Near-optimal small-depth lower bounds for small distance connectivity

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were $n^{k^{\exp(-O(d))}}$ (due to Beame, Impagliazzo, and Pitassi [BIP98]) and $n^{\Omega((\log k)/d)}$ (following from a recent formula size lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal, since a simple construction gives depth-$d$ circuits of size $n^{O(k^{2/d})}$ for this problem (and strengthening our bound even to $n^{k^{\Omega(1/d)}}$ would require proving that undirected connectivity is not in $\mathsf{NC^1}.$) Our proof is by reduction to a new lower bound on the size of small-depth circuits computing a skewed variant of the "Sipser functions" that have played an important role in classical circuit lower bounds [Sip83, Yao85, H{\aa}s86]. A key ingredient in our proof of the required lower bound for these Sipser-like functions is the use of \emph{random projections}, an extension of random restrictions which were recently employed in [RST15]. Random projections allow us to obtain sharper quantitative bounds while employing simpler arguments, both conceptually and technically, than in the previous works [Ajt89, BPU92, BIP98, Ros14]

On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝ We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Previous work of the author [Rossmann\u2708] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence of quantifier-rank poly(k). Quantitatively, this improves the result of [Rossmann\u2708], where the upper bound on quantifier-rank is a non-elementary function of k

Formulas vs. Circuits for Small Distance Connectivity

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by the recursive doubling technique) on {\bf circuits} of depth $O(\log k)$ and size $O(kn^3)$. In contrast, we show that solving this problem on {\bf formulas} of depth $\log n/(\log\log n)^{O(1)}$ requires size $n^{\Omega(\log k)}$ for all $k(n) \leq \log\log n$. As corollaries: (i) It follows that polynomial-size circuits for Distance $k(n)$ Connectivity require depth $\Omega(\log k)$ for all $k(n) \leq \log\log n$. This matches the upper bound from recursive doubling and improves a previous $\Omega(\log\log k)$ lower bound of Beame, Pitassi and Impagliazzo [BIP98]. (ii) We get a tight lower bound of $s^{\Omega(d)}$ on the size required to simulate size-$s$ depth-$d$ circuits by depth-$d$ formulas for all $s(n) = n^{O(1)}$ and $d(n) \leq \log\log\log n$. No lower bound better than $s^{\Omega(1)}$ was previously known for any $d(n) \nleq O(1)$. Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size $n$ via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance $k(n)$ Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity

Bounded-depth Frege complexity of Tseitin formulas for all graphs

We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)javax.xml.bind.JAXBElement@531a834b in depth-d Frege systems for [Formula presented], where tw(G) is the treewidth of G. This extends Håstad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)javax.xml.bind.JAXBElement@25a4b51fpoly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution

Propositional logic, complexity theory and a nontrivial hierarchy of theories of weak fragments of peano arithmetic

In dieser Arbeit betrachte ich einige Arbeiten von Paris, Wilky [key-1] und Ajati [key-3, key-4] welche einen Zusammenhang zwischen Komplexitäts- und Beweistheorie herstellen. J. Paris und A. Wilkie [key-1] betrachteten die Fragen ob jede \Delta_{0}- Teilmenge A von \mathbb{N} auch eine \Delta_{0} definierbare Zählfunktion \{|m=|A\cap n|\} besitzt. Eine damit eng verwanndte Fragestellung ist, ob das Schubfachprinzip PHP in einer schwachen Teiltheorie I\Delta_{0} der Peano Arithmetik bewiesen werden kann. I\Delta_{0} umfasst die selben Axiome wie die Peano Arithmetik. Das Axiomenschema der Induktion ist jedoch nur für beschränkte Formeln gegeben. Paris und Wilky konnten mithilfe der Forcing-Technik die Konsistenz von I\exists_{1}(F)+\exists xF:x\mapsto x-1 zeigen. Weiters konnten sie unter Verwendung der Cook-Reckhov Vermutung, die Konsistenz von I\Delta_{0}(F)+\exists xF:x\mapsto x-1 zeigen. Die Cook-Reckhov Vermutung besagt, dass ein Beweis der aussagenlogischen Form PHP_{n} von PHP ”schwer” ist. Dieser zweite Beweis benutzt einen Zusammenhang zwischen I\Delta_{0} und Frege Systemen. Ajtai [key-3] verband die Verwendung der Forcing-Technik und dieses Zusammenhangs um die Konsistenz von I\Delta_{0}(F)+\exists xF:x\mapsto x-1 , ohne der Verwendung der Cook-Reckhov Vermutung, zu zeigen. Dazu nahm er an, dass in einem nicht-standard Modell \mathcal{M} von I\Delta_{0} ein ”einfacher” Beweis von PHP_{n} für ein n existiert. Dieses \mathcal{M} beschränkte er auf die Substruktur M_{n}=\mathcal{M}\upharpoonright n , welche er dann durch Forcing zu einem M[G] erweiterte in welchem PHP auf natürliche Weise nicht wahr sein kann. Eine kombinatorische Überlegung zeigt, dass in M[G] aber das Axiomenschema der vollständigen Induktion bis n wahr ist. Damit kann man nun den ”einfachen” Beweis von PHP_{n} Schritt für Schritt prüfen, was zu einem Widerspruch führt. Später [key-4] verallgemeinerte Ajtai diese Art der Beweisführung, um zu zeigen, dass PHP\Delta_{0}\not\vdash PAR , wobei PHP\Delta_{0}=I\Delta_{0}\cup PHP und PAR folgende Aussage ist: Keine Menge mit einer ungeraden Anzahl von Elementen kann in Teilmengen mit genau zwei Elementen partitioniert werden. PAR kann weiter zum ”module p Counting Principle” CP_{p} verallgemeinert werden [key-4]. Schlussendlich zeigte Ajtai für alle Primzahlen p\neq q , dass die CP_{p} paarweise unabhängig sind. Als Konsequenz dieser Erkenntnisse bekommen wir eine Hierarchie von schwachen Theorien der Peano Arithmetik: I\exists_{1}\subset I\Delta_{0}\subset PHP\Delta_{0}\subset CP_{p}\Delta_{0}\mbox{ für alle Primzahlen }