Qualitative Spatial Reasoning: A Cognitive and Computational Approach

Qualitative Spatial Reasoning: A Cognitive and Computational Approach. Marco Ragni (ragni@informatik.uni-freiburg.de) University of Freiburg, Department of Computer Science D-79110 Freiburg, Germany Felix Steffenhagen (steffenh@informatik.uni-freiburg.de) University of Freiburg, Department of Computer Science D-79110 Freiburg, Germany Mental Models. For such relational reasoning problems, there exist several empirically validated effects in reason- ing: the number of models (indeterminacy effect), the form of premises (figural effect), the wording of conclusion and the preference effect. These effects are explained by men- tal model theory ( MMT ) (Johnson-Laird & Byrne, 1991; Johnson-Laird, 2001). According to the MMT , linguistic processes are relevant to transfer the information from the premises into a spatial array and back again, but the reason- ing process itself fully relies on model manipulation only. A mental model is an internal representation of objects and rela- tions in spatial working memory, which matches the state of affairs given in the premises. The semantic theory of mental models is based on the mathematical definition of deduction, i.e. a propositional statement ϕ is a consequence of a set of premises P , written P |= ϕ, if in each model A of P , the con- clusion ϕ is true. Abstract In recent years a lot of psychological research efforts have been made in analyzing human spatial reasoning. Psychol- ogists have implicitly used few spatial cognitive models, i.e. models of how humans conceptualize spatial information and reason about it. But only little effort has been put into the task of identifying from an algorithmic point of view the con- trol mechanism and complexity involved in spatial relational reasoning. In this paper we extend the SRM model (Ragni, Knauff, & Nebel, 2005; Ragni & Steffenhagen, 2007) by new specifications and formalization of Baddeleys Working mem- ory model. By the resulting model CR OS we are able to ex- plain a number of new psychological effects of spatial repre- sentation and reasoning by the number of mental operations involved in solving these tasks. The discussion includes con- sequences of the formalization for the role of the central exec- utive in spatial relational reasoning. Keywords: Spatial Reasoning; Computational Modelling Introduction The ability to deal with spatial and temporal information is one of the most fundamental skills of any intelligent system and important in our everyday lives. When route descrip- tions are given, spatial information is usually contained in the description. While in engineering or physics it is most com- mon to represent spatial information quantitatively, e.g. us- ing coordinate systems, human communication mainly uses a qualitative description, which specifies qualitative relation- ships between spatial entities. But how is this information processed? Where is the focus of cognitive attention in pro- cessing qualitative information? In the following we focus on relational reasoning problems, e.g. The red car is to the left of the yellow car. The yellow car is to the left of the orange car. The yellow car is to the left of the green car. The green car is to the left of the blue car. Is the blue car (necessarily) to the right of the orange car? The statements are called premises, the cars are the terms, and the question refers to a putative conclusion. A premise of the form “The red car is to the left of the yellow car” consists of (two) objects, and a (usually binary) relation like “to the left of”. More precisely, the first object (red car) is the ”to be localized object”(LO), which is placed according to the relation (left of) of the second object (yellow car 1 ), which is the “reference object” (RO) (Miller & Johnson-Laird, 1976). 1 In the following he objects are abbreviated by R,Y, O, G and B Without having an algorithmic formalization of a cognitive model, the task of testing and improving this model seems to be rather difficult, whereas the transfer of such cognitive models to AI systems seems to be even harder. Only a pre- cise computational model, which defines parameters and op- erations, makes testable predictions. Furthermore, by using empirical data, formally specifying the role of the subsystems of a cognitive model, i.e. its store systems, it is possible to identify the necessary abilities of a computational model. In this paper we formalize and analyze a combination of the preferred mental model theory and Baddeleys working memory model. Then we show how this model (i) is able to solve relational reasoning tasks, (ii) explains empirical re- sults in the literature by the number of mental operations, (iii) report an experiment, which tests a new prediction made by the CR OS , and (iv) finally give an idea how the CR OS can help in specifying the role of the central executive (CE). State of the Art Psychological Background. For joining spatial reasoning and representation, it is necessary to specify and work out the main assumptions of mental model theory ( MMT ) and Badde- leys Working Memory Model (BWMM). The mental model theory assumes that the human reason- ing process consists of three distinct phases: The model gen- eration phase, in which a first model is constructed out of