Modeling mental spatial reasoning about cardinal directions.

Cognitive Science 38 (2014) 1521–1561 Copyright © 2014 Cognitive Science Society, Inc. All rights reserved. ISSN: 0364-0213 print / 1551-6709 online DOI: 10.1111/cogs.12125

Modeling Mental Spatial Reasoning About Cardinal Directions Holger Schultheis,a Sven Bertel,b Thomas Barkowskya a b

Department of Informatics, Universit€ at Bremen Faculty of Media, Bauhaus-Universit€ at Weimar

Received 16 November 2009; received in revised form 24 April 2013; accepted 10 September 2013

Abstract This article presents research into human mental spatial reasoning with orientation knowledge. In particular, we look at reasoning problems about cardinal directions that possess multiple valid solutions (i.e., are spatially underdetermined), at human preferences for some of these solutions, and at representational and procedural factors that lead to such preferences. The article presents, first, a discussion of existing, related conceptual and computational approaches; second, results of empirical research into the solution preferences that human reasoners actually have; and, third, a novel computational model that relies on a parsimonious and flexible spatio-analogical knowledge representation structure to robustly reproduce the behavior observed with human reasoners. Keywords: Spatial mental models; Spatial reasoning; Spatio-analogical representations; Computational cognitive modeling; Scalable representation structures; Orientation knowledge; Cardinal directions

1. Introduction Mental reasoning about spatial information is an essential component of intentional, goal-directed human behavior. While one can argue that spatial perception of some sort likely exists within all living creatures, it is in the capacity of reasoning about spatial information that an organism’s potentials for carrying out beneficial actions and interactions and, ultimately, for surviving are dramatically increased. The abilities to memorize spatial features of physical objects, such as their relative locations, and to use this memory to create and enhance internal representations of the environment lie at the root of spatial reasoning, and of spatial cognition in general. For a human, reasoning about space Correspondence should be sent to Holger Schultheis, Department of Informatics, Universit€at Bremen, Enrique-Schmidt-Str. 5, Bremen 28359, Germany. E-mail: [email protected]

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H. Schultheis, S. Bertel, T. Barkowsky / Cognitive Science 38 (2014)

either takes place immediately, while being embedded in a given situation and in the presence of broad sensory input, or it can be more abstract and act on previously acquired knowledge or on hypothetical settings or courses of actions. Orientation knowledge, that is, knowledge about directions between physical objects, constitutes a fundamental type of spatial knowledge. Other fundamental types pertain to topological aspects, or to distances, among others. In this paper, we deal with a specific form of orientation knowledge, namely with knowledge about cardinal directions and the mental reasoning about them. Cardinal directions are qualitative directions that are given in terms of a global reference system; for example, north, west, south, and east are the four main cardinal directions of the compass rose. Cardinal directions are often used when referring to orientation knowledge acquired from external media such as maps or through verbal or textual communication. If one draws an inference with cardinal directions, one may often get a new piece of orientation knowledge based on the composition of other already existing pieces of knowledge. For example, think of three physical objects A, B, and C. Imagine that you know that A is located north of B and that C is situated east of A. With some basic knowledge of how to use the compass rose system you can infer that C can be located northeast of B. The questions that we will address in this paper are the following: Given a number of spatial facts that may be read off from external information sources or retrieved from memory, how do people actually reason about cardinal directions? What kind of specific mental strategies are involved in letting people find a solution to a given spatial problem that involves orientation knowledge? If multiple solutions exist, which one will be found or chosen? How can we investigate the mental processes involved in constructing a solution or in choosing one among several possible ones, and how can we describe the processes adequately? Can we identify typical characteristics of mental reasoning with orientation knowledge that provide further insights into general human spatial reasoning abilities? In the following, we will present our answers to these and related questions, and we will base them on the results of both empirical and computational investigations. Our continuing, broader aim is to propose a novel, integrated representation structure and a set of computational processes that jointly explain how the human mental spatial reasoning system operates and that are in line with empirically observable characteristics. We believe that, with respect to this broader aim of explaining how the human mental spatial reasoning system operates, the explorations into reasoning with cardinal directions and the integrated, concise descriptions of its functioning that we offer within this article address important pieces of the overall puzzle. The remainder of this paper is structured as follows: Section 2 offers a discussion of analogical representations in mental spatial reasoning (of mental models, in particular) and looks at existing descriptions of reasoning with cardinal directions in artificial intelligence and cognitive modeling. The section concludes by observing that, for reasoning problems with more than one valid solution, humans frequently seem to prefer certain valid solutions to others. We ask why this is so and, in Section 3, reflect on possible preferences in mental reasoning problems that involve cardinal direction knowledge. We present a set of theoretical considerations as to which preference patterns one may expect

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between entities is given, it is often not possible to unambiguously derive new direction relations. As an illustration, consider the following reasoning problem about direction relations between three towns, A, B, and C: Suppose one knows that (a) A is located to the west of B; and that (b) C is to the northeast of A. Which direction relation holds then between C and B? Using a sector direction system (DS) with eight cardinal directions (see Fig. 1c), four of the eight directions are in fact correct answers to this question: If equal distances are assumed, C is northwest of B (Fig. 2a); if the distance between A and C is considerably shorter than the distance between A and B, C is west of B (Fig. 2b); if the distance between A and C is a bit longer than the distance between A and B, C is north of B (Fig. 2c); if the distance between A and C is considerably longer than the distance between A and B, C is northeast of B (Fig. 2d). This means that, given only our two premises, the direction relation between C and B is not unambiguously specified. In other words, the reasoning problem is underdetermined. Similar to our example, most spatial reasoning problems do not fully specify all relations between all entities and thus the provided information is in line with several different spatial situations. However, by definition (e.g., Johnson-Laird, 1989), a mental model represents a single situation (i.e., in our case: a single geometric instantiation of the premises). Accordingly, every possible interpretation of a given problem (such as the four different situations of the three-cities problem, shown in Fig. 2a–d) each corresponds to one possible mental model. If a human reasoner—in solving such a problem—wants to keep track of or explore all possible situations, she needs to maintain every single interpretation as a separate mental model. Particularly, she needs, in principle, to maintain these mental models in WM. Taking into account that the number of possible interpretations for a spatial reasoning problem may, according to Knauff, Schlieder, and Freksa (2002), easily exceed the number of 100, it is not possible that a human reasoner can maintain all mental models in WM. Consequently, the question arises of how humans, given their memory limitations, cope with such huge amounts of information and, put differently, how humans are able to solve—and usually successfully solve—spatial problems. A number of studies (e.g., Jahn, Knauff, & Johnson-Laird, 2007; Knauff, Rauh, & Schlieder, 1995) indicate that humans handle high information load in spatial cognition by building just one specific mental model at a time to represent the spatial problem (a)

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Fig. 2. The three-cities problem with cities A, B, and C. Possible directions of C as seen from B, given that A is known to be west of B and that C is known to be northeast of A. (a) C is northwest of B; (b) C is west of B; (c) C is north of B; (d) C is northeast of B.

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3.1. Scope To make an analysis of existing preferences feasible, we will focus our efforts on a subset of all—in principle—possible cardinal direction reasoning situations. This focusing involves the following three aspects. First, we will consider eight different cardinal directions: north, northwest, west, southwest, south, southeast, east, and northeast. Although humans are, in principle, able to reason about cardinal directions on a finer scale (e.g., using northnortheast), we will restrict our study of preferences in reasoning about cardinal directions to these eight directions. The main reason for this restriction is that previous research has found that humans often use the four main cardinal directions (Gugerty & Brooks, 2004; Loftus, 1978), sometimes further refined to produce the mentioned eight direction relations (Huttenlocher, Hedges, & Duncan, 1991; Tversky & Schiano, 1989). Second, the reasoning problems taken into account will each consist of two premises about cardinal directions between three entities, and a third cardinal direction relation will need to be inferred. More formally, the considered problems have the form A dirAB B, A dirAC C, B ? C, where the direction relations between A and B and between A and C are given, and that between B and C needs to be inferred; that is, we are considering deductive reasoning problems (also called three-term series problems; Johnson-Laird, 1972). Third, for problems of the scheme in scope here, preferences will be examined only for those problems in which the two lines AB and AC form an angle (BAC) of 45° (e.g., A east B, A southeast C) or of 90° magnitude (A east B, A south C). Although these restrictions narrow down the group of the considered reasoning situations, they seem to be uncritical with respect to the aim of gaining a deeper understanding of the basic representation structure underlying human deductive cardinal direction reasoning. Regarding the first restriction, as already mentioned, the eight considered directions seem to constitute the set of directions typically used by human reasoners. The second and the third restriction can be seen as a concentration on the more elemental reasoning steps: It seems sensible to assume that reasoning about cardinal directions that goes beyond the considered problem types can be reduced to a succession and/or a combination of such elemental reasoning steps. Consequently, the class of spatial reasoning problems that we are going to address in the following will likely allow us to observe mechanisms of preference formation that are also active in the majority of deductive cardinal direction reasoning problems with which humans may be confronted. 3.2. Preferences It seems reasonable to assume that preferences in mental model construction in general depend, among others, on the amount of cognitive effort respectively required by the construction of the preferred and the alternative models. That is, the cognitive effort required to construct a preferred model can be expected to be less than that required for constructing a non-preferred one. Based on this assumption, we conjecture that preferred mental


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is, representations scale to the current task demands. Furthermore, evidence from psychological and artificial intelligence research (reviewed in Schultheis et al., 2007) indicates that mental representations of spatial information are best viewed as being composed of several distinct, knowledge type-specific representation structures. Such knowledge type-specific representation structures contribute to the scalability of spatial representations. If a certain situation provides or requires only knowledge about orientations between objects, only a representation structure specialized for representing orientation knowledge will be employed. If the consideration of further types of knowledge such as topology becomes necessary, further representation structures that are specialized for representing topological knowledge will be added to the overall spatial representation. Thus, knowledge type-specific representations constitute basic building blocks of the overall spatial representation. They allow the demand-triggered scaling of the overall representation to simpler as well as more complex spatial situations. In this article, the focus is on one of the basic representational building blocks, a representation structure for cardinal direction knowledge. As such a representation structure holds only spatial knowledge about orientations (i.e., in particular, no visual information), we assume that it enables instances of SMMs. Before we will start to describe our own modeling approach, the following three sections will give an overview about reasoning with cardinal directions, first, from an artificial intelligence point of view; second, regarding existing models; and, third, with respect to a core property of human spatial reasoning. 2.3. Reasoning about cardinal directions in artificial intelligence When acquiring knowledge about a spatial relation, humans usually abstract from precise metric values. For example, in the case of a cardinal direction between two locations, people will normally not represent the exact angle of the direction to a global reference direction (e.g., north), but instead conceptualize the direction relation in a more abstract form. In mental reasoning contexts, specific instances of cardinal direction knowledge (e.g., the direction of Rome, Italy, with respect to Paris, France) are usually held symbolically with respect to a given schema rather than as a precise numerical, angular value: Rome is located southeast of Paris. Due to the underlying schematization process, the direction values that may instantiate the relation between two locations are typically grouped into meaningful direction categories: They are expressed in the form of qualitative spatial relations (Hernandez, 1994). In artificial intelligence, qualitative spatial relations are formally dealt with in qualitative spatial reasoning (QSR; see e.g., Cohn, 1997; Vieu, 1997, for an overview). Orientation knowledge (of which cardinal direction knowledge is a specific case) plays a central role in QSR, in addition to other types of spatial knowledge such as topological, distance, or shape knowledge. In the following, we will focus on cardinal direction relations as part of extrinsic, geocentric reference systems that permit expressing the common directional distinctions of the compass rose. Other types of orientation relations are used in formal systems proposed by Freksa (1992), Schlieder (1993), R€ohrig (1994), or Zimmermann and Freksa (1996), among others.


