## Blocksworld (Color)

The colored Blocksworld domain is a variant of the traditional Blocksworld presented above. As in the traditional Blocksworld, colored Blocksworld consists of two types of objects, tables and blocks. Again, the domain has exactly one table and each problem instance has some number of blocks. The actions of the domain are still “pick-up-block-from” and “put-down-block-on”, and this domain also comes in two flavors: goal and reward. The major difference from the traditional Blocksworld domain is that each block in the colored Blocksworld domain is assigned a color, and the goal configuration is specified in terms of block colors rather than specific blocks. Thus, in general, there are many different valid goal configurations. Goal conditions are expressed with existential quantification. For example, the PPDDL fragment

 ``` (:goal (and (exists (?b1) (is-green ?b1)) (exists (?b2) (and (is-blue ?b2) (on-top-of ?b1 ?b2))))) ```
states that the goal is to have any green block on top of any blue block.

The noise in the colored Blocksworld domain is the same as in the traditional Blocksworld domain. That is, the colored Blocksworld domain incorporates probabilistic effects by adding a “slip” probability. Each time a block is picked up or put down, the block will slip and fall onto the table with probability 0.25.

Notice that although the goal configuration is existentially quantified and hence not precisely specified, the same basic policy that can be used to solve the traditional Blocksworld can be used to solve the colored Blocksworld. To solve a colored Blocksworld problem, unstack all of the blocks and then, in a bottom up fashion, choose a block that satisfies a color constraint and place it in the appropriate position.

The colored Blocksworld domain aims to add complexity to the traditional Blocksworld domain by incorporating existential quantification into the goal configuration. The indeterminacy of the goal in the colored Blocksworld domain can make the planning problem considerably harder than its traditional counterpart. Thus, a colored Blocksworld problem may be impossible for a given planner to solve in a reasonable amount of time, whereas that same planner would have no problem on a traditional Blocksworld problem of the same size.2

A generation program for random colored Blocksworld domains was provided to participants and the competition problems were generated from this same program.

Håkan L. S. Younes
2005-12-06