A Quasi-Random Approach to Matrix Spectral Analysis

Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of Hermitian matrices. The result is a completely new, efficient and stable, parallel algorithm to compute an approximate spectral decomposition of any Hermitian matrix. The algorithm can be implemented by Boolean circuits in $O(\log^2 n)$ parallel time with a total cost of $O(n^{\omega+1})$ Boolean operations. This Boolean complexity matches the best known rigorous $O(\log^2 n)$ parallel time algorithms, but unlike those algorithms our algorithm is (logarithmically) stable, so further improvements may lead to practical implementations. All previous efficient and rigorous approaches to solve the Eigen-Problem use randomization to avoid bad condition as we do too. Our algorithm makes further use of randomization in a completely new way, taking random powers of a unitary matrix to randomize the phases of its eigenvalues. Proving that a tiny Gaussian perturbation and a random polynomial power are sufficient to ensure almost pairwise independence of the phases $(\mod (2\pi))$ is the main technical contribution of this work. This randomization enables us, given a Hermitian matrix with well separated eigenvalues, to sample a random eigenvalue and produce an approximate eigenvector in $O(\log^2 n)$ parallel time and $O(n^\omega)$ Boolean complexity. We conjecture that further improvements of our method can provide a stable solution to the full approximate spectral decomposition problem with complexity similar to the complexity (up to a logarithmic factor) of sampling a single eigenvector.Comment: Replacing previous version: parallel algorithm runs in total complexity $n^{\omega+1}$ and not $n^{\omega}$. However, the depth of the implementing circuit is $\log^2(n)$: hence comparable to fastest eigen-decomposition algorithms know

New Acceleration of Nearly Optimal Univariate Polynomial Root-findERS

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By proposing and incorporating some novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.Comment: 89 pages, 5 figures, 2 table

Path Checking for MTL and TPTL over Data Words

Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are quantitative extensions of linear temporal logic, which are prominent and widely used in the verification of real-timed systems. It was recently shown that the path checking problem for MTL, when evaluated over finite timed words, is in the parallel complexity class NC. In this paper, we derive precise complexity results for the path-checking problem for MTL and TPTL when evaluated over infinite data words over the non-negative integers. Such words may be seen as the behaviours of one-counter machines. For this setting, we give a complete analysis of the complexity of the path-checking problem depending on the number of register variables and the encoding of constraint numbers (unary or binary). As the two main results, we prove that the path-checking problem for MTL is P-complete, whereas the path-checking problem for TPTL is PSPACE-complete. The results yield the precise complexity of model checking deterministic one-counter machines against formulae of MTL and TPTL

The Complexity of Bisimulation and Simulation on Finite Systems

In this paper the computational complexity of the (bi)simulation problem over restricted graph classes is studied. For trees given as pointer structures or terms the (bi)simulation problem is complete for logarithmic space or NC$^1$, respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and S\'antha. Furthermore, if only one of the input graphs is required to be a tree, the bisimulation (simulation) problem is contained in AC$^1$ (LogCFL). In contrast, it is also shown that the simulation problem is P-complete already for graphs of bounded path-width

Minimisation of Multiplicity Tree Automata

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor editing changes from previous versio

Quasi-Linear Cellular Automata

Simulating a cellular automaton (CA) for t time-steps into the future requires t^2 serial computation steps or t parallel ones. However, certain CAs based on an Abelian group, such as addition mod 2, are termed ``linear'' because they obey a principle of superposition. This allows them to be predicted efficiently, in serial time O(t) or O(log t) in parallel. In this paper, we generalize this by looking at CAs with a variety of algebraic structures, including quasigroups, non-Abelian groups, Steiner systems, and others. We show that in many cases, an efficient algorithm exists even though these CAs are not linear in the previous sense; we term them ``quasilinear.'' We find examples which can be predicted in serial time proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log t, log t log log t and log^2 t. We also discuss what algebraic properties are required or implied by the existence of scaling relations and principles of superposition, and exhibit several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica

An investigation of messy genetic algorithms

Genetic algorithms (GAs) are search procedures based on the mechanics of natural selection and natural genetics. They combine the use of string codings or artificial chromosomes and populations with the selective and juxtapositional power of reproduction and recombination to motivate a surprisingly powerful search heuristic in many problems. Despite their empirical success, there has been a long standing objection to the use of GAs in arbitrarily difficult problems. A new approach was launched. Results to a 30-bit, order-three-deception problem were obtained using a new type of genetic algorithm called a messy genetic algorithm (mGAs). Messy genetic algorithms combine the use of variable-length strings, a two-phase selection scheme, and messy genetic operators to effect a solution to the fixed-coding problem of standard simple GAs. The results of the study of mGAs in problems with nonuniform subfunction scale and size are presented. The mGA approach is summarized, both its operation and the theory of its use. Experiments on problems of varying scale, varying building-block size, and combined varying scale and size are presented