John H. Reif

Logical computation using algorithmic self-assembly of DNA triple-crossover molecules

Recent work has demonstrated the self-assembly of designed periodic two-dimensional arrays composed of DNA tiles, in which the intermolecular contacts are directed by ‘sticky’ ends. In a mathematical context, aperiodic mosaics may be formed by the self-assembly of ‘Wang’ tiles, a process that emulates the operation of a Turing machine. Macroscopic self-assembly has been used to perform computations; there is also a logical equivalence between DNA sticky ends and Wang tile edges. This suggests that the self-assembly of DNA-based tiles could be used to perform DNA-based computation. Algorithmic aperiodic self-assembly requires greater fidelity than periodic self-assembly, because correct tiles must compete with partially correct tiles. Here we report a one-dimensional algorithmic self-assembly of DNA triple-crossover molecules that can be used to execute four steps of a logical (cumulative XOR) operation on a string of binary bits.

Complexity of the mover's problem and generalizations

This paper concerns the problem of moving a polyhedron through Euclidean space while avoiding polyhedral obstacles.

New lower bound techniques for robot motion planning problems

We present new techniques for establishing lower bounds in robot motion planning problems. Our scheme is based on path encoding and uses homotopy equivalence classes of paths to encode state. We first apply the method to the shortest path problem in 3 dimensions. The problem is to find the shortest path under an Lp metric (e.g. a euclidean metric) between two points amid polyhedral obstacles. Although this problem has been extensively studied, there were no previously known lower bounds. We show that there may be exponentially many shortest path classes in single-source multiple-destination problems, and that the single-source single-destination problem is NP-hard. We use a similar proof technique to show that two dimensional dynamic motion planning with bounded velocity is NP-hard. Finally we extend the technique to compliant motion planning with uncertainty in control. Specifically, we consider a point in 3 dimensions which is commanded to move in a straight line, but whose actual motion may differ from the commanded motion, possibly involving sliding against obstacles. Given that the point initially lies in some start region, the problem of finding a sequence of commanded velocities which is guaranteed to move the point to the goal is shown to be non-deterministic exponential time hard, making it the first provably intractable problem in robotics.

Social potential fields : A distributed behavioral control for autonomous robots

Abstract A Very Large Scale Robotic (VLSR) system may consist of from hundreds to perhaps tens of thousands or more autonomous robots. The costs of robots are going down, and the robots are getting more compact, more capable, and more flexible. Hence, in the near future, we expect to see many industrial and military applications of VLSR systems in tasks such as assembling, transporting, hazardous inspection, patrolling, guarding and attacking. In this paper, we propose a new approach for distributed autonomous control of VLSR systems. We define simple artificial force laws between pairs of robots or robot groups. The force laws are inverse-power force laws, incorporating both attraction and repulsion. The force laws can be distinct and to some degree they reflect the ‘social relations’ among robots. Therefore we call our method social potential fields. An individual robots motion is controlled by the resultant artificial force imposed by other robots and other components of the system. The approach is distributed in that the force calculations and motion control can be done in an asynchronous and distributed manner. We also extend the social potential fields model to use spring laws as force laws. This paper presents the first and a preliminary study on applying potential fields to distributed autonomous multi-robot control. We describe the genetic framework of our social potential fields method. We show with computer simulations that the method can yield interesting and useful behaviors among robots, and we give examples of possible industrial and military applications. We also identify theoretical problems for future studies.

Parallel tree contraction and its application

Abstract : Trees play a fundamental role in many computations, both for sequential as well as parallel problems. The classic paradigm applied to generate parallel algorithms in the presence of trees has been divide-conquer; finding a 1/3 - 2/3 separator and recursively solving the two subproblems. A now classic example is Brents work on parallel evaluation of arithmetic expressions. This top-down approach has several complications, one of which is finding the separators. We define dynamic expression evaluation as the task of evaluating the expression with no free preprocessing. If we apply Brents method, finding the separators seems to add a factor of log n to the running time. We give a bottom-up algorithm to handle trees. That is, all modifications to the tree are done locally. This bottom-up approach which we call CONTRACT has two major advantages over the top-down approach: (1) the control structure is straight forward and easier to implement facilitating new algorithms using fewer processors and less time; and (2) problems for which it was too difficult or too complicated to find polylog parallel algorithms are now easy.

Kinodynamic motion planning

Kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints. In the general problem, given a robot system, we must find a minimal-time trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration. We consider the simplified case of a point mass under Newtonian mechanics, together with velocity and acceleration bounds. The point must be flown from a start to a goal, amidst polyhedral obstacles in 2D or 3D. Although exact solutions to this problem are not known, we provide the first provably good approximation algorithm, and show that it runs in polynomial time

Programming DNA Tube Circumferences

Synthesizing molecular tubes with monodisperse, programmable circumferences is an important goal shared by nanotechnology, materials science, and supermolecular chemistry. We program molecular tube circumferences by specifying the complementarity relationships between modular domains in a 42-base single-stranded DNA motif. Single-step annealing results in the self-assembly of long tubes displaying monodisperse circumferences of 4, 5, 6, 7, 8, 10, or 20 DNA helices.

Motion planning in the presence of moving obstacles

This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP hard even if B has no velocity modulus bound, where in both cases B has 6 degrees of freedom. To prove these results we use a unique method of simulation of a Turing machine which uses time to encode configurations (whereas previous lower bound proofs in robotics used the system position to encode configurations and so required unbounded number of degrees of freedom). We also investigate a natural class of dynamic problems which we call asteroid avoidance problems: B, the object we wish to move, is a convex polyhedron which is free to move by translation with bounded velocity modulus, and the polyhedral obstacles have known translational trajectories but cannot rotate. This problem has many applications to robot, automobile, and aircraft collision avoidance. Our main positive results are polynomial time algorithms for the 2-D asteroid avoidance problem with bounded number of obstacles as well as single exponential time and nO(log n) space algorithms for the 3-D asteroid avoidance problem with an unbounded number of obstacles. Our techniques for solving these asteroid avoidance problems are novel in the sense that they are completely unrelated to previous algorithms for planning movement in the case of static obstacles. We also give some additional positive results for various other dynamic movers problems, and in particular give polynomial time algorithms for the case in which B has no velocity bounds and the movements of obstacles are algebraic in space-time.

The bridge test for sampling narrow passages with probabilistic roadmap planners

Probabilistic roadmap (PRM) planners have been successful in path planning of robots with many degrees of freedom, but narrow passages in a robots configuration space create significant difficulty for PRM planners. This paper presents a hybrid sampling strategy in the PRM framework for finding paths through narrow passages. A key ingredient of the new strategy is the bridge test, which boosts the sampling density inside narrow passages. The bridge test relies on simple tests of local geometry and can be implemented efficiently in high-dimensional configuration spaces. The strengths of the bridge test and uniform sampling complement each other naturally and are combined to generate the final hybrid sampling strategy. Our planner was tested on point robots and articulated robots in planar workspaces. Preliminary experiments show that the hybrid sampling strategy enables relatively small roadmaps to reliably capture the connectivity of configuration spaces with difficult narrow passages.

A logarithmic time sort for linear size networks

A randomized algorithm that sorts on an <italic>N</italic> node network with constant valence in <italic>O</italic>(log <italic>N</italic>) time is given. More particularly, the algorithm sorts <italic>N</italic> items on an <italic>N</italic>-node cube-connected cycles graph, and, for some constant <italic>k</italic>, for all large enough <italic>α</italic>, it terminates within <italic>kα</italic> log <italic>N</italic> time with probability at least 1 - <italic>N</italic><supscrpt>-<italic>α</italic></supscrpt>.