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financial compensation of €8 per hour. The participants’ ages ranged from 20 to 29 years, with a mean age of 24.1 years. 6.3.2. Procedure The experiment consisted of a learning phase followed by a test phase. The learning phase was identical to the learning phase of Experiment 1. The test phase comprised 52 reasoning trials, all of which were four-term series problems. These 52 trials consisted of 4 practice trials, 16 prediction-relevant 45° problems, 16 additional 45° problems, and 16 90° problems. The four practice trials were meant to familiarize participants with the reasoning task and were always the first four trials each participant worked on. The remaining 48 trials were presented in randomized order. Each trial started with a screen displaying the first premise in the form of a verbal statement. Participants were instructed to press the “space” key on a standard “QWERTZ” keyboard as soon as they had understood the premise. After pressing “space,” the first premise disappeared and a second premise was shown. This procedure repeated until the participant had been seen all three premises (e.g., C is east of R; R is northwest of J; R is southwest of K). After the third premise, a question was presented that asked for a previously not specified cardinal direction between two entities mentioned in the premises (e.g., Viewed from C: where can J be located?). Participants had to specify the location of one of the two objects with respect to the other (e.g., J with respect to C) as in the first experiment. As the arrangement of the numbers on the screen did not have a significant effect in Experiment 1, numbers were circularly arranged for all participants. 6.3.3. Materials All employed reasoning problems comprised three premises that introduced three cardinal direction relations between four entities such that the direction of one entity to all other entities was specified. For describing the experimental problems, we will assume that the entity that appears in all premises is labeled A. The 16 prediction-relevant problems were constructed starting from the 45° problems employed in Experiment 1 (Fig. 3): Each of the original 45° problems was extended by a third premise that contained a fourth entity. In particular, the cardinal direction between the fourth entity and A that was introduced by the new premise was identical to the main cardinal direction that was already present in the original problem. Put differently, each prediction-relevant problem specified that two different entities were in the same main cardinal direction relation to A. Such extending the original 45° problems yielded eight prediction-relevant 45° problems. To obtain more prediction-relevant data, these eight problems were presented twice (in different surface form, see below) to each participant resulting in 16 prediction-relevant problems per participant. The task of the participants was to infer the cardinal direction relation between one of the two entities that are related by a main cardinal direction to A to the entity that is related by an intermediate direction to A. Importantly, the inference question always involved that entity, which lay in a main cardinal direction from A, that was introduced in the later premise.


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QSR that deal with spatially extended entities frequently require shape abstraction primitives such as convex-hull operators (Randell, Cui, & Cohn, 1992). 2.4. Existing models of reasoning about cardinal directions We will now have a look at a few existing computational approaches to modeling human mental reasoning about cardinal directions. Established cognitive architectures employed for symbolic modeling tasks, including ACT-R (Anderson et al., 2004) and Soar (Newell, 1990), are typically deficient with respect to two characteristics that are crucial for cognitively plausible modeling of spatial reasoning (e.g., Anderson & Lebiere, 2003): First, they do not possess explicit facilities for spatial or VMI-related representation structures and processes in general and, second, they do not provide spatio-analogical representation and reasoning techniques. While recent efforts regarding both ACT-R (Gunzelmann & Lyon, 2011) and Soar (Lathrop, Wintermute, & Laird, 2011) address the former issue, the latter still awaits sufficient attention. It seems at least not wholly unrelated to these two deficits that, based on the established architectures, not many attempts have been made to date to adequately model mental reasoning with cardinal directions. Gugerty and Rodes (2007) provide us with a description of one of the few existing approaches to extend ACT-R by more appropriate WM structures for reasoning with cardinal directions: Their model enhances ACT-R’s memory structures by adding a simple visual short-term memory (VSTM) buffer equipped with slots that can hold coarse egocentric location information. The model’s purpose is to describe how humans determine cardinal directions between two objects in the environment based on spatial information conveyed by a map-like display. Inside the VSTM, spatial information can be processed in a basic spatio-analogical manner as information can be shifted from one memory slot to (spatially) neighboring slots. Ragni, Knauff, and Nebel (2005) present another spatio-analogical approach to modeling spatial mental reasoning. While their SRM (spatial reasoning by models) system does not explicitly deal with cardinal directions, but instead focuses on linear ordering information, it offers explanations of human spatial reasoning based on the theory of SMM (Johnson-Laird, 1983). Knowledge within SRM is stored within a two-dimensional array, which, to some extent, permits a spatio-analogical modeling of mental reasoning about spatial arrangements, such as among objects in tabletop scenarios. 2.5. Underdeterminacy and preferences in mental spatial reasoning Spatial reasoning problems often come in the form of a set of spatial constraints (e.g., given in verbal formats) that frequently do not unambiguously specify a single spatial situation. If the problem only weakly constrains the space of possible solutions, multiple situations exist that each satisfies a reasoning problem’s set of constraints. This property of spatial problems is particularly prominent in reasoning about directions. For instance, Frank (1996) observes that, when only direction but no distance information

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between entities is given, it is often not possible to unambiguously derive new direction relations. As an illustration, consider the following reasoning problem about direction relations between three towns, A, B, and C: Suppose one knows that (a) A is located to the west of B; and that (b) C is to the northeast of A. Which direction relation holds then between C and B? Using a sector direction system (DS) with eight cardinal directions (see Fig. 1c), four of the eight directions are in fact correct answers to this question: If equal distances are assumed, C is northwest of B (Fig. 2a); if the distance between A and C is considerably shorter than the distance between A and B, C is west of B (Fig. 2b); if the distance between A and C is a bit longer than the distance between A and B, C is north of B (Fig. 2c); if the distance between A and C is considerably longer than the distance between A and B, C is northeast of B (Fig. 2d). This means that, given only our two premises, the direction relation between C and B is not unambiguously specified. In other words, the reasoning problem is underdetermined. Similar to our example, most spatial reasoning problems do not fully specify all relations between all entities and thus the provided information is in line with several different spatial situations. However, by definition (e.g., Johnson-Laird, 1989), a mental model represents a single situation (i.e., in our case: a single geometric instantiation of the premises). Accordingly, every possible interpretation of a given problem (such as the four different situations of the three-cities problem, shown in Fig. 2a–d) each corresponds to one possible mental model. If a human reasoner—in solving such a problem—wants to keep track of or explore all possible situations, she needs to maintain every single interpretation as a separate mental model. Particularly, she needs, in principle, to maintain these mental models in WM. Taking into account that the number of possible interpretations for a spatial reasoning problem may, according to Knauff, Schlieder, and Freksa (2002), easily exceed the number of 100, it is not possible that a human reasoner can maintain all mental models in WM. Consequently, the question arises of how humans, given their memory limitations, cope with such huge amounts of information and, put differently, how humans are able to solve—and usually successfully solve—spatial problems. A number of studies (e.g., Jahn, Knauff, & Johnson-Laird, 2007; Knauff, Rauh, & Schlieder, 1995) indicate that humans handle high information load in spatial cognition by building just one specific mental model at a time to represent the spatial problem (a)

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Fig. 2. The three-cities problem with cities A, B, and C. Possible directions of C as seen from B, given that A is known to be west of B and that C is known to be northeast of A. (a) C is northwest of B; (b) C is west of B; (c) C is north of B; (d) C is northeast of B.


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situation, rather than all possible models. For any given situation, there seems to be a pronounced preference for one (or a few) particular model(s) among all possible ones. Such models have been termed preferred mental models and are assumed to (a) arise from the inclination to mentally process information in an efficient way (i.e., preferred mental models are thought to be easier and/or faster to build than other models) and (b) constitute one major knowledge processing strategy in enabling humans to consistently and successfully reason about space even when problems are weakly constrained. The tendency to reason with preferred mental models thus seems to be a crucial, integral aspect of human spatial cognition and, therefore, should be a similarly integral aspect of any computational model of spatial reasoning about directions. As each mental model supports a specific conclusion, preferred conclusions have commonly been used to identify preferred mental models. Research on preferred mental models in spatial cognition has only comparatively recently begun to be carried out in a systematic fashion and until now has only touched on few selected areas of reasoning about spatial information. Although it might be reasonable to conjecture from the existing studies that reasoning with preferred mental models is a general phenomenon and also does play a vital role in reasoning with cardinal directions, no experiments have so far been reported in support of such conjecture. In addition, even if this conjecture were accepted, it remains an open question as to which of all possible models are in fact the preferred ones, and if the preferences existed solely within or also across individuals. Yet to know which models are preferred is of decisive importance for an evaluation of accurateness of computational models: Only if the computational model prefers the same mental models as humans usually do, and under the same conditions, the model may be assumed to be cognitively plausible. Consequently, a main focus of our subsequent report is on preferred mental models: theoretically, empirically, and with respect to computational modeling.

