+ q r g. A corollary is a complete classification of the Count(q) versus Count(p) problem. Another corollary shows that the pigeonhole principle for injective maps does not follow from any of the Count(q) principles. This solves an open question [Ajtai 94]. 1 Introduction The most fundamental questions in the theory of the complexity of calculations are concerned with complexity classes in which `counting' is only possible in a quite restricted sense. Thus it is not surprising that many elementary counting principles are unprovable in systems of Bounded Arithmetic. These are axiom systems where the induction axiom schema is restricted to predicates of low syntactic complexity. For a good basic reference see [Krajicek 95]. The status of...
Uniformly Generated Submodules Permutation Modules
, 1998
"... This paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a series of nontraditional problems in the representation theory of S n . Most of our ..."
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Cited by 3 (3 self)
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This paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a series of nontraditional problems in the representation theory of S n . Most of our
A Fractal which violates the Axiom of Determinacy
, 1994
"... By use of the axiom of choice I construct a symmetrical and selfsimilar subset A ` [0; 1] ` R. Then by an elementary strategy stealing argument it is shown that A is not determined. The (possible) existence of fractals like A clarifies the status of the controversial Axiom of Determinacy. ..."
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Cited by 2 (0 self)
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By use of the axiom of choice I construct a symmetrical and selfsimilar subset A ` [0; 1] ` R. Then by an elementary strategy stealing argument it is shown that A is not determined. The (possible) existence of fractals like A clarifies the status of the controversial Axiom of Determinacy.
Bootstrapping the Primitive Recursive Functions by 47 Colors
, 1994
"... I construct a concrete colouring of the 3 element subsets of N. ..."
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Cited by 1 (0 self)
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I construct a concrete colouring of the 3 element subsets of N.
On Resolution Complexity of
, 2002
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Dissertation Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Dissertation Series publications. Copies may be obtained by contacting: BRICS