Background Literature Review for An Introduction to
Least Commitment Planning, by Daniel S. Weld
Written by Andrew Nierman
I begin with a summary of the Weld article as it provides a good overview of planning, and introduces most of the requisite terminology for the rest of the background. Following this, I will detail prior and subsequent work. A bibliography is also included.
The paper's focus is entirely on planning, in the sense that Weld does not entertain the notion of a hybrid system in which planning is intermixed with gaining information from the environment which could then change the plan. This idea of sensing/perceiving details from the environment and then acting conditionally upon those percepts is known as replanning. Weld also restricts his attention to generative, refinement planners. There are generative and case-based planners, which may be either refinement planners, or transformational planners. Generative planners construct plans from scratch, whereas case-based planners construct their plans from previously constructed plans or partial plans. A refinement planner simply adds constraints or actions to a plan, and a transformational planner can interleave this refinement action with retracting these constraints and actions as well. Weld notes that the algorithms presented can be extended to these other classes of planners.
Weld states that the planning problem has three parts:
This definition is somewhat abstract. For instance, the performance and scope of applicability of the planner is largely a function of which representation language is used. Obviously a planner will be able handle a larger variety of situations if it is given a more expressive language, but this also leads to a more intractable algorithm (as well as an algorithm which is more difficult to write).
Weld then outlines the simplifying assumptions that he makes in the formulation of his planning algorithms:
These four assumptions greatly simplify the planning process. For instance, the agent, after receiving the initial world state, can make its plan to achieve its goal, and execute this plan with its eyes closed, knowing that there are no external forces or dynamic elements of the world that will cause the plan to fail.
Weld discusses the STRIPS representation language (it is the representation language used in his POP algorithm). One important aspect of the STRIPS language is that it makes an assumption called the closed-world assumption in order to deal with the frame problem. In essence, anything that is not explicitly stated is assumed to be false. So, STRIPS preconditions and goal states consist of a conjunction of positive literals. The effects may include negated literals, but these simply serve to cancel out literals in the actions precondition list.
Weld then makes the distinction between world space vs. plan space planners, progressive vs. regressive planners, and total-order vs. partial order planning. In a world space planner, the search nodes can be viewed as world states that are fully instantiated. The arcs connecting these nodes (world states) are the actions that move the agent from one world state to another. When represented in this way, this planning problem is amenable to the many forms of search algorithms that have been developed. Search through the world space can either occur from the initial state forward (progressive planner) or from the goal backwards (regressive planner). It is not always the case that one is clearly better than the other, but it is often the case that goals have fewer preconditions to satisfy than there are potential actions to take. In this case, utilizing a regressive approach would cut down on the branching factor. In a plan space planner, the search nodes are partially specified plans, and arcs are refinement operations on these partial plans. With this formulation, one could use partial ordering of plans as well. These partially ordered plans delay ordering decisions unless they are essential (and thus they are least commitment planners). These partially ordered plans can then be linearized into a total order plan. In effect, one partial order plan represents many total order plans, so it is easy to see why this could be more efficient.
Causal links are introduced when we begin considering partial order planners. These simply designate which precondition of a specified operator is met by a specific post condition of another operator. This also brings up the notion of threats. There are some cases where we need to restrict the order of the overall partially ordered plan in order to ensure that preconditions are not clobbered (in the vernacular of the planner community) or undone before the action actually takes place. The way in which this is done depends upon the representational language being used, but in the POP algorithm this can be accomplished through promotion and demotion. Either forcing the threatening action to come before the supporter of the precondition in question or after the action with this precondition has occurred. In more expressive language there may be more methods to resolve threats (as we will see with UCPOP).
UCPOP is an extended version of POP that handles action schemata with variables, conditional effects, disjunctive preconditions, and universal quantification. I will discuss each of these briefly and outline the increased expressivity that they provide to the planning algorithm over the more simplistic POP algorithm.
The first of these extensions allows for variables in the plan. This simply serves to extend the idea of least commitment planning. Now, not only can we reserve commitment with respect to the ordering of operators, we can also reserve commitment on the actual binding of variables. Both of these forms of least commitment will serve to reduce search time (in general). This also serves an engineering purpose: it obfuscates the need for duplication of similar action rules. Algorithmically, we must also worry about how to instantiate these variables (unify them). In order to ensure that we have a solution, the variable will have to be instantiated, but as Weld points out, we can get this for free by following some simple requirements: the initial state should contain no variables, and the and all of the variables mentioned in the effects of an operator must also appear in the precondition of that operator.
