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In computer science, functional orbits – i.e., tracks left by the iterations of a function – in a finite domain naturally appear at a high or a low level. This paper introduces a Coq logical orbit framework, the purpose of which is to help rigorously developing software systems with some complex data structures from specification to implementation. The result is a Coq library of orbit concepts formalized as definitions, lemmas and theorems. Most of them are inspired by our previous work in geometric modelling, where combinatorial hypermaps were used to describe surface subdivisions appearing in computational geometry problems, e.g., building convex hulls or Delaunay diagrams. Now, this domain remains a reference for us, but our results are drastically extended and usable in other computer science areas. The proof of Floyd's cycle-detection algorithm, known as “the tortoise and the hare”, confirms that point. The library contains operations to observe, traverse and update orbits – addition, deletion, mutation, transposition – with proofs of their behavior. It focuses on the important case where the involved function is a partial injection. In this case, it defines a connectivity relationship and evaluates the variation of the number of connected components during updates. [ABSTRACT FROM AUTHOR]

We address the problem of practical manipulation planning for rearrangement tasks of many movable objects. We study a special case of the rearrangement task, where the only allowed manipulation is pushing. We search for algorithms that can provide practical planning time for most common scenarios. We present a hierarchical classification of manipulation problems into several classes, each characterized by properties of the plans that can solve it. Such a classification allows one to consider each class individually, to analyze and exploit properties of each class, and to suggest individual planning methods accordingly. Following this classification, we suggest algorithms for two of the defined classes. Both items have been tested in a simulated environment, with up to 32 movable objects and 66 combined DOF. We present the simulations results as well as some experimental results using a real platform. [ABSTRACT FROM PUBLISHER]