Pavel Hrubes

Arithmetic Complexity in Ring Extensions

Given a polynomial f with coefficients from a field F, is it easier to compute f over an extension ring R than over F? We address this question, and show the following. For every polynomial f , there is a noncommutative extension ring R such that F is in the center of R and f has a polynomial-size formula over R. On the other hand, if F is algebraically closed, no commutative extension ring R can reduce formula or circuit complexity of f . To complete the picture, we prove that over any field, there exist hard polynomials with zero-one coefficients. (This is a basic theorem, but we could not find it written explicitly.) Finally, we show that low-dimensional extensions are not very helpful in computing polynomials. As a corollary, we obtain that the elementary symmetric polynomials have formulas of size n O(log log n) over any field, and that division gates can be efficiently eliminated from circuits,

On lengths of proofs in non-classical logics

Abstract We give proofs of the effective monotone interpolation property for the system of modal logic K , and others, and the system I L of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubes, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubes, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations.

A lower bound for intuitionistic logic

Abstract We give an exponential lower bound on the number of proof-lines in intuitionistic propositional logic, I L , axiomatised in the usual Frege-style fashion; i.e., we give an example of I L -tautologies A 1 , A 2 , … s.t. every I L -proof of A i must have a number of proof-lines exponential in terms of the size of A i . We show that the results do not apply to the system of classical logic and we obtain an exponential speed-up between classical and intuitionistic logic.

Short Proofs for the Determinant Identities

We study arithmetic proof systems

Random Formulas, Monotone Circuits, and Interpolation

{\mathbb P}_c({\mathbb F})

Short proofs for the determinant identities

and

Circuits with medium fan-in

{\mathbb P}_f({\mathbb F})

On the Real τ-Conjecture and the Distribution of Complex Roots.

operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field

On Isoperimetric Profiles and Computational Complexity.

{\mathbb F}

On the nonnegative rank of distance matrices

. We establish a series of structural theorems about these proof systems, the main one stating that