Conditional effects are also allowed in UCPOP. From an engineering standpoint, this simply makes it easier to code the operators for the planner. By using conditional effects one may reuse operators, and simply have conditional effects to differentiate the resultant action. But more importantly, conditional effects extend the idea of least commitment planning. The example given by Weld is that of the move operator and the move-to-table operator. Without conditional effects, we needed both of these operators for similar tasks. But if we want to achieve a goal of clear(A) for instance, then we would have to commit to a location for the block on top of A, because we need to know whether we are satisfying the clear(A) precondition with move or with move-to-table. This clearly would violate the idea of least commitment planning, and conditional effects alleviate this problem. Allowing conditional effects also introduces another type of threat resolution called confrontation. With this technique one simply adds the negation of a conditional effects antecedent to the agenda. With the antecedent negated, the action based on that conditional cannot take place.
Disjunctive preconditions (and disjunctive antecedents) are also allowed in UCPOP. Weld cautions against prolific use of these disjunctions though, as they will cause the search space to explode. Weld does not allow disjunction in the effects though. This would complicate the planner and we would have to have some way to reason about these partially known effects (such as probabilistic methods).
The final extension that is made is that of universal quantification. This also greatly simplifies the writing of operators, and extends the planners representational abilities. One assumption that is added is that the world we are describing is finite. The universal quantifiers are actually implemented by expanding them into what is known as the base. Because of the finite world assumption, this base is also finite. Weld gives the briefcase example to show the increased expressivity allowed. When the briefcase is moved it is now much easier to say that everything inside of the briefcase has also moved.
Weld concludes the paper with a brief foray into advanced topics that are currently under research. These include building planners that handle more expressive languages (while still maintaining some level of tractability), dealing with incomplete information or dynamic domains, and planning with multiple cooperating agents.
It is not my aim to surprise or shock you.... But the simplest way I can summarize is to say that there are now in the world machines that think, that learn and create. Moreover, their ability to do these things is going to increase rapidly until - in a visible future - the range of problems they can handle will be coextensive with the range to which the human mind has been applied.
GPS was to be a domain independent framework for solving problems. GPS did not live up to its claims, but it did serve to lay the groundwork for later ideas in planning. The basic idea behind GPS was to break a problem into its constituent parts by determining what action needed to take place for each goal and subgoal. This technique, known as means-ends analysis, went as follows:
I want to take my son to nursery school. Whats the difference between what I have and what I want? One of distance. What changes distance? My automobile. My automobile wont work. What is needed to make it work? A new battery. What has new batteries? An auto repair shop....
and so on.
The STRIPS planner was developed by Fikes and Nilsson in 1972. STRIPS was a world space regression planner. The STRIPS language was used in many future systems (such as the POP algorithm) because it lead to a greater degree of tractability as compared with more expressive languages like Situation Calculus or full First Order Logic (FOL). The STRIPS closed world assumption (that was discussed in the Weld paper) was a method of dealing with the frame problem. The frame problem can be stated in many different ways, but the underlying problem is the inability to express how things change in the world. Here are a few examples of various formulations of the frame problem:
We can see that these problems can be handled by various means, depending upon the language being used. For instance, Weld handles the ramification problem with UCPOPs universal quantifiers, while STRIPS handles the frame problem with its closed-world assumption.
The HACKER system developed by Sussman (1974) was a direct descendant of STRIPS. In fact, it was in this system that the infamous Sussman anomaly was discovered. None of the planners of the time could solve the Sussman anomaly. The root of the problem was that these planners were non-interleaved planners. These planners could solve subgoals (such as ON(A, B), or ON(B, C) ), but they could only combine these subgoals by placing all of the steps for one subgoal before or after all of the steps for another subgoal. If one of the subgoals had to undo part of the other subgoal, then an overall solution would not be achieved.
Subsequent systems such as WARPLAN by Warren (1974), and INTERPLAN by Tate
(1974), tried to solve the Sussman anomaly by arbitrarily interleaving
plan steps. (WARPLAN was also interesting because it was implemented in
Prolog, and took only 100 lines of code).
In 1975, Sacerdoti developed NOAH, which pioneered the technique of searching through plan space rather than world space, and the construction of partially ordered plans. The search through plan space has several advantages. Namely, there is no necessary connection between the order of planning and the order of execution. One can make the obvious/important decisions first, possibly reducing the branching factor. And independence in sub problems can be exploited. But, NOAH didnt actually take advantage of these intrinsic properties of partial ordered planning. Sacerdoti stated:
The basic operations of NOAH were developed in an ad hoc fashion. No attempt has been made to justify the transformations that they perform, or to enable them to generate all transformations. However, it should be possible to define an algebra of plan transformations... a body of formal theory about the ways in which interacting subgoals can be dealt with.
NONLIN was developed in 1977 by Tate (the author of INTERPLAN). NONLIN was the first system to use causal links. With this additions, threats could be detected and dealt with in a formal manner. NONLIN also had a top-level backtracking structure that would enable it to find plans where NOAH would fail.
