Efficiency and Minimal Change in Spatial Belief Revision

Efficiency and Minimal Change in Spatial Belief Revision Antje Krumnack (antje.krumnack@psychol.uni-giessen.de) Leandra Bucher (leandra.bucher@psychol.uni-giessen.de) Jelica Nejasmic (jelica.nejasmic@psychol.uni-giessen.de) Markus Knauff (markus.knauff@psychol.uni-giessen.de) Experimental Psychology and Cognitive Science, Justus Liebig University Giessen, Otto-Behaghel-Strasse 10F, 35394 Giessen, Germany Abstract In the light of new information it is sometimes necessary to change existing beliefs regarding the state of the world. During such a belief revision reasoners have to decide which beliefs to retain and which ones to retract in order to regain consistency within their belief states. Drawn from the conception that spatial belief revision is based on the alteration of mental models, we discuss the influence of cognitive effort and minimal change when constructing and re- vising mental models. We offer several possibilities how min- imal change can be defined in this context and provide a com- putational model along with a cost function to describe the re- vision processes. Furthermore, we present empirical evidence for the suitability of the model. Results indicate that there is a significant influence of cognitive efficiency principles on spa- tial belief revision. Keywords: Spatial reasoning; mental models; belief revision; mental models; computational framework; cost function. Introduction Imagine two friends describe you the location of a new restau- rant. The first one says: ”The post office is to the left of the book store and the restaurant is to the right of the book store.” The second friend interrupts: ”No, the restaurant is to the left of the post office.” At which point the first one agrees: ”Yes, you are right, the restaurant is to the left of the post of- fice.” In which order do you think the three buildings are to be found? Most likely you had an initial belief about how the buildings were arranged (post office-book store- restaurant) which you then had to change in the light of the contradict- ing information. How reasoners change their beliefs when given new information as well as the underlying processes is topic of belief revision research. The investigation of spa- tial belief revision constitutes a new research topic (see also Bucher, Krumnack, Nejasmic, & Knauff, 2011). In this pa- per we want to discuss the question, what is the most efficient way to change a spatial belief? assume that the major strategy employed in spatial rea- soning is the successive construction of a ”simulation” or ”model” of the ”state of affairs”, which contains all the in- formation given concerning the situation to be considered. New information, such as a reasoning problem’s conclusion, is generated or evaluated by inspecting and varying the pos- sible models (Johnson-Laird & Byrne, 1991). There exist numerous empirical findings that spatial reasoning in par- ticular relies on the construction, inspection, and variation of spatial mental models (Knauff, Rauh, & Schlieder, 1995; Rauh, Schlieder, & Knauff, 1997; Knauff, Rauh, Schlieder, & Strube, 1998). In the case of unambiguous descriptions of a spatial situation, from which only a unique spatial arrange- ment can constructed (i.e. determinate descriptions), there is evidence that verbatim information from the premises de- scribing a determinate arrangement is not reliably retrievable from memory (Mani & Johnson-Laird, 1982), which supports the assumption that mental models rather than sentences are stored in memory. Based on these findings we propose the following three steps for the process of spatial belief revision. First, given a determinate description of a spatial arrangement, a mental model of the spatial arrangement is constructed according to the information provided. Second, given a statement that is in conflict with the information encoded in the mental model, this inconsistency has to be detected as a prerequisite for be- lief revision. We assume that inconsistency detection happens by model inspection (e.g. Knauff et al., 1995; Johnson-Laird, Legrenzi, & Girotto, 2004). Finally, the mental model is re- vised with the goal to create a model that is consistent with the new information. In the light of the above considerations we want to define belief revision as following: Definition Belief Revision: Belief revision is a model change where a proposition x, inconsistent with a given model M, is integrated into M under the requirement that the revised model be consistent and represents the information of x. Using this definition we consider the role of cognitive ef- fort and minimal change during spatial belief revision. To that goal we discuss how minimal change can be defined in this context. We provide a computational model along with a cost function to describe the revision processes and present empirical evidence for the influence of cognitive efficiency principles on spatial belief revision and the suitability of the model. Construction, inspection and revision of spatial mental models Consider the following two statements, also called premises: 1. A is to the left of B 2. C is to the right of B