Quantification of reachable attractors in asynchronous discrete dynamics

Motivation: Models of discrete concurrent systems often lead to huge and complex state transition graphs that represent their dynamics. This makes difficult to analyse dynamical properties. In particular, for logical models of biological regulatory networks, it is of real interest to study attractors and their reachability from specific initial conditions, i.e. to assess the potential asymptotical behaviours of the system. Beyond the identification of the reachable attractors, we propose to quantify this reachability. Results: Relying on the structure of the state transition graph, we estimate the probability of each attractor reachable from a given initial condition or from a portion of the state space. First, we present a quasi-exact solution with an original algorithm called Firefront, based on the exhaustive exploration of the reachable state space. Then, we introduce an adapted version of Monte Carlo simulation algorithm, termed Avatar, better suited to larger models. Firefront and Avatar methods are validated and compared to other related approaches, using as test cases logical models of synthetic and biological networks. Availability: Both algorithms are implemented as Perl scripts that can be freely downloaded from http://compbio.igc.gulbenkian.pt/nmd/node/59 along with Supplementary Material.Comment: 19 pages, 2 figures, 2 algorithms and 2 table

Dynamics of the entanglement spectrum in spin chains

We study the dynamics of the entanglement spectrum, that is the time evolution of the eigenvalues of the reduced density matrices after a bipartition of a one-dimensional spin chain. Starting from the ground state of an initial Hamiltonian, the state of the system is evolved in time with a new Hamiltonian. We consider both instantaneous and quasi adiabatic quenches of the system Hamiltonian across a quantum phase transition. We analyse the Ising model that can be exactly solved and the XXZ for which we employ the time-dependent density matrix renormalisation group algorithm. Our results show once more a connection between the Schmidt gap, i.e. the difference of the two largest eigenvalues of the reduced density matrix and order parameters, in this case the spontaneous magnetisation.Comment: 16 pages, 8 figures, comments are welcome! Version published in JSTAT special issue on "Quantum Entanglement In Condensed Matter Physics

Fast-kick-off monotonically convergent algorithm for searching optimal control fields

This is the published version, also available here: http://dx.doi.org/10.1103/PhysRevA.84.031401.This Rapid Communication presents a fast-kick-off search algorithm for quickly finding optimal control fields in the state-to-state transition probability control problems, especially those with poorly chosen initial control fields. The algorithm is based on a recently formulated monotonically convergent scheme [T.-S. Ho and H. Rabitz, Phys. Rev. E 82, 026703 (2010)]. Specifically, the local temporal refinement of the control field at each iteration is weighted by a fractional inverse power of the instantaneous overlap of the backward-propagating wave function, associated with the target state and the control field from the previous iteration, and the forward-propagating wave function, associated with the initial state and the concurrently refining control field. Extensive numerical simulations for controls of vibrational transitions and ultrafast electron tunneling show that the new algorithm not only greatly improves the search efficiency but also is able to attain good monotonic convergence quality when further frequency constraints are required. The algorithm is particularly effective when the corresponding control dynamics involves a large number of energy levels or ultrashort control pulses

Quasi-continuous Interpolation Scheme for Pathways between Distant Configurations

A quasi-continuous interpolation (QCI) scheme is introduced for characterizing physically realistic initial pathways from which to initiate transition state searches and construct kinetic transition networks. Applications are presented for peptides, proteins, and a morphological transformation in an atomic cluster. The first step in each case involves end point alignment, and we describe the use of a shortest augmenting path algorithm for optimizing permutational isomers. The QCI procedure then employs an interpolating potential, which preserves the covalent bonding framework for the biomolecules and includes repulsive terms between unconstrained atoms. This potential is used to identify an interpolating path by minimizing contributions from a connected set of images, including terms corresponding to minima in the interatomic distances between them. This procedure detects unphysical geometries in the line segments between images. The most difficult cases, where linear interpolation would involve chain crossings, are treated by growing the structure an atom at a time using the interpolating potential. To test the QCI procedure, we carry through a series of benchmark calculations where the initial interpolation is coupled to explicit transition state searches to produce complete pathways between specified local minima.This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/H042660/1]This document is the unedited Author’s version of a Submitted Work that was subsequently accepted for publication in the Journal of Chemical Theory and Computation, copyright © American Chemical Society after peer review. To access the final edited and published work see http://dx.doi.org/10.1021/ct300483

Probably Approximately Correct MDP Learning and Control With Temporal Logic Constraints

We consider synthesis of control policies that maximize the probability of satisfying given temporal logic specifications in unknown, stochastic environments. We model the interaction between the system and its environment as a Markov decision process (MDP) with initially unknown transition probabilities. The solution we develop builds on the so-called model-based probably approximately correct Markov decision process (PAC-MDP) methodology. The algorithm attains an $\varepsilon$-approximately optimal policy with probability $1-\delta$ using samples (i.e. observations), time and space that grow polynomially with the size of the MDP, the size of the automaton expressing the temporal logic specification, $\frac{1}{\varepsilon}$, $\frac{1}{\delta}$ and a finite time horizon. In this approach, the system maintains a model of the initially unknown MDP, and constructs a product MDP based on its learned model and the specification automaton that expresses the temporal logic constraints. During execution, the policy is iteratively updated using observation of the transitions taken by the system. The iteration terminates in finitely many steps. With high probability, the resulting policy is such that, for any state, the difference between the probability of satisfying the specification under this policy and the optimal one is within a predefined bound.Comment: 9 pages, 5 figures, Accepted by 2014 Robotics: Science and Systems (RSS

Three Simulation Algorithms for Labelled Transition Systems

Algorithms which compute the coarsest simulation preorder are generally designed on Kripke structures. Only in a second time they are extended to labelled transition systems. By doing this, the size of the alphabet appears in general as a multiplicative factor to both time and space complexities. Let $Q$ denotes the state space, $\rightarrow$ the transition relation, $\Sigma$ the alphabet and $P_{sim}$ the partition of $Q$ induced by the coarsest simulation equivalence. In this paper, we propose a base algorithm which minimizes, since the first stages of its design, the incidence of the size of the alphabet in both time and space complexities. This base algorithm, inspired by the one of Paige and Tarjan in 1987 for bisimulation and the one of Ranzato and Tapparo in 2010 for simulation, is then derived in three versions. One of them has the best bit space complexity up to now, $O(|P_{sim}|^2+|{\rightarrow}|.\log|{\rightarrow}|)$, while another one has the best time complexity up to now, $O(|P_{sim}|.|{\rightarrow}|)$. Note the absence of the alphabet in these complexities. A third version happens to be a nice compromise between space and time since it runs in $O(b.|P_{sim}|.|{\rightarrow}|)$ time, with $b$ a branching factor generally far below $|P_{sim}|$, and uses $O(|P_{sim}|^2.\log|P_{sim}|+|{\rightarrow}|.\log|{\rightarrow}|)$ bits