Complexity in Spatial Reasoning

Complexity in Spatial Reasoning Marco Ragni (ragni@informatik.uni-freiburg.de) Department of Computer Science, Georges-Kohler-Allee 79110 Freiburg, Germany Thomas Fangmeier (thomas.fangmeier@cognition.uni-freiburg.de) Center for Cognitive Science, Friedrichstrase 50 79098 Freiburg, Germany Lara Webber (lara.webber@cognition.uni-freiburg.de) Center for Cognitive Science, Friedrichstrase 50 79098 Freiburg, Germany Markus Knauff (knauff@cognition.uni-freiburg.de) Center for Cognitive Science, Friedrichstrase 50 79098 Freiburg, Germany The hammer is to the right of the pliers. The screwdriver is to the left of the pliers. The wrench is in front of the screwdriver. The saw is in front of the pliers. Abstract We introduce a unified approach to account for the problems people have in spatial reasoning. This approach combines two theories: the mental model theory which aims to explain the deduction process, and the relational complexity theory which explains the processing complexity of the spatial relations needed in order to conceptualize the reasoning problem. We propose that a combination of these two theories can account for some of various errors found in spatial reasoning. We present two experiments in which we demonstrate that participants use the principle of first free fit to construct preferred mental models. We then formally implement these findings in the Spatial Reasoning by Models computational framework. Which relation holds between the wrench and the saw? Keywords: Spatial reasoning; Mental Models; complexity; computational framework Introduction Everyday spatial reasoning is strongly connected to the extensive use of spatial relations which locate one object with respect to others. Examples of such relations include binary relations such as “to the left of”, or “in front of”, and even more complex relations like the ternary relation “in- between”. A typical reasoning problem dealing with such spatial relations is to infer relations between objects from an incomplete description of a spatial configuration of objects. Hence, in the deductive reasoning process implicit relations between a series of objects are to be inferred from assertions describing the spatial configuration. An easy example is provided by the following problem: The first four assertions are called premises, while the question refers to a possible conclusion that can be drawn from the premises. The mental model theory (MMT), proposed by Johnson-Laird and Byrne (1991), suggests that people draw conclusions by constructing and inspecting a spatial array that represents the state of affairs described in the premises. This reasoning process consists of three distinct stages: comprehension, description, and validation (see next section for explanation). According to the MMT, linguistic processes are only relevant to transfer the information from the premises into a spatial array and back again, but the reasoning process itself relies only on non- linguistic processes. A limitation of the MMT so far is that this theory does not explain the difficulty humans have with complex relations. The MMT explains the complexity of reasoning problems, but neglects the construction complexity of the models. This is, we believe, where the relational complexity theory (RCT) introduced by Halford (1993) comes into play. Different models representing a spatial description can be measured in terms of cognitive economicity (Halford, 1993; Goodwin & Johnson-Laird, 2005). But a weakness of the RCT is that it does not explain how the reasoning process itself works. In other words both theories, RCT and MMT are limited to some extent, or—more positively—can complement each other. This paper suggests an integration of RCT and MMT by providing a formal framework which is based on a particular specification of the MMT, namely the theory of preferred