3. Which model preferences are to be expected and why? We have so far presented arguments, some based on analogical conjectures from related domains, which suggest that human mental reasoning about directional knowledge may be influenced by model preferences. If this holds to be true, one should find preference effects in data of human performance in directional reasoning tasks. In the following, we will again focus on reasoning tasks that involve knowledge about cardinal directions between objects. In particular, we will look for answers to these three questions: (a) Can preference effects be found?; if so: (b) Which models are actually preferred?; and (c) How and when are model preferences formed? The current section details preferences that can be expected to exist in reasoning about cardinal directions and theoretical considerations, which give rise to these expectations. Section 4 then presents an empirical study conducted to validate our assumptions. Finally, in Section 5, we will address the third question and report on a computational model that describes and explains mental processing during reasoning about cardinal directions.

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3.1. Scope To make an analysis of existing preferences feasible, we will focus our efforts on a subset of all—in principle—possible cardinal direction reasoning situations. This focusing involves the following three aspects. First, we will consider eight different cardinal directions: north, northwest, west, southwest, south, southeast, east, and northeast. Although humans are, in principle, able to reason about cardinal directions on a finer scale (e.g., using northnortheast), we will restrict our study of preferences in reasoning about cardinal directions to these eight directions. The main reason for this restriction is that previous research has found that humans often use the four main cardinal directions (Gugerty & Brooks, 2004; Loftus, 1978), sometimes further refined to produce the mentioned eight direction relations (Huttenlocher, Hedges, & Duncan, 1991; Tversky & Schiano, 1989). Second, the reasoning problems taken into account will each consist of two premises about cardinal directions between three entities, and a third cardinal direction relation will need to be inferred. More formally, the considered problems have the form A dirAB B, A dirAC C, B ? C, where the direction relations between A and B and between A and C are given, and that between B and C needs to be inferred; that is, we are considering deductive reasoning problems (also called three-term series problems; Johnson-Laird, 1972). Third, for problems of the scheme in scope here, preferences will be examined only for those problems in which the two lines AB and AC form an angle (BAC) of 45° (e.g., A east B, A southeast C) or of 90° magnitude (A east B, A south C). Although these restrictions narrow down the group of the considered reasoning situations, they seem to be uncritical with respect to the aim of gaining a deeper understanding of the basic representation structure underlying human deductive cardinal direction reasoning. Regarding the first restriction, as already mentioned, the eight considered directions seem to constitute the set of directions typically used by human reasoners. The second and the third restriction can be seen as a concentration on the more elemental reasoning steps: It seems sensible to assume that reasoning about cardinal directions that goes beyond the considered problem types can be reduced to a succession and/or a combination of such elemental reasoning steps. Consequently, the class of spatial reasoning problems that we are going to address in the following will likely allow us to observe mechanisms of preference formation that are also active in the majority of deductive cardinal direction reasoning problems with which humans may be confronted. 3.2. Preferences It seems reasonable to assume that preferences in mental model construction in general depend, among others, on the amount of cognitive effort respectively required by the construction of the preferred and the alternative models. That is, the cognitive effort required to construct a preferred model can be expected to be less than that required for constructing a non-preferred one. Based on this assumption, we conjecture that preferred mental


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models in reasoning about cardinal directions (a) are prototypically triangular and (b) include one of the four main directions as the inferred direction when possible. Each of these points will be made explicit in the following paragraphs. 3.2.1. Triangularity Any mental model of a cardinal direction reasoning problem of the types described above will correspond to a triangle in which the objects involved in the problem (e.g., the cities, in our example) are the corners, and in which the lines that represent the direction relations between the objects are the sides of the triangle. Yet, as Fig. 2 shows, the triangles resulting from constructing specific mental models may be notably different from one another. In particular, the four triangles differ in the degree to which they can be considered to be prototypical triangles. Starting from early childhood, triangle instances are considered better examples of their category, if they are isosceles and their height roughly equals their width (Levenson, Tirosh, & Tsamir, 2011; Martin, Lukong, & Reaves, 2007). Degenerate triangles constitute the most extreme deviation from these properties of prototypical triangles. Thus, models corresponding to situations b and d of Fig. 2—being very close to degenerate triangles—instantiate less prototypical triangles than models corresponding to situations a and c, which both exhibit the characteristics usually associated with prototypical triangles. Taking into account that prototypical members of a category have been shown to be processed faster than atypical members (Rosch, 1975), it seems reasonable to assume that mental models resulting in a prototypical triangle are processed (e.g., constructed) faster than mental models resulting in a non-prototypical triangle. As a consequence, mental models that correspond to situations similar to a and c in Fig. 2 should be faster, that is, more efficient, to mentally construct than models corresponding to situations b and d. Thus, our first prediction is that preferred mental models will correspond to situations with prototypically shaped triangles. 3.2.2. Main cardinal directions Our second prediction regarding the precise form of preferred mental models is based on two observations. First, as already mentioned above, humans tend to use the four main cardinal directions (north, south, etc.) more often than any of the intermediate directions such as northeast (Gugerty & Brooks, 2004; Loftus, 1978). Second, main cardinal directions are processed faster than intermediate directions. Evidence for a preference for the main direction axes in human information processing can be found as early as in neural responses in primary visual cortex (e.g., Bauer & Dow, 1989; DeValois, Yund, & Hepler, 1982). A preference for the main axes over intermediate axes seems to be deeply rooted in the human information processing apparatus. Combining both observations, we propose that the types of the direction relations available as conclusions are one factor influencing the preferred conclusion: Main direction relations are preferred over more intermediate direction relations. It is important to note that the two principles proposed to be governing preferred mental model construction do not unambiguously specify a single mental model in each reasoning situation. To achieve an unambiguous specification, one needs to assign priorities

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to the principles of triangularity and preference for main directions. We assume that, in case of conflict, the preference for more typical triangles will possess precedence over the preference for main cardinal directions. This last assumption concludes our theoretical considerations of possible origins for preferences in reasoning about cardinal directions. In the next section, we will present an empirical investigation that puts these considerations to a test.

4. Experiment 1: Preferences in reasoning about cardinal directions As argued in the preceding sections, there are good reasons to suspect that preferred mental models play a role in human reasoning with spatial problems that involve knowledge about orientations between objects, including knowledge about cardinal directions. In this section, we describe an experiment that addresses preferences in configurations with three objects; two binary cardinal directions are given as premises and the third direction needs to be inferred. In line with the considerations given above, we restrict the experimental investigation to reasoning problems in which the two given directions form an angle of 45° or 90° magnitude. Consequently, we will investigate the 16 problems that are schematically displayed in Fig. 3. The first eight problems possess 45° angles and can be constructed by selecting all neighboring direction relation pairs of the eight-sector model (e.g., (east, northeast), (northeast, north), etc.). Similarly, the latter eight problems possess angles of 90° and can be constructed by selecting all direction relation pairs in the eight-sector model that have exactly one direction between them (e.g., (east, north), (northeast, northwest), etc.). The experiment’s aim is to examine whether, and if so, which, preferred mental models in human direction reasoning exist for these 16 problems. The experimental method

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Fig. 3. Schematic illustrations of the 16 types of reasoning problems used in the experiment.


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follows previous studies that investigated preferred mental models for other types of spatial knowledge (e.g., Jahn et al., 2007; Knauff et al., 1995; Rauh et al., 2005). 4.1. Method 4.1.1. Participants Seventeen undergraduate and 13 graduate students of various degree programs at the University of Bremen, 15 female and 15 male, volunteered to take part in the experiment. Each received a financial compensation of €8 per hour. The participants’ ages ranged from 22 to 38 years, with a mean age of 28.4 years. 4.1.2. Procedure The experiment started with a learning phase, which ensured that the participant understood the cardinal directions as used in the subsequent test phase. Each trial of the learning phase consisted of a statement, a question, and a feedback screen. On the first screen, a verbal statement regarding the direction relation between two entities (e.g., X is southeast of Y)1 was presented. After the participant indicated that she had understood and memorized the statement by pressing a specific button on a keyboard, the statement disappeared and the second screen was shown. On the second screen, the participant was asked to determine—by selecting one of a number of keys—the screen position of one of the given entities with respect to the other. To this end, the reference entity was displayed at the center of the screen surrounded by the numbers 1–8 in a counterclockwise order, with number 1 positioned directly above the reference entity. Each number represented one of the eight directions (1 = north, 2 = northwest,. . ., 8 = northeast), and the subjects were instructed to press the number key that corresponded to the direction relation of the free entity to the reference entity as given in the verbal statement from the first screen. Depending on whether this direction was given correctly, the third screen simply either showed the message correct! or wrong! (with an indication of the position that would have been correct). Each of the eight possible directions was presented at least two times and was assumed to have been sufficiently learned/understood by the participant once she had it correctly determined on two successive occurrences. The learning phase ended as soon as all eight directions had been sufficiently learned. After the learning phase, the test phase began. The test phase consisted of 140 trials, respectively organized into three blocks of 12, 64, and 64 trials. As the first block was included in the test phase with the only aim of familiarizing the participants with the task, responses to it were later excluded from statistical analysis. Each trial began with a screen on which the problem was presented in the form of two verbal statements on the direction relations between two entities each, and three entities all together (e.g., A is east of G, G is southwest of N) and a corresponding question (e.g., Viewed from A: where can N be located?). Participants were instructed to consider this problem description until they thought that they had found a solution to it.2 As soon as they indicated this by pressing the space bar, the answer screen appeared. On this screen, participants had to identify the position of one of the entities with respect to the other (e.g., N with respect to A) just

