Universal Plans: One Action Sequence to Solve Them All!

TL;DR

The notion of a universal plan, which when executed, is guaranteed to solve all planning problems in a category, regardless of the obstacles, initial state, and goal set, is introduced.

Abstract

This paper introduces the notion of a universal plan, which when executed, is guaranteed to solve all planning problems in a category, regardless of the obstacles, initial state, and goal set. Such plans are specified as a deterministic sequence of actions that are blindly applied without any sensor feedback. Thus, they can be considered as pure exploration in a reinforcement learning context, and we show that with basic memory requirements, they even yield optimal plans. Building upon results in number theory and theory of automata, we provide universal plans both for discrete and continuous (motion) planning and prove their (semi)completeness. The concepts are applied and illustrated through simulation studies, and several directions for future research are sketched.

Figures (4)

• Figure 1: Applying a finite action sequence $(\rightarrow, \rightarrow, \rightarrow, \rightarrow, \downarrow)$ in a grid environment (white cells are free space, grey cells are obstacles, and is the start state). (a) The robot tries to move four times to the right, but is kept still by the obstacle, then moves down. (b) The robot is not obstructed, and actually reaches the goal .
• Figure 2: Discretization of the planar planning problem. (a) The discretization $X_\textrm{grid}(x_I, m) \subset (x_I + 2^{-m} (\mathbb{Z} \times \mathbb{Z}))$ at scaling resolution $2^{-m}$. The horizontal and vertical lines connecting the vertices are drawn here to indicate possible transitions, but are not included in $X_\textrm{grid}(x_I, m)$. The dark blue line indicates a possible plan taking the initial state (red dot) to the goal state (green dot). (b) & (c) Two possible ways the same grid can emerge from two different initial states $x_I, x_I' \in \textrm{int}X$.
• Figure 3: Determining the sufficient scaling factor $\eta(X,r)$ in an environment whose complement is a set with positive reach. (a) Radius $r$ of the goal area $B_r(x_G)$, and $p = \textrm{reach}(\mathbb{R}^2 \setminus X)$. (b) The grid $V(X, x_I, m) := X_\textrm{grid}(x_I, m) \cap X_{p/2}$ in which $m = p/4$. Note that $x_I \notin V(X, x_I, m)$, but $x_I \in B_{p/2}(z_p)$ in which $z_p$ () minimizes the distance from $x_I$ to the set $X_{p/2}$.
• Figure 4: Typical executions of plans believed to be universal: (a) Digits of $\pi$ applied to search a 100x100 grid; 5548 of 7124 states were visited in 47785 total steps; all examples involve going from a red initial state in the upper left to a green goal state or region in the lower right. (b) Same plan as in (a); 5220 of 7128 states were visited in 48291 steps. (c) Applying Champernowne's number results in 11903745 steps; the progress after 50000 steps is shown. (d) Using $\pi$ digits, starting at position 1200000, to search a maze in 213675 steps. (e) Using $\pi$ digits starting at 12000000 to solve a continuous planning problem in 970 steps, using the proven method from Section \ref{['sec:scale']}. (f) An alternative method, which seems more efficient, but is not yet proven to be complete; digits of $\pi$ starting with 3400000 were used.

Theorems & Definitions (29)

• definition 1: Connected Grid
• definition 2: Robot grid search problem
• definition 3: Rich and normal plans
• definition 4: Deterministic Finite Automaton
• definition 5: Reachable state; Essential class
• proposition 1: Rich plans converge to essential classes (Compton)
• lemma 1: Essential class of finite trajectories
• proof
• corollary 1: Rich plan tries every action sequence from every state
• proof