Reformulating Theories Of Action For Efficient Planning

Domain theories are used in a wide variety of fields of computer science as a means of representing properties of the domain under consideration. These fields include artificial intelligence, software engineering, VLSI design, cryptography, and distributed computing. In each ease, the advantages of using theories include the precision of task specification and the ability to verify results. A great deal of effort has gone into the development of tools to make the use of theories easier. This effort has met with some success. However, a fundamental problem remains: the choice of symbolic formulation for a theory, including both the choice of features for describing the environment and the design of abstractions that encode the actions. This paper describes fundamental research on the algebraic structure of the representations of domain theories. The perspective of this work is to view a problem’s state space as though it were physical space, and the actions in the state space as though they were physical motions. A domain theory should then state the laws of motion within the space. Following the analogy with physics, a representation is a coordinate system, and theories are transformed by changing coordinates. This permits symbolic computational techniques to be used to transform theories and find useful decompositions. A system has been implemented using Mathematiea and GAP that performs these computations. The mathematical basis for this approach is given, and the computations are illustrated by examples. 1 Formulations of Theories of Action Each theory can be represented in a large number of different ways that vary in their computational effectiveness. A good choice of symbols and a good choice of formulation using those symbols are absolutely necessary for effective symbolic computation. A simple example is given by the following three representations for the two-disk Towers of Hanoi. Representation TOHI: Let the nine states of the 2-disk Towers of Hanoi be { (b, s), < b < 3,1 <s < 3 ], whereb and s are the numbers of the pegs the big and small disks are on, respectively. Let the two actions be: X: (b, s) ~ (b, s (rood(3)) Y: (b, s) ~ (b+ 1 (mod(3)) as X moves the small disk right one peg (wrapping around from peg 3 to peg 1), and Y moves the large disk one peg to the left (wrapping around from peg 1 to peg 3). X is always executable, but Y can be executed only in three states. Representation TOH2: Let the states be the same TOH1, and let the six possible actions be: X1 = move the top disk from peg 1 to peg 2 Y1 --move the top disk from peg 1 to peg 3 X2 = move the top disk from peg 2 to peg 3 Y2 = move the top disk from peg 2 to peg 1 X3 = move the top disk from peg 3 to peg 1 Y3 = move the top disk from peg 3 to peg 2 Each of these actions is executable in four states. Representation TOH3: Let the states be the same as TOH1, and let the two possible actions be: X: (b, s) ~ (b, s (rood(3)) Z: (b, s) ~ (b + 1 (rood(3)) + 1, s + 1 (rood(3)) X is as before, and Z is a macro-action that moves both disks left one peg. Each of these representations can be implemented in terms of every other. For example, Z in TOH3 is implemented as a disjunction of sequences of actions from TOHI: Z = [XXY, XYX, YXX} or from TOH2: Z = {X1Y1X2, X2Y2X3, X3Y3X1 }. One of the most important properties of a representation for planning is the mutual interference of its actions. The actions in TOH3 are independent controls; X solves the position of the small disk, and Z solves the big disk. X does not changethe position of the big disk, and Z does not change the relative positions of the disks. The disks can be solved in either order. This representation captures the important property of the Towers of Hanoi task: the disks can be solved in any order. In representations TOH1 and TOH2, this property was obscured by details of the implementations of the disk moves. Even for this simple puzzle there are many possible formulations. The three formulations given above differ in that the first indexes the moves according to the disk moved, whereas the second indexes the moves by the peg moved from (or to, for the inverse 36 From: AAAI Technical Report WS-96-07. Compilation copyright © 1996, AAAI (www.aaai.org). All rights reserved. moves.) The third representation eliminates subgoal interference, which is present in the first two representations. One advantage of the first and third formulation is clear: they scales up to theories for more disks, because their actions will be included unchanged in those theories. New moves will be added for the new disks. However, the actions in the second formulation must be redefined as more disks are added, so this theory does not scale up. The result is that analysis and synthesis of the two-disk problem performed using the first or third formulation can be reused when larger problems are attempted, but analysis and synthesis from the second formulation must be discarded when larger problems are attempted. Furthermore, the properties of these two representations scale up to all Towers of Hanoi problems, e.g., the family of representations based on TOH3 have no subgoal interference. The third representation is clearly the best for efficient planning in this domain. Computer science is not the first field to be faced with the problem of properly formulating theories. Throughout he history of science, it has always been desirable to formulate theories in as general a way as possible, so that important regularities are identified and separated from details particular to individual situations. In particular, physics has had a great deal of success in formulating theories of wide generality yet high predictive accuracy. In this paper, we will see that many of the mathematical structures employed in the statement of physical theories can be usefully generalized to the statement of abstract theories. We will begin with a brief discussion of the properties of physical theories in the next two sections. The remainder of the paper discusses the appropriate mathematics for analyzing these properties, and gives examples of the analysis and reformulation of theories. 2 Invariants of Laws The ability to formulate any law of nature depends on the fact that the predictions given by the law, together with certain initial conditions, will be the same no matter when or where the results of the predictions are observed. In physical theories, the fact that absolute time and location are never relevant is essential for the statement of laws; without this fact, general laws could not be stated, and the complexity of the world would eliminate the possibility of intelligent comprehension of the environment. This irrelevance is stated in terms of the invariance of laws under translation in time and space. Such invariance is so self-evident that it was not even stated clearly until less than a century ago. It was Einstein who recognized the importance of invariance in the formulation of physical law, and brought it to the forefront of physics. Before Einstein, it was natural to first formulate physical law and then derive the laws of invariance. Now, the reverse is true. As the eminent physicist Wigner states, "It is now natural for us to try to derive the laws of nature and to test their validity by means of the laws of invariance .... "(Wigner, 1967, p.5). This is especially clear in the development of quantum mechanics. Invariance is important not only in physics. As Dijkstra states, "Since the earliest days of proving the correctness of programs, predicates on the programs’s state space have played a central role. This role became essential when non-deterministic systems were considered .... I know of only one satisfactory way of reasoning about such systems: to prove that none of the atomic actions falsifies a special predicate, the so-called ’global invariant’." (Dijkstra, 1985.) In other words, as the system moves in its state space, the global invariant is a law of motion. Dijkstra goes on to point out the central difficulty in the use of invariants: "That solves the problem in principle; in each particular case, however, we have to choose how to write down the global invariant. The choice of notation influences the ease with which we can show that, indeed, none of the atomic actions falsifies the global invariant." We need a mathematics of invariance to help us formulate invariants.