Proof complexity is the study of non-deterministic computational models, called proof systems, for proving that a given formula of propositional logic is unsatisfiable. As one of the subfields of computational complexity theory, the main questions of study revolve around the amount of resources needed to prove the unsatisfiability of various formulas in different proof systems. This line of inq...

During the last decade, an active line of research in proof complexity has been to study space complexity and timespace trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving. For the polynomial calculus proof system, the only previously known space lower bound is for CNF formulas of unbounded width in [Alekhnovic...

The joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. ...

It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using (s d)O(n) arithmetic operations, where n and s are the numbers of variables and constraints and d is the maximal degree of the polynomials involved. We associate to each of these problems an intrinsic system degree which becomes in wor...

This paper continues the functional approach to the P-versus-NP problem, begun in [1]. Here we focus on the monoid RM 2 of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We construct a machine model for the functions in RM 2 , and evaluation functions. We prove that RM 2 is not finitely generated, and use this to show separation resu...

The first hard problem we will examine is what is known as Satisfiability or SAT. As input, we are given a set of n boolean variables X = {x1, x2, . . . , xn} (i.e., each variable can be set to either true or false). We are then given a boolean formula over these variables of the following form (noting that this is just a specific example where X = {x1, x2, x3, x4}): (x1 ∨ x2 ∨ x3) ∧ (x2 ∨ x3 ∨...

We study a particular case of integer polynomial optimization: Minimize a polynomial F̂ on the set of integer points described by an inequality system F1 ≤ 0, . . . , Fs ≤ 0, where F̂ , F1, . . . , Fs are quasiconvex polynomials in n variables with integer coefficients. We design an algorithm solving this problem that belongs to the time-complexity class O(s) · lO(1) · dO(n) · 2O(n 3), where d ≥ ...

We continue to investigate binary sequence (fu) over {0, 1} defined by (−1)fu = ( (u−u)/p p ) for integers u ≥ 0, where ( · p ) is the Legendre symbol and we restrict ( 0 p ) = 1. In an earlier work, the linear complexity of (fu) was determined for w = p − 1 under the assumption of 2p−1 6≡ 1 (mod p2). In this work, we give possible values on the linear complexity of (fu) for all 1 ≤ w < p− 1 un...

In this paper we study the complexity of factorization of polynomials in the free noncommutative ring F〈x1, x2, . . . , xn〉 of polynomials over the field F and noncommuting variables x1, x2, . . . , xn. Our main results are the following: • Although F〈x1, . . . , xn〉 is not a unique factorization ring, we note that variabledisjoint factorization in F〈x1, . . . , xn〉 has the uniqueness property....