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as they did in the learning phase. After they had pressed an answer key, that is, any number key between 1 and 8, the answer screen was replaced by a transition screen with the message: To work on the next problem, press the space bar. Subjects were allowed to take breaks whenever the transition screen was shown. All trials within each block were presented in random order. 4.1.3. Materials For each of the 16 problems under investigation (see Fig. 3), several different ways exist of presenting the problem to the participants. Consider, for instance, problem 1 of Fig. 3: One way of presenting this situation verbally—assuming that entity A is at the intersection of the two lines, entity B at the lower right, and entity C at the upper center —is to state that A is west of B and A is southwest of C. Alternatively, the problem can be presented as B is east of A, C is northeast of A. This change in relational terms that are used in the premises arises from a change of the order in which the entities are named and, thus, we will refer to this property of a problem’s presentation as entity order. Although, from a formal point of view, different entity orders describe the same problem, it seems unjustified to assume (without supporting data on this) that humans will always construct the same mental model for formally identical problems, regardless how they are presented. Also, the spatial arrangements of both the numerical answer keys for the positional questions and of the numbers on the screen surrounding the reference entity may potentially influence preferences for mental model. To explicitly investigate effects of all these factors on preferences, we controlled them in the following ways: First, we presented each of the 16 problems with all four possible entity orders in the premises. Accordingly, problem 1, for instance, was presented as (a) B is east of A, A is southwest of C; (b) A is west of B, A is southwest of C; (c) B is east of A, C is northeast of A; and (d) A is west of B, C is northeast of A to every participant. Second, the arrangement of the numbers on the answer screen was varied across participants. Half of the participants saw a rectangular number pattern and the other half saw a circular pattern. Finally, participants did not give answers through a (rectangularly arranged) numerical keypad, nor did they even see one; they instead used the (linearly arranged) 1–8 number keys of a standard QWERTZ keyboard. Taking into account the four different entity orders for each of the 16 problems produces 64 distinct order-problem combinations, all of which were presented to every participant. Because of the experiment’s aim to investigate model preferences, it seemed sensible to present each of the 64 combinations twice. Assuming for the moment that model preferences exist in fact, we would expect preferred models to afford a correct conclusion, that is, to contain as the third, inferred relation, one of the actually physically possible relations. For this reason, it is the set of valid answers that will be of most value for an investigation of potential preferences. As the participants were expected to make some errors (they did, indeed; see below), it seemed advisable to present each combination twice to keep the number of values missing due to errors within acceptable range for the subsequent data analysis. Accordingly, both the second and the third blocks each comprised all 64 order-problem combinations. To avoid memory effects, letters used for


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designation of the entities in the second block differed from letters used in the third block. 4.1.4. Design Taken together, the above considerations yielded a 2 (arrangement) 9 4 (entity order) 9 16 (problem) design. The first factor (arrangement) was between subjects (15 subjects per factor level), and the second and third factors were within subjects. 4.1.5. Hypotheses On the basis of our predictions above, we hypothesized that preferences for mental models would exist for each of the 16 problems. Specifically, we assumed that (a) models representing the situation in the form of a prototypical triangle and (b) models featuring one of the four main cardinal directions as the inferred relation would be preferred. (c) The preference for prototypical triangles would possess precedence over the preference for main cardinal directions. This set of hypothesized preferences unambiguously specifies a unique preferred model for each investigated reasoning problem. All of these uniquely predicted preferred models are depicted in Fig. 4. 4.2. Results Given the difficulty of the task, accuracy was generally high (ranging from 70% to 100% with a mean of 93%). Nevertheless, participants made incorrect inferences. As argued above, only correct answers help in identifying potentially preferred mental models; all error trials were thus excluded from further analyses. For every participant and every order-problem combination, the first correct answer was entered into the analyses.3 If, for instance, participant P gave a correct answer to some order-problem combination C during the second block, this answer was used for statistical analysis. If, however, P’s 1

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answer to C was wrong in the second but correct in the third block, P’s answer from the third block was analyzed. Each answer was classified as either conforming to our hypotheses or as not conforming to our hypotheses, resulting in a dichotomous-dependent variable. Although this procedure yielded a nearly complete data matrix (64 order-problem combinations 9 30 subjects), there were still some missing values. These missing values were replaced by the mode of (classified) answers to the same problem across different orders. All of the analyses reported in the following are based on the resulting data matrix. To check whether the arrangement of numbers on the answer screen had any effect on the (existence of) preferences, we first compared the two arrangement groups employing an extension of the v2 test statistic to repeated measure designs (Tideman, 1979). The test yielded v2(1) = 0.1839, which indicates that preferences in the two groups were not significantly different (p > .65). As a result, we combined the data of both groups to investigate the preferential structure for the different order-problem combinations. If no preferences existed, each of the possible correct conclusions should be equally likely. That is, for a 45° problem (problems 1–8 in Fig. 3), one would expect to observe the particular conclusion predicted by our hypotheses to be produced by about 25% of the subjects and, analogously, by about 33% of the subjects for a 90° problem (problems 9–16 in Fig. 3). Fig. 5a shows for each of the 64 order-problem combinations the respective numbers of subjects who reported the conclusion in line with our hypotheses and of those who constructed different, correct conclusions. Clearly, in all 64 combinations, the large majority of reported conclusions were in line with our hypotheses: At least 25 of 30 subjects (i.e., 83.3%) reported hypotheses-conform conclusions in each of the 64 combinations. Binomial tests employed for each order-problem combination revealed that the observed frequencies deviated significantly from the frequencies expected if no preferences existed (all ps < 3.826 9 10 8). Even taking the large number of tests into account, the p value of all 64 tests together is guaranteed to be below .00001. To further illustrate how participants performed on different problems, entity orders, and problem types (45° and 90°), percentages of hypothesized preferred conclusions broken down according to these factors were computed across all correct responses. As can be seen in Fig. 6, the strong preference for the hypothesized conclusion is evident across all problems, entity orders, and problem types. Consequently, preferences for specific mental models are evident for all order-problem combinations. Furthermore, the preferences exhibited were the same regardless of the concrete order-problem combination analyzed and, in particular, were in accord with our prior hypotheses regarding preferences; that is, prototypical triangles and main directions are preferred, with precedence in this order. In summary, the preceding analyses yielded three main results: First, humans prefer certain mental models to others in reasoning about cardinal directions. Second, these model preferences stem from preferences for prototypical triangles and for reasoning with main directions. Third, the observed preferences are quite robust and remain the same regardless of the type of problem, the order in which the relevant entities are presented, or the way in which the cardinal directions are visualized in the external world.

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Fig. 5. Experimental (a) and model (b) results for the 64 order-problem combinations. Portions of bars shown in dark color indicate a choice of predicted preferred solutions. Portions of bars in light color indicate a choice of other correct solutions.

5. The model Having established the existence and nature of preferences in human reasoning about cardinal directions, we developed and implemented a computational model of the representations and processes involved in human reasoning about cardinal directions. Model construction was guided by the principles underlying scalable representation structures, and relying on these principles is at the core of the model’s ability to explain the observed reasoning behavior. The model’s properties are described and discussed in more detail in the remainder of this section. 5.1. Standard models of mental representation of directions Integrated mental representations of directional knowledge can be conceptually and computationally modeled in many ways. Most frequently, one-, two-, or three-dimensional arrays have been suggested, with entities stored in individual array cells. Array cells may have standard sizes, or multiples thereof (Glasgow & Papadias, 1992); they may be organized in rectilinear (Khenkhar, 1991), circular (Funt, 1980), or hexagonal grids; or otherwise. For inferring of direction relations between entities, an external reference system is imposed onto the array (e.g., north is up), effectively transposing the problem to an inferring of direction relations between array cells. A main advantage of array-based models lies in their conceptual simplicity, which means that implementing them in computer simulations is rather straightforward. Regarding model validity, the psychological literature has frequently advocated an existence of one or several spatial arrays in (spatial) WM, which store knowledge, for example, during linear or two-dimensional relational reasoning (Arend et al., 2003), or in mental maps of virtual environments (Johns & Blake, 2001). However, it remains unclear whether in these cases, array structures are employed as models for mental representations because

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they truly are the most cognitively adequate structure to be used or, simply, because they are the most adequate structure available among a standard repertoire of representational metaphors in computation. For the modeling of directional reasoning, a fixed resolution of arrays of any sort eventually leads to problems of expressiveness as it sets the degree of direct neighborhood for an array cell. Consequently, for a fixed neighborhood topology, only a number of directions up to the number of neighboring cells can be conceptually distinguished (see Fig. 7 for an example). At a quick glance, it may appear that having too few neighbors to distinguish direction relations could be easily patched by dynamically shifting to a higher resolution neighborhood topology. However, as existing array-based models usually integrate directional, topological, and distance knowledge within the same grid structure, a direction-induced shift to a higher resolution neighborhood often cannot be made without also changing either content or representational structure for knowledge about distance and topology.


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An array metaphor can help to derive specific mental construction principles: Criteria such as first fit or first free fit have been proposed to establish which particular array cell (s) will be filled when spatial (e.g., relational) knowledge is added (Boeddinghaus, Ragni, Knauff, & Nebel, 2006; Ragni et al., 2005). These criteria can be empirically validated. While such validations can surely help increase knowledge about mental representation structures in spatial reasoning, we maintain—based on the discussion of grid structures above—that the employed validation methodology may be employed more effectively if human behavior were in fact tested against more plausible representation structures. In the following, we will propose such a representation structure for mental reasoning with cardinal orientation knowledge. The assumption of a mental representation structure such as an array that is fixed (e.g., in terms of spatial resolution) and, at the same time, able to gradually accommodate more representational content yet introduces another problem: One can easily construct examples of reasoning situations in which the gradual filling with content of any fixed-resolution array (e.g., through an integration of spatial knowledge fragments, one at a time) will require a massive reorganization, either of content (e.g., by copying content within the array) or of array structure, once that fixed, implementation-depended capacity thresholds (e.g., based on the number of contained entities, relations, etc.) are reached. Such threshold-based reorganization currently does not seem to be suggested by human cognition research; its occurrence should otherwise be well observable in the form of systematic discontinuities in the growth patterns of mental representation structures (and, for instance, show itself through step functions in measured reaction times, error rates, etc.; cf. Bertel, Sima, & Lindner, 2009). 5.2. A circular representation structure for directions Guided by the principles underlying scalable representation structures outlined above, we developed our representation structure for direction knowledge to match the subsequent design requirements: It should (a) be as parsimonious as possible (i.e., be cognitively economic, see above); (b) follow on current behavioral and neuroscientific research on reasoning with SMM (i.e., be plausible); (c) include only directional knowledge (i.e.,

Fig. 7. Three grid-based neighborhood topologies that differ in the number of distinguishable directions. Distances can be seen to also increase with some shifts to larger neighborhoods.

