We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-oo upper bound: there are proofs of size n O(d(log(n)) 2=d) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnja s cij's ...

Solution. In a finite connected graph with n ≥ 2 vertices, the domain for the vertex degrees is the set {1, 2, . . . , n − 1} since each vertex can be adjacent to at most all of the remaining n−1 vertices and the existence of a degree 0 vertex would violate the assumption that the graph be connected. Therefore, treating the n vertices as the pigeons and the n−1 possible degrees as the pigeonhol...

Abstract. We consider the proof complexity of the minimal complete fragment of standard deep inference, denoted KS. To examine the size of proofs we employ atomic flows, diagrams that trace structural changes through a proof but ignore logical information. As results we obtain a polynomial simulation of dag-like cut-free Gentzen and Resolution, along with some extensions. We also show that thes...

Pór and Wood conjectured that for all k, l ≥ 2 there exists n ≥ 2 with the following property: whenever n points, no l + 1 of which are collinear, are chosen in the plane and each of them is assigned one of k colours, then there must be a line (that is, a maximal set of collinear points) all of whose points have the same colour. The conjecture is easily seen to be true for l = 2 (by the pigeonh...

We introduce uniform versions of monotone and deep inference proof systems in the setting of bounded arithmetic, relating the size of propositional proofs to forms of proof-theoretic strength in weak fragments of arithmetic. This continues the recent program of studying the complexity of propositional deep inference. In particular this work is inspired by previous work where proofs of the propo...

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S 2 + iWPHP(Σ b 1), and use it to derive Fermat’s little theorem and Euler’s criterion for the Legendre symbol in S 2 + iWPHP(PV ) extended by the pigeonhole principle PHP(PV ). We ...

The Kolakoski sequence (Kn) is perhaps one of the most famous examples of selfdescribing sequences for which some problems are still open. In particular, one does not know yet whether the density of 1’s in this sequence is equal to 12 . This work, which does not answer this question, provides explicit bounds for the main sequences related to (Kn). The proofs rest on a new identity involving the...

Abstract. We consider the proof complexity of the minimal complete fragment, KS, of standard deep inference systems for propositional logic. To examine the size of proofs we employ atomic flows, diagrams that trace structural changes through a proof but ignore logical information. As results we obtain a polynomial simulation of versions of Resolution, along with some extensions. We also show th...

We begin with an appetizer before starting the lecture proper — an example that demonstrates that randomized one-way communication protocols can sometimes exhibit surprising power. It won’t surprise you that the Equality function — with f(x,y) = 1 if and only if x = y — is a central problem in communication complexity. It’s easy to prove, by the Pigeonhole Principle, that its deterministic one-...