In 1987, Chapman formalized Sacerdotis ideas and built on the original NOAH planner to create TWEAK. Chapman provided the formal theory that Sacerdoti had talked about. Chapman proved that TWEAK was both sound and complete. TWEAK was to be the basis for the SNLP and POP (Soderland & Weld, 1991) algorithms.
These planners were quite useful in simple domains, but their limited expressiveness (due to their basis on the STRIPS language) limited their applicability. The improvements to POP resulting in the UCPOP algorithm (as described above) helped to extend the expressiveness. Increased expressiveness could be achieved even over and above the UCPOP algorithm (by implementing planners based on full FOL for instance). But with increased expressiveness (and even expressiveness on the level of UCPOP) comes tractability problems. One method for dealing with efficiency issues is that of hierarchical planning. If we want to perform a task, it often makes sense to break the task into abstract components (sub problems). For instance, if we want to launch a rocket, we will initially formulate a plan that has as its components: prepare booster rocket, prepare capsule, load cargo, etc. (Not on the level of: insert bolt A, into hole B....) These abstract components can then be refined into more detailed sub-plans. This process can definitely lead to more efficient plan computation, but there are some assumptions that need to be made in order for this to work. The problem must be able to be decomposed into sub problems in the first place. Also, once solutions to sub problems have been found, their recombination must result in the overall solution. Often times this will not be the case, especially if the sub problems have a lot of interaction.
Another area that the paper did not deal with was that of conditional or contingency planning. In this sort of planning, a conditional plan is constructed for each possible situation that could occur. Then there must be some way for the agent to perform a sensing action to find out which part of the plan to execute. In this form of planning, the plan is still constructed beforehand, and simply executed. But, execution monitoring is also available. This defers the job of dealing with conditions until they actually arise. The agent can then do replanning to achieve a goal from an unforeseen/unexpected state. Conditional planning and execution monitoring are two extremes on a continuous spectrum, and in reality a system could lie somewhere in the middle. One issue that Weld does not address in the paper is that of disjunctive outcomes for the actions, when more than one outcome is reasonably likely. This could be implemented with a sensing action to determine which of the outcomes occurred, and then a conditional plan could be constructed accordingly. This does complicate the issue though. The designer must determine which outcomes need to be considered. Also, we would like the agent to be able to operate in complex domains with incomplete and incorrect information. In this case, the agent may need to assess likelihoods and costs of various outcomes. The goal then is to construct a plan that maximizes the probability of success, minimizes cost, and ignores contingencies that are easy to deal with, or that are unlikely (leave these to replanning). Ideas such as these have lead to the development of conditional planners such as CNLP by Peot and Smith, 1992 and C-Buridan by Draper, Weld, and Hanks, 1994. C-Buridan was built on top of Buridan which was a planner with a probabilistic model of the world, in which actions could have uncertain effects. In C-Buridan, sensing actions were added, and contingent action (like that available in CNLP) was made possible. The input to the system is a probability distribution over world states, a goal expression, action descriptions, and a probability threshold. C-Buridan will produce a contingent plan that makes the goal expression true with a probability no less than the threshold.
Allen, James; Hendler, James; Tate, Austin; eds. Readings in Planning. Morgan Kaufman Publishers, Inc. 1990.
Chapman, David. "Planning For Conjunctive Goals." Artificial Intelligence, 32(3), pp. 333-377, 1987.
Draper, Denise; Hanks, Steve; Weld, Daniel. "Probabilistic Planning with Information Gathering Contingent Execution." Proceedings AIPS-94, pp. 31-36.
Draper, Denise; et. al. "An Approach to Planning with Incomplete Information." KR-92.
Fikes, Richard; Nilsson, Nils. "STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving." Artificial Intelligence, 2(3-4), pp. 189-208, 1971.
Golden, Keith; Weld, Daniel. "Representing Sensing Actions: The Middle Ground Revisited." KR 96.
Kushmerick, Nicholas; Hanks, Steve; Weld, Daniel. "An Algorithm for Probabilistic Planning." Artificial Intelligence, 76(1-2), pp. 239-286, 1995.
Norvig, Peter. Paradigms of Artificial Intelligence Programming: Case Studies in Common Lisp. Morgan Kaufman Publishers, Inc. 1992.
Russell, Stuart; Norvig, Peter. Artificial Intelligence: A Modern Approach , pp. 337-416, 1994.
Weld, Daniel S. "An Introduction to Least Commitment Planning." AI Magazine , pp. 27-61, Winter 1994.
Wilkins, David. Practical Planning: Extending the Classical AI Planning Paradigm. Morgan Kaufman Publishers, Inc. 1988.