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it should not be an abstracted, raster-based version of a VMI or, even less plausible, of an external picture); (d) be robust and flexible in the accommodation of additional knowledge (e.g., adding new spatial knowledge fragments should not require major reorganization of structure or content); (e) eventually allow for the integration of its content with that of other modal WM structure (e.g., of topological or distal knowledge, or with regard to mental imagery); and (f) allow for the reproduction of empirical results (e.g., regarding observed human preferences in mental model construction, as described above). In the remainder of this section, we will first detail the novel proposal for a representation structure for mental reasoning with cardinal directions. We will then describe how mental reasoning can be modeled and explained based on this structure, in particular mental reasoning with cardinal directions, and how such models can reproduce empirical results. As explained above, the proposed structure constitutes one of the basic building blocks of more comprehensive spatial representations. Accordingly, we have implemented the proposed structure as one member of an extended family of spatio-analogical representations formats (we call them SARF) that are WM components within our larger computational cognitive architecture for spatial reasoning, Casimir (Schultheis & Barkowsky, 2011). In Casimir, the individual SARF components interface with other WM, long-term memory, and perceptual components. As the component in focus here is concerned with cardinal direction knowledge, we will refer to it as SARFCD. 5.2.1. Entities In our context, an entity designates a mental concept of an object in space. For mental reasoning tasks with cardinal directions in SARFCD, an entity is conceptualized as a point (e.g., a city on a world map, or a specific house on the map of a town). Entities will be denoted as starting with an uppercase letter. In SARFCD, an entity E is modeled as an object with (a) a label (e.g., Paris); (b) an association with a currently assigned DS which specifies the set of direction relations which may involve E (e.g., the set of north, east, south, west, for a standard four-sector model); (c) a cyclic direction list (DL) with counterclockwise ordering and of length |DS| (e.g., of length 4 for a four-sector model) that holds references to objects with which E is currently in some DS direction relation; and (d) a direction root (DR), which associates a specific position within DL with a specific relation from DS (thus, by way of DL’s ordering, implicitly associating all positions within DL with respective relations from DS). See Fig. 8 for an illustration. Note that, in SARFCD, an entity is only created when knowledge about a cardinal direction relation about the entity is introduced. That is, the Paris entity depicted in Fig. 8 may be the result of SMM construction process initiated by the subsequent introduction of two knowledge fragments: (a) Paris is west of Prague and (b) Paris is north of Algiers. We will subsequently describe this construction process. 5.2.2. Direction relations In its current implementation, SARFCD provides for any hierarchical ordering of sector-based DSs in terms of resolution (see Fig. 9 for an example of ordered cardinal DSs). Upon introducing a knowledge fragment (e.g., Paris is west of Prague), the system first


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DL (direction list): (Prague, nil, nil, Algiers) DR (direction root): west Fig. 8. An exemplary SARFCD entity Paris with its properties. The two spatial knowledge fragments encoded in the direction list are, first, that Paris is west of Prague and, second, that Paris is north of Algiers.

checks whether any of the entities in the fragment already exist in the SMM. Let us assume for the moment that neither Paris nor Prague so far exists. An entity Paris gets created and its DR will be set to west. To be as parsimonious as possible, SARFCD assumes that knowledge employed in mental reasoning with direction is always encoded at the coarsest possible resolution level (i.e., it uses the coarsest fitting DS that contains the relation in question). In our current example, the coarsest system with a relation west is the two-member system with west and east, and Paris gets associated to it. The entity’s DL gets instantiated as a cyclic list of length 2 and an association with Prague is entered at the head position (see Fig. 10). In addition, a Prague entity will be created; we will later return to how a propagation of direction relations along an integrated structure serves to either implicitly or explicitly establish associations for this second entity. Upon introducing the second knowledge fragment Paris is north of Algiers, SARFCD’s direction mapping components establishes whether the new direction relation (north) is compatible with the DSs that are so far associated with the entities concerned. As this is not the case for Paris, processes operating on the hierarchy of DSs track the coarsest system in which both the entity’s new and existing relations can be expressed. Paris gets associated to this system; the DL is modified accordingly (see Fig. 8 for the result and Fig. 11 for a sketch of the gradual development of the Paris entity over a sequence of introducing the first and then the second knowledge fragment). It is important to note that the resolution of an entity’s DS is determined based on the introduced knowledge about that entity alone, and that resolution changes in one entity’s system are inconsequential for the resolutions of other entities held within the same SMM. The final, integrated configuration with all three entities and both relations is depicted in Fig. 12a. Again, note the differences in the respectively associated DSs: Each entity is associated to its coarsest fitting DS. 5.3. Reasoning with the representation structure For the SARFCD model, we assume that humans employ two main methods to infer directions between entities: a direct inference and a transitive inference. The direct inference method checks whether a direction relation between the two entities has already been explicitly introduced into the SMM. If this is the case, the relation is retrieved. In contrast, the transitive inference method tries to establish a sequence of two or more direct relations that indirectly connect the two entities in question. If such a sequence is found, a new

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Fig. 9. Three direction systems with cardinal directions in a partial ordering by resolution.

direct direction relation between the two entities is inferred by successively traversing the sequence and by pairwise composition of the relations found along the sequence. It seems cognitively more plausible to assume such a need-based, transitive inference procedure than it would be to assume an explicit, complete store of binary direction relations between all spatial entities in memory, independent of tasks. Need-based transitive inferences of directions between spatial entities have been shown to exist elsewhere (e.g., across representational levels in spatial hierarchies, cf. Stevens & Coupe, 1978). Fig. 12b shows the integrated WM structure after a direction relation between Prague and Algiers has been transitively inferred through a sequence Prague – Paris – Algiers. We already mentioned above that, in the absence of distance knowledge, a single direction relation often cannot be unambiguously inferred from the composition of two direction relations. This is indeed the case for inferring the direction relation between Prague and Algiers from a composition of the two direction relations already contained in our little example scenario. Fig. 12b indicates three valid results of this inference, while Fig. 13 displays examples of possible geometric interpretations. 5.3.1. Converse of directions For a given direction relation and a given DS, SARFCD can derive the converse relation. This is important to maintain local consistency within the integrated representation structure: If Paris is to the west of Prague, Prague should be to the east of Paris. While a host of empirical research has shown that mental representation of spatial knowledge can well be non-symmetrical (e.g., in mental representations employed in wayfinding, a distance from object A to object B does not need to correspond to the distance from B to A; Sadalla, Burroughs, & Staplin, 1980), it is clear that directions can be symmetrical. The SARFCD model thus offers an option to automatically derive and enter into the structure the converse relation of any direct relation that gets entered into the structure. Fig. 12a and b are results of processes for which such automatic generation is employed.


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We argue that a mental mapping between relations and their respective converses is probably more likely to occur in “artificial” domains, such as in reasoning with cardinal directions, than in domains where mental representations result from the exploration of spatial environments.

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H. Schultheis, S. Bertel, T. Barkowsky / Cognitive Science 38 (2014) Paris Prague Paris

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Fig. 13. Possible geometric interpretations of the transitive inference results of Fig. 12b: (1) Prague is northeast of Algiers; (2) Prague is north of Algiers; (3) Prague is east of Algiers.

5.3.2. Modeling inferences with preferences Let us now take a closer look at how SARFCD produces inferences with the same preferences in SMM construction as we have found in the inferences produced by human reasoners. We will consider the involved concrete model construction mechanisms as applied to our Paris, Prague, and Algiers example. As an initial situation, take the state of affairs represented in Fig. 12a. The direction relations between Paris and Algiers, and those between Paris and Prague are readily available in the SMM and can simply be read off. The direction from Prague to Algiers, however, is not and needs to be inferred. Inferences in SARFCD occur task and need based, when the SMM is queried for information that is not explicitly available (yet). The triggered inference process takes into account the DS in which both entities currently are. In exploiting the SMM’s analogical representation structure, SARFCD now tries to establish a triangular conclusion by adding directional information to the next available locations in the DLs of the Prague and Algiers entities (see Fig. 14, panel 2). A check of the potential conclusion for its triangularity (see also Appendix A) is then performed based on the DLs of all three involved entities. Each of the three entities’ DL now contains entries for the other two entities. Let the distance between two entities, e1 and e2, in a DL be the number of transitions that are required to reach e2 starting from e1. To check for triangularity, this inter-entity distance is computed for each of the three entity nodes and then divided by the number of directions that are distinguished in the current DS of the respective entity node. The resulting ratios are added up to yield an overall ratio that characterizes the potential conclusion. If the ratio is found to be greater than ½, the sides of the triangle do not form a match; therefore, the currently coarser one of the two entities’ DSs is refined and the triangle construction and checking process starts again. The procedure is repeated until a valid triangle emerges and the cardinal direction relation from Prague to Algiers can be simply read off. As illustrated in the case of the example in Fig. 14, the process leads to the construction of the preferred conclusion, and it does so by simply relying on and economically exploiting the spatio-analogical properties of SARFCD’s knowledge representation format. Neither do externally derived goal descriptions exist, nor are multiple conclusions considered simultaneously and the preferred selected. In fact, no explicit selection process


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Fig. 14. Inferring the preferred direction relation from Prague to Algiers. The coarser one of the two entities’ direction systems gets refined at each stage (2–5) until a triangular solution is found (6). The solution first found through this process is the preferred one.

exists; the preferred conclusion simply emerges from the SMM construction process. The preferences for prototypical triangles as well as for main cardinal directions, and the precedence relation between them, are incidental to the interplay between knowledge representations and reasoning processes in SARFCD. 5.4. Modeling results Above, we defined a set of six target properties for our model. Let us now compare SARFCD to these.

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1. The model should be as parsimonious as possible (i.e., be cognitively economic). We have shown in the last section how the use of spatio-analogical properties of the employed knowledge representation structures allowed us to create a concise, economical model that possess very little processing overhead. 2. The model should follow on current behavioral and neuroscientific research on reasoning with SMM (i.e., be plausible). Our main argument here is that the proposed structure is much more plausible than previously existing approaches as it can robustly and flexibly incorporate additional knowledge bits without the need to reorganize content or representation structure at some point. This is an important improvement over other approaches. Second, with respect to selecting circular primitives for SARFCD’s knowledge representation structure, we argue that this selection is similarly plausible, for instance, given the evidence for topographically organized cortical maps in parietal cortex that exhibit robust retinotopic mapping of remembered angles (Sereno, Pitzalis, & Martinez, 2001). 3. The model should include only directional knowledge (i.e., it should not be an abstracted, raster-based version of a VMI or, even less plausibly, of an external picture). This certainly holds for SARFCD: Only directional information is included. The model does not explicitly or implicitly assume any raster- or image-based mental representation format. 4. The model should be robust and flexible in the accommodation of additional knowledge (e.g., adding new spatial knowledge fragments should not require major reorganization of structure or content): see our answer to target (1), above. 5. As the proposed representation structure is supposed to constitute one of several basic building blocks of a more comprehensive spatial representation, the model should eventually allow for the integration of its content with that of other WM structures (e.g., of topological or distal knowledge). While this clearly constitutes a major focus for our future research, we nevertheless believe that SARFCD will turn out to be well capable of being integrated with other models due to its parsimonious design and reliance on analogical processing. 6. The model should allow for the reproduction of empirical results (e.g., regarding observed human preferences in mental model construction). SARFCD does in fact reproduce all human preferences observed in the study. To achieve a more finegrained assessment as to what extent SARFCD can account for the empirical observations (see Fig. 5a) in the cardinal direction reasoning task, we conducted 100 model runs for each of the 64 order-problem combinations. The only parameter used to fit the data was the probability4 with which the model produces conclusions that do not correspond to the preferred models indicated in Fig. 4. The best fit was achieved employing a probability of 0.05 and the model results obtained using this probability are displayed in Fig. 5b. As can be seen, the model results closely resemble the empirical results. This impression was corroborated by a v2 goodness of fit test that indicated a very tight fit of the model to the data (v2(63) = 20.15).


