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ABSTRACT

This research note contains an extension of a previous work by Cabalar and Santos (2011) that formalised several spatial puzzles formed by strings and holes. That approach explicitly ignored some configurations and actions that were irrelevant for the studied puzzles but are physically possible and may become crucial for other spatial reasoning problems. In particular, the previous work did not consider the formation of string loops or the situations where a holed object is partially crossed by another holed object. In this paper, we remove these limitations by treating string loops as dynamic holes that can be created or destroyed by a pair of elementary actions, respectively picking or pulling from strings. We explain how string loops can be recognised in a data structure representing the domain states and define a notation to represent crossings through string loops. The resulting formalism is dual in the sense that it also allows understanding any hole as a kind of (sometimes rigid) closed string loop.

7. Concluding remarks

This paper discussed a solution to the challenging problem of formally describing a particular characteristic of flexible objects such as strings: their capacity of making loops that can be used (and reasoned about) as holes in spatial reasoning processes. This solution resolves two issues left open in our previous work (reported in [11]), namely, the representation of states where a holed object partially crosses another hole and the creation of string loops. In this paper, we have described the identification of string loops in lists of string crossings, together with the actions related to the creation and unwinding of string loops. In possession of these actions, the framework in [11] can now be used to reason about spatial puzzles where the manipulation of loops is an essential part of the solution. Future research shall be conducted mainly in two fronts: the consideration of actions related to winding (and unwinding) knots and the deployment of these ideas in real application domains. The latter may include tasks such as autonomous needle steering or the actual manipulation of (and reasoning about) real world objects by a humanoid robot, such as the Darpa Robotics Challenge6 that has as one of its goals the implementation of a humanoid robot with the “ability to manipulate and use a diverse assortment of tools designed for humans”.