Boundary Region Relations

We are interested in the problem of how an agent organizes its sensorimotor experiences in order to create a spatial representation. Our approach to solve this problem is the Spatial Semantic Hierarchy (SSH), where multiple levels of spatial representation coexists. At the SSH topological level, space is represented by places and connectivity relation among them. Places are arranged into streets so that the topological representation looks like the street network of a city. Grouping places into regions allows an agent to reason e ciently about its spatial knowledge. Different types of regions can be de ned as the agent travels in the environment. Using the language of Causal Theories, we give a formal account of how an agent establishes boundary region relations while navigating its environment. Introduction The basic problem we are interested in solving is how an agent creates its spatial representation from its sensorimotor experiences. Our approach to solve this problem is the Spatial Semantic Hierarchy (SSH) (Kuipers & Byun 1988; Kuipers et al. 1993; Kuipers 1996; Kuipers & Byun 1991; Kuipers 1978; Kuipers & Levitt 1988). The SSH is an ontological hierarchy, where each level of the hierarchy has its own ontology abstracting the ontology of the levels below it. In this paper we are primarily concerned with the SSH topological level. At this level, space is represented by places and connectivity relations among them. Places This work has taken place in the Qualitative Reasoning Group at the Arti cial Intelligence Laboratory, The University of Texas at Austin. Research of the Qualitative Reasoning Group is supported in part by NSF grants IRI-9504138 and CDA 9617327, by NASA grant NAG 9-898, and by the Texas Advanced Research Program under grants no. 003658-242 and 003658-347. A boundary is a sequence of one or more directed streets. A boundary region is the set of places de ned to be on one side of a boundary. A boundary relation establishes for a given place whether it belongs to the boundary, or to one of the regions associated with the boundary. are arranged into streets so that the topological map looks like the street network of a city. When people solve route nding problems using a map, they group places into regions. Regions are then used to guide the search for a route between two speci c places. For example, in order to nd a route from Austin to Boston, we might rst nd a route from Texas to Massachusetts, and then use this route to nd the actual route from Austin to Boston. In order for an autonomous agent to use this hierarchical planning strategy, it has to create the appropriated space representation from its sensorimotor experiences. In this paper we describe how an agent establishes boundary region relations while navigating its environment (see footnote (1)). Once a su cient number of boundary relations have been accumulated, they provide a useful topological routending heuristic. For example, to nd a route from A to B, if there exists a street s such that A belongs to the right of s and B belongs to the left of s, look for routes from A to s and from s to B. The idea of using boundary relations in the context of the SSH was informally proposed in (Kuipers 1978; Kuipers & Levitt 1988). In this paper we give a formal ground to those ideas. Using the formalism of causal theories (McCain & Turner 1997) we describe how an agent deduces di erent boundary relations while navigating its environment. As it will be computationally expensive and cognitively ungrounded to assume that the agents knows the relation between every place and every boundary, we are interested in de ning the di erent states of partial knowledge associated with boundary relations. Moreover, as we do not rely on metrical information, our formalization captures the following default: in order for an agent to go from one side to the other of a boundary, the agent has to cross that boundary. We analyze how the boundary relations are a ected when this default is not true, that is, when the agent misses the boundary. We use the term topological map to refer to the SSH topological level. The paper is organized as follows: we rst review the ideas behind the Spatial Semantic Hierarchy (SSH) as well as we present the language of Causal theories. In particular, we de ne the language in which the topological map is described. Then we present our theory describing how the agent assimilates boundary relations. Finally, we de ne the boundary relations entailed by the environment given the set of actions executed by the agent. Background In this section we describe the main ideas behind the Spatial Semantics Hierarchy (SSH) as well as the language of Causal theories (McCain & Turner 1997). We describe in detail the SSH topological level as we are interested in de ning how boundary regions are associated with it. Causal theories will be used then to formally specify how boundary relation are established. The Spatial Semantic Hierarchy The Spatial Semantic Hierarchy (SSH) (Kuipers & Byun 1988; Kuipers et al. 1993; Kuipers 1996; Kuipers & Byun 1991) is an ontological hierarchy of representations for knowledge of large-scale space . Each level of the hierarchy has its own ontology (the set of objects and relations it uses for describing the world) and its own set of inference and problem-solving methods. The objects, relations, and assumptions required by each level are provided by those below it. Next we describe the di erent SSH levels. The sensorimotor level of the agent provides continuous sensors and e ectors, but not direct access to the global structure of the environment, or the robot's position or orientation within it. At the control level of the hierarchy, the ontology is an egocentric sensorimotor one, without knowledge of xed objects or places in an external environment. A distinctive state is de ned as the local maximum found by a hill-climbing control strategy, climbing the gradient of a selected feature, or distinctiveness measure. Trajectory-following control laws take the robot from one distinctive state to the neighborhood of the next, where hill-climbing can nd a local maximum, reducing position error and preventing its accumulation. The ontology at the SSH causal level consists of views, distinctive states, actions and schemas. A view is a description of the sensory input obtained at a locally distinctive state. An action denotes a sequence of one or more control laws which can be initiated at a locally distinctive state, and terminates after a hill climbing control law with the robot In large-scale space the structure of the environment is revealed by integrating local observations over time, rather than being perceived from a single vantage point. at another distinctive state. A schema is a tuple ((V; dp); A; (V ; dq)) representing the (temporally extended) event in which the robot takes a particular action A, starting with view V at the distinctive state dp, and terminating with view V 0 at distinctive state dq. In addition, we require that dp 6= dq. At the topological level of the hierarchy, the ontology consists of places, streets and regions, with connectivity and containment relations among them. Relations among the distinctive states and trajectories de ned by the control level, and among their summaries as schemas at the causal level, are e ectively described by the topological network. Using the network representation, navigation among distinctive states is not dependent on the accuracy, or even the existence, of metrical knowledge of the environment. At the metrical level of the hierarchy, the ontology for places, paths, and sensory features is extended to include metrical properties such as distance, direction, shape, etc. Geometrical features are extracted from sensory input, and represented as annotations on the places and paths of the topological network. The SSH Topological Level. As mentioned above, the ontology at the SSH topological level consists of places, streets and regions, with connectivity and containment relations among them. A street is an ordered sequence of places. Associated with each street there are two directions, pos and neg, that discriminate between the two directions one might be facing along a street. For example, in gure (1a), street s2 consists of three places A,B and C. When facing the positive direction of s2, the places are ordered as A;B;C. When facing in the opposite direction, the places are ordered as C;B;A. We use the following schemas to indicate the relations above: 1. inStreet(p,s) : place p is in street s. 2. nextS(p,q,s,dir) : place q is the next place when traveling from p on the direction dir of street s. In addition we require that nextS(p; q; s; pos) nextS(q; p; s; neg): At each place, an order is de ned among the di erent streets containing the place. This order speci es the next street the agent will face if it rotates to the right or For example, we do not allow turns of 360 degrees. The causal theory we are to de ne uses an underlying propositional language. However, we use schemas to present our theory. Schemas allow us to see the theory as a manysorted rst-order causal theory in which the domain closure and unique names assumptions are made. By assuming a nite set of constants for the di erent sorts in the theory, it is possible to ground the theory to produce an equivalent propositional theory. We have sorts for places, streets, regions, directions, truth values, uents, actions, and time. We also require that the next place is unique, that is, nextS(p; q; s; d) ^ nextS(p; r; s; d) q = r.