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6. SARFCD and arrays: Competing predictions Above we have shown (a) that there are a number of theoretical considerations that favor SARFCD over an array-like representation structure and (b) that SARFCD nicely accounts for human behavior in cardinal direction reasoning. Yet unconsidered is the question to what extent SARFCD is superior to an array-like representation structure in accounting for human behavior in reasoning about cardinal directions. The aim of this section is to address this question by empirically testing the differing predictions of SARFCD and an array-based reasoning model on a novel type of cardinal direction reasoning problem. The employed array-based model is a slightly adapted version of the SRM model proposed by Ragni et al. (2005). This model was chosen for the following reasons: First, the SRM is a directional reasoning model that can be applied to reasoning about cardinal directions in nearly its original form. Second, as SARFCD, the SRM is explicitly intended to implement spatial reasoning with mental models. Third, for the three-term series problems considered so far, the SRM predicts the same preferences as SARFCD and, thus, the SRM constitutes a competitor that is on par with SARFCD. Recently, a computational model called PRISM (preferred inferences in reasoning with SMM) has been proposed by Ragni and Knauff (2013) as a successor to the SRM. As PRISM’s mechanisms for generating inferences in deductive spatial reasoning are identical to those in the SRM, all descriptions and results expounded below for the SRM equally hold for PRISM. In the following, we briefly describe the SRM and the predictions of both models before we present a new experiment that tests these predictions. 6.1. The SRM The SRM was originally proposed to account for reasoning about the four direction relations “to the left of,” “to the right of,” “in front of,” and “behind” (Ragni et al., 2005). All related entities and the relations between these entities are represented by a spatial array. The SRM assumes that at the outset of a reasoning problem, the array is empty and that the array is gradually filled by entities as direction relations between entities become known. If, for example, the SRM first learns that A is to the left of B, it will place A into the center field of the array and B into an adjacent cell (Fig. 15b) such that the relation between the cells containing A and B represents the relation given in the premise (Fig. 16a). If the second premise states that C is to the left of B, the SRM will, starting from the cell containing B, search for an empty cell that is in the direction that corresponds to the given relation and place C into that cell (Fig. 15c). Placing a new object into the first free cell that is in the correct direction has been termed the first-freefit principle by Ragni et al. (2005) and this principle constitutes an integral part of the SRM. In adapting the SRM to reasoning about cardinal directions, we assume that relations between array cells represent eight cardinal directions, as shown in Fig. 16a, but that the

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processes working on the spatial array remain the same. Accordingly, the two premises A is southwest of B and A is west of C lead to the representation shown in Fig. 16b. Note that the relation between B and C that would be inferred on the basis of the representation in the SRM is the same as in SARFCD. 6.2. Model predictions As in the previous example, the SRM predicts the same preferences as SARFCD for all 45° problems. Predictions about preferences differ, however, if a third premise containing a fourth entity is added. Consider the information that D is east of A, for instance. If the SRM has to integrate this information into the representation depicted in Fig. 16b, the representation shown in Fig. 16c will be the result. Accordingly, the SRM will infer that the relation between D and B is that D is southeast of B. It seems important to note how the preferences of the SRM arise from the structural properties of its array representation. Though the fourth premise only contains information about the relation between two objects (A and D), integrating this information into the array also requires specifying the relation between D and C. That is, the comparatively rigid structure of the array forces the SRM to represent relations between entities for which no information has been provided or inferred. In particular, this incidental specification of the relation between D and C determines the spatial relation between D and B such that an intermediate cardinal direction is inferred by the SRM. Being based on the principles underlying scalable representation structures, SARFCD allows representing given information without being forced to also commit to relations that have not yet been specified. Provided with the three premises of the above example, the representation in SARFCD will be as shown in Fig. 17a. Utilizing its inference processes as described above (see also Fig. 14), SARFCD will infer that D is south of B when asked for the relation between these two entities (Fig. 17b). Thus, for the considered situation the SRM predicts that an intermediate cardinal direction will be inferred and SARFCD predicts that a main cardinal direction will be inferred.

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As in this example, the SRM generally predicts that adding a fourth entity (D) to a 45° problem such that two of the four entities (C and D) have the same main cardinal direction relation to a third entity (A) leads to a preference of inferring an intermediate direction between the fourth entity and the remaining entity (B), while SARFCD predicts that main cardinal directions will be the preferred inference result. In the next section, we report an experiment that investigates which of these two different predictions is more in accord with human reasoning behavior. 6.3. Experiment 2: Checking model predictions This experiment employed four-term series problems that were constructed by extending the three-term series problems employed in Experiment 1 (Fig. 3). To examine the above identified differences in model predictions it would, in principle, have been sufficient to consider only prediction-relevant 45° problems, that is, extended 45° degree problems that are of the type discussed above. However, such problems come with particular structural properties that could have potentially distorted the experimental results. Against this background we decided to counterbalance these structural properties across

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reasoning trials by embedding the prediction-relevant 45° problems in a number of additional extended 45° and 90° problems. 6.3.1. Participants Thirty-one students of various degree programs at the University of Bremen, 23 female and 8 male, volunteered to take part in the experiment. Each received course credit or a


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financial compensation of €8 per hour. The participants’ ages ranged from 20 to 29 years, with a mean age of 24.1 years. 6.3.2. Procedure The experiment consisted of a learning phase followed by a test phase. The learning phase was identical to the learning phase of Experiment 1. The test phase comprised 52 reasoning trials, all of which were four-term series problems. These 52 trials consisted of 4 practice trials, 16 prediction-relevant 45° problems, 16 additional 45° problems, and 16 90° problems. The four practice trials were meant to familiarize participants with the reasoning task and were always the first four trials each participant worked on. The remaining 48 trials were presented in randomized order. Each trial started with a screen displaying the first premise in the form of a verbal statement. Participants were instructed to press the “space” key on a standard “QWERTZ” keyboard as soon as they had understood the premise. After pressing “space,” the first premise disappeared and a second premise was shown. This procedure repeated until the participant had been seen all three premises (e.g., C is east of R; R is northwest of J; R is southwest of K). After the third premise, a question was presented that asked for a previously not specified cardinal direction between two entities mentioned in the premises (e.g., Viewed from C: where can J be located?). Participants had to specify the location of one of the two objects with respect to the other (e.g., J with respect to C) as in the first experiment. As the arrangement of the numbers on the screen did not have a significant effect in Experiment 1, numbers were circularly arranged for all participants. 6.3.3. Materials All employed reasoning problems comprised three premises that introduced three cardinal direction relations between four entities such that the direction of one entity to all other entities was specified. For describing the experimental problems, we will assume that the entity that appears in all premises is labeled A. The 16 prediction-relevant problems were constructed starting from the 45° problems employed in Experiment 1 (Fig. 3): Each of the original 45° problems was extended by a third premise that contained a fourth entity. In particular, the cardinal direction between the fourth entity and A that was introduced by the new premise was identical to the main cardinal direction that was already present in the original problem. Put differently, each prediction-relevant problem specified that two different entities were in the same main cardinal direction relation to A. Such extending the original 45° problems yielded eight prediction-relevant 45° problems. To obtain more prediction-relevant data, these eight problems were presented twice (in different surface form, see below) to each participant resulting in 16 prediction-relevant problems per participant. The task of the participants was to infer the cardinal direction relation between one of the two entities that are related by a main cardinal direction to A to the entity that is related by an intermediate direction to A. Importantly, the inference question always involved that entity, which lay in a main cardinal direction from A, that was introduced in the later premise.

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Similar to the prediction-relevant problems, the additional 45° and 90° problems were constructed by adding a new premise to the 16 reasoning problems employed in the first experiment. This new premise not only introduced a new entity but also related this entity to A with a new cardinal direction such that the 16 problems shown in Fig. 18 were obtained. Inference required determining the relation between the entity associated with the middle line in each problem depiction and an entity associated with one of the outer lines. Given the two outer lines in each problem, two inference questions were possible for each of the additional 45° and 90° problems. Presenting each of the additional problems with both of their possible inference questions resulted in 16 additional 45° and 90° problems. As in Experiment 1, many different ways of presenting the problems to the participants are possible by varying entity order and premise order. Furthermore, due to the nature of the newly constructed reasoning problems, there is a third important aspect, partly confounded with premise order, that needs to be taken into account in the scope of problem presentation. This aspect arises from the fact that two of the three premises allow answering the inference question. Thus, the position of the two premises (first and second, first and third, or second and third) that supports the required inference may also be considered an important aspect of problem presentation. A full factorial combination of all three presentation factors would have resulted in a prohibitive number of reasoning trials. Therefore, using a pseudo-random assignment of presentation modes, problem presentation was counterbalanced across all 48 experimental trials such that (a) the position of the two relevant premises was equally often (1,2), (1,3), and (2,3) and (b) each entity order appeared equally often. Balancing of premise position was only possible by complementing the prediction-relevant problems with additional reasoning problems and, in fact, this was one main purpose of including the additional problems. Remember that in prediction-relevant problems, the inference question has to refer to that entity, which lies in a main cardinal direction from A, that is introduced in the later premise. Due to this requirement, the position of relevant premises can never be (1,2) in prediction-relevant problems. Due to constraints on the premise order in the prediction-relevant problems and the desire to balance premise position, a balanced premise order could only be approximated across all trials. The approximation was generally good, yielding presentation modes in which the different premises appeared nearly equally often in all of the three positions across trials. 6.3.4. Results In line with the aim of the experiment, the analysis of the experimental results focused mainly on the prediction-relevant 45° problems. One participant did not follow instructions and was excluded from analyses. Four further participants were excluded, because it could not be ruled out with certainty that they had answered the reasoning problems by guessing: Exact multinomial tests (see, e.g., Read & Cressie, 1988) revealed that the observed frequencies of incorrect answers, answers predicted by the SRM, answers predicted by SARFCD, and answers not predicted by either model did not


