The weak pigeonhole principle for function classes in S^1_2

It is well known that S^1_2 cannot prove the injective weak pigeonhole principle for polynomial time functions unless RSA is insecure. In this note we investigate the provability of the surjective (dual) weak pigeonhole principle in S^1_2 for provably weaker function classes.Comment: 11 page

Resolution over Linear Equations and Multilinear Proofs

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using the (monotone) interpolation by a communication game technique we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [RazTzameret06]. Specifically, we show the following: 1. Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations. 2. Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the \emph{functional} pigeonhole principle. The latter are different than previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.Comment: 44 page

Colourful TFNP and Propositional Proofs

Recent work has shown that many of the standard TFNP classes - such as PLS, PPADS, PPAD, SOPL, and EOPL - have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C-PLS, C-PPADS, C-PPAD, C-SOPL, and C-EOPL. While C-PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that: - Each of the classes C-PLS, C-PPADS, and C-SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C-PLS is characterized by depth-1.5 Frege (Res(polylog(n)). - The classes C-PPAD and C-EOPL coincide exactly with the uncoloured classes PPADS and SOPL, respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively. - Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C-PLS ? C-PPADS = C-SOPL, generalizing the intersection theorem PLS ? PPADS = SOPL. However, while it is known in the uncoloured world that PLS ? PPAD = EOPL = CLS, we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C-PLS^O ? C-PPAD^O ? C-EOPL^O. To prove our results, we introduce an abstract multivalued proof system - the Blockwise Calculus - which may be of independent interest

Nonlocal Position Changes of a Photon Revealed by Quantum Routers

Since its publication, Aharonov and Vaidman's three-box paradox has undergone three major advances: i). A non-counterfactual scheme by the same authors in 2003 with strong rather than weak measurements for verifying the particle's subtle presence in two boxes. ii) A realization of the latter by Okamoto and Takeuchi in 2016. iii) A dynamic version by Aharonov et al. in 2017, with disappearance and reappearance of the particle. We now combine these advances together. Using photonic quantum routers the particle acts like a quantum "shutter." It is initially split between Boxes A, B and C, the latter located far away from the former two. The shutter particle's whereabouts can then be followed by a probe photon, split in both space and time and reflected by the shutter in its varying locations. Measuring the former is expected to reveal the following time-evolution: The shutter particle was, with certainty, in boxes A+C at t1, then only in C at t2, and finally in B+C at t3. Another branch of the split probe photon can show that boxes A+B were empty at t2. A Bell-like theorem applied to this experiment challenges any alternative interpretation that avoids disappearance-reappearance in favor of local hidden variables.Comment: Revised versio

Hardness measures and resolution lower bounds

Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this report we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. We also extend these measures to all clause-sets (possibly satisfiable).Comment: 43 pages, preliminary version (yet the application part is only sketched, with proofs missing

Iterated lower bound formulas: a diagonalization-based approach to proof complexity

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally. We use the approach to give an explicit sequence of CNF formulas {φn} such that VNP ≠ VP iff there are no polynomial-size IPS proofs for the formulas φn. This provides a natural equivalence between proof complexity lower bounds and standard algebraic complexity lower bounds. Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas φn themselves assert the non-existence of short IPS proofs for formulas encoding VNP ≠ VP at a different input length. Our result also has meta-mathematical implications: it gives evidence for the difficulty of proving strong lower bounds for IPS within IPS. For any strong enough propositional proof system R, we define the *iterated R-lower bound formulas*, which inductively assert the non-existence of short R proofs for formulas encoding the same statement at a different input length, and propose them as explicit hard candidates for the proof system R. We observe that this hypothesis holds for Resolution following recent results of Atserias and Muller and of Garlik, and give evidence in favour of it for other proof systems