Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Fault Tolerance in Cellular Automata at High Fault Rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes F and graph classes G, an (F, G)-system is a boolean dynamical system such that all local transition functions lie in F and the underlying graph lies in G. Let F be a class of boolean functions which is closed under composition and let G be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If F contains the self-dual functions and G contains the planar graphs then the fixed-point existence problem for (F, G)-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If F contains the self-dual functions and G contains the graphs having vertex covers of size one then the fixed-point existence problem for (F, G)-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24 versio

Noise Sensitivity of Boolean Functions and Applications to Percolation

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given. Consider, for example, bond percolation on an $n+1$ by $n$ grid. A configuration is a function that assigns to every edge the value 0 or 1. Let $\omega$ be a random configuration, selected according to the uniform measure. A crossing is a path that joins the left and right sides of the rectangle, and consists entirely of edges $e$ with $\omega(e)=1$. By duality, the probability for having a crossing is 1/2. Fix an $\epsilon\in(0,1)$. For each edge $e$, let $\omega'(e)=\omega(e)$ with probability $1-\epsilon$, and $\omega'(e)=1-\omega(e)$ with probability $\epsilon$, independently of the other edges. Let $p(\tau)$ be the probability for having a crossing in $\omega$, conditioned on $\omega'=\tau$. Then for all $n$ sufficiently large, $P\{\tau : |p(\tau)-1/2|>\epsilon\}<\epsilon$.Comment: To appear in Inst. Hautes Etudes Sci. Publ. Mat

Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems

We present dichotomy theorems regarding the computational complexity of counting fixed points in boolean (discrete) dynamical systems, i.e., finite discrete dynamical systems over the domain {0,1}. For a class F of boolean functions and a class G of graphs, an (F,G)-system is a boolean dynamical system with local transitions functions lying in F and graphs in G. We show that, if local transition functions are given by lookup tables, then the following complexity classification holds: Let F be a class of boolean functions closed under superposition and let G be a graph class closed under taking minors. If F contains all min-functions, all max-functions, or all self-dual and monotone functions, and G contains all planar graphs, then it is #P-complete to compute the number of fixed points in an (F,G)-system; otherwise it is computable in polynomial time. We also prove a dichotomy theorem for the case that local transition functions are given by formulas (over logical bases). This theorem has a significantly more complicated structure than the theorem for lookup tables. A corresponding theorem for boolean circuits coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on Theoretical Computer Science (ICTCS'2007

Noise sensitivity in bootstrap percolation

Answering questions of Itai Benjamini, we show that the event of complete occupation in 2-neighbour bootstrap percolation on the d-dimensional box [n]^d, for d\geq 2, at its critical initial density p_c(n), is noise sensitive, while in k-neighbour bootstrap percolation on the d-regular random graph G_{n,d}, for 2\leq k\leq d-2, it is insensitive. Many open problems remain.Comment: 16 page

Hardness Results for Dynamic Problems by Extensions of Fredman and Saks’ Chronogram Method

We introduce new models for dynamic computation based on the cell probe model of Fredman and Yao. We give these models access to nondeterministic queries or the right answer +-1 as an oracle. We prove that for the dynamic partial sum problem, these new powers do not help, the problem retains its lower bound of  Omega(log n/log log n). From these results we easily derive a large number of lower bounds of order Omega(log n/log log n) for conventional dynamic models like the random access machine. We prove lower bounds for dynamic algorithms for reachability in directed graphs, planarity testing, planar point location, incremental parsing, fundamental data structure problems like maintaining the majority of the prefixes of a string of bits and range queries. We characterise the complexity of maintaining the value of any symmetric function on the prefixes of a bit string

A study of temporary captures and collisions in the Circular Restricted Three-Body Problem with normalizations of the Levi-Civita Hamiltonian

The dynamics near the Lagrange equilibria $L_1$ and $L_2$ of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and the design of trajectories for space missions. In this paper we investigate the temporary captures using the tube manifolds of the horizontal Lyapunov orbits originating at $L_1$ and $L_2$ of the CR3BP at energy values which have not been considered so far. After showing that the radius of convergence of any Hamiltonian normalization at $L_1$ or $L_2$ computed with the Cartesian variables is limited in amplitude by $\|1-\mu-x_{L_1}\|$ ($\mu$ denoting the reduced mass of the problem), we investigate if regularizations allow us to overcome this limit. In particular, we consider the Hamiltonian describing the planar three-body problem in the Levi-Civita regularization and we compute its normalization for the Sun-Jupiter reduced mass for an interval of energy which overcomes the limit of Cartesian normalizations. As a result, for the largest values of the energy that we consider, we notice a transition in the structure of the tubes manifolds emanating from the Lyapunov orbit, which can contain orbits that collide with the secondary body before performing one full circulation around it. We discuss the relevance of this transition for temporary captures.Comment: 17 pages, 9 figures. v1: original preprint. v2: author's accepted manuscript. To appear in International Journal of Non-Linear Mechanic