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differ significantly from frequencies expected by chance for these participants (all ps > 0.1). 6.3.4.1. Accuracies: Accuracy on the prediction-relevant problems ranged from 56% to 100% with a mean of 84%. Accuracy on all reasoning trials ranged from 60% to 94% with a mean of 82%. The observed accuracies indicate that the new reasoning problems were considerably more difficult than the problems employed in the first experiment: Several participants were unable to clearly perform above chance level and those that did had lower average accuracy than the accuracy achieved in Experiment 1. In fact, accuracy in Experiment 2 is significantly worse across both the prediction-relevant problems (t(54) = 2.8109, p < .01 two-tailed) and all problems (t(54) = 5.3539, p < .0001 two-tailed) than accuracy across all problems in Experiment 1. 6.3.4.2. Preferences: Preferences were analyzed in terms of percentages of different types of correct responses. Three types of correct responses were distinguished: responses as predicted by SARFCD, responses as predicted by the SRM, and responses not predicted by any of the two models (NP). If human reasoners had no preferences, one would expect to observe responses of type SARFCD, SRM, and NP in 25%, 25%, and 50% of all cases. Fig. 19 displays the observed mean percentages for these three types of responses. As can been seen, responses of type SARFCD are significantly more frequent than expected by chance (t(25) = 6.227, p < .001 two-tailed), responses of type NP are significantly less frequent than expected by chance (t(25) = 8.787, p < .001 two-tailed), and the frequency of responses of type SRM is not significantly different from chance level (t(25) = 0.922, p > .35 two-tailed). Thus, the conclusion that was predicted to be preferred by SARFCD is the only conclusion that was observed significantly more often than would be expected by chance. Moreover, the conclusion predicted to be preferred by SARFCD was inferred significantly more often than the conclusion predicted to be preferred by the SRM (t(25) = 3.666, p < .01 twotailed). Consequently, the observed preferences are in accord with the predictions of SARFCD but not with the predictions of the SRM. At the same time the participants’ preferences are not as clear as in Experiment 1. This raises the question whether our group of participants may consist of two subgroups such that one subgroup exhibits preferences as predicted by the SRM and the other subgroup exhibits preferences as predicted by SARFCD. To investigate this, we categorized each participant into one of three groups based on the relation of the frequencies for SRM-predicted and SARFCD-predicted answers: If a participant gave more SRM-predicted than SARFCDpredicted answers, she was categorized as belonging to the srm group; if a participant gave more SARFCD-predicted than SRM-predicted answers, she was categorized as belonging to the sarf group; if there was no difference in frequencies between the two types of answers, the participant was categorized as belonging to the tie group. Following this procedure, 6, 18, and 2 participants belonged to the srm, sarf, and tie groups, respectively. In particular, the number of participants in the srm group is significantly lower

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(p < .05, binomial test) than expected if the participants’ propensity to prefer the SRMpredicted answer was equal to the propensity to prefer the SRAFCD-predicted answer. It remains unclear whether the answer frequencies of participants in the srm group are due to the fact that these participants reason as described by the SRM or due to some other cause such as, for example, noise arising from the reasoning task’s difficulty. However, even if participants in the srm group actually reason as suggested by the SRM, our data indicate that this type of reasoning constitutes a minority in the population. In sum, the experimental results corroborate the theoretical considerations outlined above in suggesting that SARFCD constitutes a cognitively more plausible account of cardinal direction reasoning than an array-like representation structure.

7. Conclusion and contributions Reasoning about spatial information, that is, inferring new spatial information from existing information, is one of the more intricate and powerful means of acquiring knowledge about the setup of the environment and, ultimately, a basis for effectively acting in it. As such, spatial reasoning constitutes a particularly interesting and important aspect of human cognition. Despite its importance, however, much is still largely unknown of the precise mechanisms underlying human spatial reasoning. Although recent research in computational cognitive modeling (Gugerty & Rodes, 2007; Gunzelmann & Lyon, 2011; Lyon, Gunzelmann, & Gluck, 2008; Lathrop et al., 2011; Ragni et al., 2005) started to explore the mechanisms involved in human spatial reasoning, their major part still awaits further elucidation. To shed light on some of these mechanisms, we here investigated—theoretically, empirically, and computationally—the ones involved in human reasoning about cardinal directions. Starting from a number of basic assumptions and based on results previously reported in the literature, we identified core properties of spatial reasoning about cardinal directions; most notably, the preferences that should be observable in cardinal direction reasoning. A subsequent experimental study tested and corroborated these theoretically derived reasoning preferences. Guided by information on the properties of human cognition in general (e.g., cognitive economy) and human spatial cognition more specifically, we developed a computational model of cardinal direction reasoning called SARFCD (spatio-analogical representation format for reasoning about cardinal directions) that successfully accounted for the found preferences. Furthermore, the model accurately predicted reasoning behavior in a novel reasoning situation; in particular, more accurately than a plausible competitor model. SARFCD employs a graph-like analogical representation structure in which the nodes represent objects and the links represent cardinal directions between the objects represented by the nodes. SARFCD has a number of noteworthy properties which mirror the requirements underlying its design (see above) and which set it aside from existing computational models of directional reasoning. First, the representation structure adheres tightly to the principle of cognitive economy, that is, the structure’s complexity is only as high as required by the current reasoning


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situation. New nodes are only added to the structure if new information about the cardinal direction between one node and another comes to be known. Furthermore, the DS associated with a node (i.e., the number and type of cardinal direction relations that can hold between this node and others) is always as coarse as possible. As a result, both the number of nodes and the number of possible cardinal directions are kept at a minimum in SARFCD. Given SARFCD’s success in accounting for human spatial reasoning behavior, our work supports and complements existing findings on the economic nature of human spatial information processing (Huff & Schwan, 2012). Second, the representation structure follows on current behavioral and neuroscientific research on spatial reasoning. Generally, SARFCD is built around an analogical representation structure and, thus, is in line with the common assumption that analogical representations subserve spatial reasoning (Johnson-Laird, 1983; Tversky, 1991). Importantly, however, the plausibility of the analogicity exhibited by SARFCD is not limited to such general assumptions. For instance, the results of Sereno et al. (2001) provide evidence that human processing of spatial information involves a neighborhood-preserving representation of directions. In four experiments, Sereno et al. (2001) asked participants to remember locations of targets that were presented around a central fixation cross. Among other things, the conducted experiments independently varied the distance and the direction of the target from the fixation cross. Using fMRI, Sereno et al. (2001) were able to identify a certain area in superior parietal cortex that exclusively and topographically coded for target direction: Regardless of target distance, the activation of certain brain regions was consistently coupled to certain target directions. In particular, the regions coding for different directions were systematically related to each other such that adjacent target directions activated adjacent brain regions (see fig. 3 in Sereno et al., 2001). Thus, the directional neighborhood structure employed in SARFCD effectively mimics organizational principles of spatial (working) memory representations found in parietal cortex. Third, SARFCD represents knowledge about cardinal directions only. No other type of spatial information needs to be known to build up the representation and no other type of spatial information can be derived from it. At first sight, this may seem like an unnecessarily harsh restriction to impose on models of human spatial representations. Yet, as we have argued elsewhere (Schultheis & Barkowsky, 2011; Schultheis et al., 2007), assuming such specialized representations is cognitively and computationally plausible. Furthermore, with such a specialized representation, adding information about cardinal directions to the representation does not imply anything about other types of relations. This is a distinct advantage over the array-type representation structures used in previous approaches in which an object placement necessarily introduces not only direction but also distance information. Fourth, as another marked difference from array representations of spatial information, SARFCD does not get brittle on the continued addition of direction information. Regardless of the order or the kind of information added, it can be accommodated without reorganizing the whole representation: Local changes (e.g., adding a node, changing the DSs of single nodes) suffice to allow the structure to represent the additional information.

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Fifth, though SARFCD is highly specialized to represent only cardinal direction knowledge, it was designed to allow its integration with representations for other types of knowledge (e.g., topology or distance). By keeping track of and matching the identities of objects represented in the specialized representations, the latter can be combined to form an integrated spatial representation (see Schultheis et al., 2007, for details). Sixth, SARFCD deepens our understanding of how human preferences in reasoning about cardinal directions come about. As explained above, these preferences non-trivially arise from the interactions of the model’s core assumptions. After this discussion, let us briefly return to one of the more fundamental questions which motivated the present research and which we raised in the introduction: Are there aspects of reasoning about cardinal directions that provide further insights into human spatial reasoning, in general? In our opinion, SARFCD indeed suggests an important general property of spatial reasoning. When looking at the workings of SARFCD, it is evident that the model’s behavior—in particular, regarding reasoning preferences—largely arises from the impact that general characteristics of cognition (e.g., cognitive economy, prototypicality, modality) have on the reasoning process. As such, the presented work suggests that, though spatial reasoning may employ specific types of representation structures, it is still subject to the same processing principles that generally govern mental processing of information. Consequently, one fascinating topic for future work is to investigate to what extent this influence of general processing principles exists for other types of spatial information and for reasoning about them. We will start this investigation by, first, developing representation structures for topological and distance information and, second, by giving a more detailed account of the processes involved in integrating all three specific representation structures during reasoning with spatial knowledge of different types. Furthermore, we plan to more closely examine the expected dynamic change of these representation structures when the task at hand requires increasingly visual representations (as compared to strictly spatial ones), such as, for example, when shape information is involved.

Acknowledgments The authors gratefully acknowledge support by the German Research Foundation (DFG) through the project R1-[ImageSpace], SFB/TR 8 Spatial Cognition, as well as by the University of Bremen and the Bauhaus-Universit€at Weimar.

Notes 1. The experiment’s language was German. To give the reader a better understanding of the experimental task, all statements are reproduced in English.


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2. Instructions asked participants for “eine m€ogliche Richtungsbeziehung,” that is, “a possible direction relation.” 3. Analyses of the second correct answer yielded qualitatively identical results. 4. Although the results of Experiment 1 indicate strong preferences, it is also clear that participants occasionally drew conclusions that differ from these preferences. More generally, recent experimental results have shown that general patterns of preferences can be subject to modulation by, for example, individual differences (Schultheis & Barkowsky, 2013). At the moment, it is unclear which mechanisms mediate such modulations and it remains for future research to extend SARFCD accordingly. In its current version, SARFCD approximates such mechanisms by assuming that the conclusions arising from the processing in SARFCD (i.e., those conclusions that correspond to the conclusions preferred by humans) are sometimes varied to yield a non-preferred conclusion. The probability with which such variations occur is a free parameter of SARFCD and has been estimated in fitting the results of Experiment 1.

