A Logical Account of Causal and Topological Maps

The Spatial Semantic Hierarchy (SSH) is a set of distinct representations for large scale space, each with its own ontology and each abstracted from the levels below it. At the control level, the agent and its environment are modeled as continuous dynamical systems whose equilibrium points are abstracted to a discrete set of distinctive states. The control laws whose execution defines trajectories linking these states are abstracted to actions, giving a discrete causal graph representation for the state space. The causal graph of states and actions is in turn abstracted to a topological network of places and paths (i.e. the topological map). Local metrical models of places and paths can be built within the framework of the control, causal and topological levels while avoiding problems of global consistency. Most of the SSH's ideas have been traditionally described in procedural terms inspired by implementations of the SSH. This description has various problems when used to implement physical agents. First, some assumptions are not explicitly stated or are difficult to meet by current sensory technology (e.g. sensory information is not rich enough to distinguish one place from another). Second, some important SSH concepts (i.e. paths) are not properly defined or understood. Third, sometimes it is not clear what the representation states and consequently it is hard to apply it to new domains. In this dissertation we propose a formal semantics for the SSH causal, topological and local metrical theories. Based on this semantics, we extend the SSH in the following important ways: (i) we include distinctive states as objects of the theory and handle perceptual aliasing, (ii) we define the models associated with the SSH causal and topological levels, (iii) we extend the SSH topological theory to handle convergent and self intersecting paths as well as hierarchical maps, (iv) we show how to combine causal, topological and noisy local metrical information, (v) based on the previous enhancements, we define an algorithm to keep track of different topological maps consistent with the agent's experiences. This algorithm supports different exploration-strategies and facilitates map disambiguation when the case arises.