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 constant-depth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discuss

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 tree-resolution proofs or is "di#cult" i.e requires exponential size tree-resolution proofs. It is shown that the class of tautologies which are hard (for tree-resolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Actually it is

For every prime m ≥ 2, we give a family of tautologies that require super-polynomial size constant-depth Frege proofs from Countm axioms, and whose algebraic translations have constant-degree, polynomial size polynomial calculus refutations over Zm. This shows that constant-depth Frege systems with counting axioms modulo m do not polynomially simulate constant-depth 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

Abstract. We show that constant-depth 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 constant-depth Frege with counting axioms systems. 1

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 pigeon-hole 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...

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 non-traditional problems in the representation theory of S n . Most of our

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.

I construct a concrete colouring of the 3 element subsets of N.