References Anderson, J. R., Bothell, D., Byrne, M. D., Douglass, S., Lebiere, C., & Qin, Y. (2004). An integrated theory of the mind. Psychological Review, 111(4), 1036–1060. Anderson, J. R., & Lebiere, C. (2003). The Newell test for a theory of cognition. Behavioral and Brain Sciences, 26, 587–640. Arend, I., Colom, R., Botella, J., Contreras, M. J., Rubio, V., & Santacreu, J. (2003). Quantifying cognitive complexity: Evidence from a reasoning task. Personality and Individual Differences, 35(3), 659–669. Baddeley, A. D. (1986). Working memory. New York: Oxford University Press. Baddeley, A. D., & Logie, R. H. (1999). Working memory: The multiple component model. In A. Miyake & P. Shah (Eds.), Models of working memory (pp. 28–61). New York: Cambridge University Press. Bauer, R., & Dow, M. (1989). Complementary global maps for orientation coding in upper and lower layers of the monkey’s foveal striate cortex. Experimental Brain Research, 76, 503–509. Berendt, B., Barkowsky, T., Freksa, C., & Kelter, S. (1998). Spatial representation with aspect maps. In C. Freksa, C. Habel, & K. F. Wender (Eds.), Spatial cognition—An interdisciplinary approach to representing and processing spatial knowledge (pp. 313–336). Berlin: Springer. Bertel, S., Sima, J. F., & Lindner, M. (2009). Scalable representation structures for visuo-spatial reasoning— Dynamic explorations into knowledge types. In U. Kurup & B. Chandrasekaran (Eds.), Papers from the 2009 AAAI Fall Symposium (FS-09-05) (pp. 2–7). Menlo Park, CA: Association for the Advancement of Artificial Intelligence. Boeddinghaus, J., Ragni, M., Knauff, M., & Nebel, B. (2006). Simulating spatial reasoning using ACT-R. In D. Fum, F. Del Missier & A. Stocco (Eds.), Proceedings of the Seventh International Conference on Cognitive Modeling (ICCM 2006) (pp. 62–67). Mahwah, NJ: Lawrence Erlbaum Associates. Cohn, A. G. (1997). Qualitative spatial representation and reasoning techniques. In G. Brewka, C. Habel, & B. Nebel (Eds.), KI-97: Advances in artificial intelligence (pp. 1–30). Berlin: Springer. Collins, A. M., & Quillian, M. R. (1969). Retrieval time from semantic memory. Journal of Verbal Learning and Verbal Behavior, 8, 240–247. DeValois, R. L., Yund, E. W., & Hepler, N. (1982). The orientation and direction selectivity of cells in macaque visual cortex. Vision Research, 22, 531–544. Finke, R. (1989). Principles of mental imagery. Cambridge, MA: MIT Press.

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Frank, A. U. (1996). Qualitative spatial reasoning: Cardinal directions as an example. International Journal of Geographical Information Science, 10(3), 269–290. Freksa, C. (1992). Using orientation information for qualitative spatial reasoning. In A. U. Frank, I. Campari, & U. Formentini (Eds.), Theories and methods of spatio-temporal reasoning in geographic space (pp. 162–178). Berlin: Springer. Funt, B. (1980). Problem-solving with diagrammatic representations. Artificial Intelligence, 13, 201–230. Glasgow, J., Narayanan, H., & Chandrasekaran, B. (Eds.) (1995). Diagrammatic reasoning: Computational and cognitive perspectives. Cambridge, MA: MIT Press. Glasgow, J., & Papadias, D. (1992). Computational imagery. Cognitive Science, 16, 355–394. Gugerty, L., & Brooks, J. (2004). Reference frame misalignment and cardinal direction judgments: Group differences and strategies. Journal of Experimental Psychology: Applied, 10(2), 75–88. Gugerty, L., & Rodes, W. (2007). A cognitive model of strategies for cardinal direction judgments. Spatial Cognition & Computation, 7(2), 179–212. Gunzelmann, G., & Lyon, D. R. (2011). Representations and processes of human spatial competence. Topics in Cognitive Science, 3, 741–759. Hernandez, D. (1994). Qualitative representation of spatial knowledge. Berlin: Springer. Huff, M., & Schwan, S. (2012). Do not cross the line: Heuristic spatial updating in dynamic scenes. Psychonomic Bulletin & Review, 19, 1065–1072. Huttenlocher, J., Hedges, L. V., & Duncan, S. (1991). Categories and particulars: Prototype effects in estimating spatial location. Psychological Review, 98, 352–376. Jahn, G., Knauff, M., & Johnson-Laird, P. N. (2007). Preferred mental models in reasoning about spatial relations. Memory & Cognition, 35, 2075–2087. Johns, C., & Blake, E. (2001). Cognitive maps in virtual environments: Facilitation of learning through the use of innate spatial abilities. In A. Chalmers & V. Lalioti (Eds.), Proceedings of the 1st international conference on Computer graphics, virtual reality and visualization (pp. 125–129). New York: ACM. Johnson-Laird, P. N. (1972). The three-term series problem. Cognition, 1, 57–82. Johnson-Laird, P. N. (1983). Mental models. Cambridge, MA: Harvard University Press. Johnson-Laird, P. N. (1989). Mental models. In M. I. Posner (Ed.), Foundations of cognitive science (pp. 469–499). Cambridge, MA: MIT Press. Khenkhar, M. (1991). Object-oriented representation of depictions on the basis of cell matrices. In O. Herzog & C.-R. Rollinger (Eds.), Text understanding in LILOG (pp. 645–656). Berlin: Springer. Knauff, M., Rauh, R., & Schlieder, C. (1995). Preferred mental models in qualitative spatial reasoning: A cognitive assessment of Allen’s calculus. In J. F. Lehman & J. D. Moore (Eds.), Proceedings of the Seventeenth Annual Conference of the Cognitive Science Society (pp. 200–205). Mahwah, NJ: Erlbaum. Knauff, M., Schlieder, C., & Freksa, C. (2002). Spatial cognition: From rat-research to multifunctional spatial assistance systems. K€ unstliche Intelligenz, 4, 5–9. Kosslyn, S. M. (1980). Image and mind. Cambridge, MA: Harvard University Press. Kosslyn, S. M. (1994). Image and brain – The resolution of the imagery debate. Cambridge, MA: MIT Press. Kosslyn, S. M., & Sussman, A. L. (1995). Roles of imagery in perception: Or, there is no such thing as immaculate perception. In M. S. Gazzaniga (Ed.), The cognitive neurosciences (pp. 1035–1042). Cambridge, MA: MIT Press. Lathrop, S. D., Wintermute, S., & Laird, J. E. (2011). Exploring the functional advantages of spatial and visual cognition from an architectural perspective. Topics in Cognitive Science, 3, 796–818. Levenson, E., Tirosh, D., & Tsamir, P. (2011). Preschool geometry: Theory, research, and practical perspectives. Rotterdam: Sense Publishing. Ligozat, G. (1998). Reasoning about cardinal directions. Journal of Visual Languages and Computing, 9, 23– 44. Loftus, G. (1978). Comprehending compass directions. Memory and Cognition, 34, 416–422.


1559

Lyon, D. R., Gunzelmann, G., & Gluck, K. A. (2008). A computational model of spatial visualization capacity. Cognitive Psychology, 57, 122–152. Martin, T., Lukong, A., & Reaves, R. (2007). The role of manipulatives in arithmetic and geometry tasks. Journal of Education and Human Development, 1, 1–14. McNamara, T. P. (1986). Mental representations of spatial relations. Cognitive Psychology, 18, 87–121. Newell, A. (1990). Unified theories of cognition. Cambridge, MA: Harvard University Press. Ragni, M., & Knauff, M. (2013). A theory and a computational model of spatial reasoning with preferred mental models. Psychological Review, 120, 561–588. Ragni, M., Knauff, M., & Nebel, B. (2005). A computational model for spatial reasoning with mental models. In B. G. Bara, L. Barsalou, & M. Bucciarelli (Eds.), Proceedings of the 27th annual cognitive science conference (pp. 1797–1802). Mahwah, NJ: Lawrence Erlbaum. Randell, D. A., Cui, Z., & Cohn, A. G. (1992). A spatial logic based on regions and connection. In B. Nebel, C. Rich & W. R. Swartout (Eds.), Proc 3rd Int. Conf. on Knowledge Representation and Reasoning, Boston, October, 1992 (pp. 165–176). San Mateo, CA: Morgan Kaufman. Rauh, R., Hagen, C., Knauff, M., Kuss, T., Schlieder, C., & Strube, G. (2005). Preferred and alternative mental models in spatial reasoning. Spatial Cognition & Computation, 5, 239–269. Read, T. R. C., & Cressie, N. A. C. (1988). Goodness-of-fit statistics for discrete multivariate data. Berlin: Springer. R€ ohrig, R. (1994). A theory for qualitative spatial reasoning based on order relations. In B. Hayes-Roth & R. E. Korf (Eds.), Proceedings of the 12th National Conference on Artificial Intelligence, AAAI ‘94 (pp. 1418–1423). Palo Alto, CA: AAAI Press. Rosch, E. (1975). Cognitive representations of semantic categories. Journal of Experimental Psychology: General, 104, 192–223. Sadalla, E. K., Burroughs, W. J., & Staplin, L. J. (1980). Reference points in spatial cognition. Journal of Experimental Psychology: Human Learning and Memory, 6(5), 516–528. Schlieder, C. (1993). Representing visible locations for qualitative navigation. In N. Piera Carrete & M. G. Singh (Eds.), Qualitative reasoning and decision technologies (pp. 523–532). Barcelona, Spain: CIMNE. Schultheis, H., & Barkowsky, T. (2011). Casimir: An architecture for mental spatial knowledge processing. Topics in Cognitive Science, 3, 778–795. Schultheis, H., & Barkowsky, T. (2013). Variable stability of preferences in spatial reasoning. Cognitive Processing, 14, 209–211. Schultheis, H., Bertel, S., Barkowsky, T., & Seifert, I. (2007). The spatial and the visual in mental spatial reasoning: An ill-posed distinction. In T. Barkowsky, M. Knauff, G. Ligozat & R. D Montello (Eds.), Spatial cognition V – reasoning, action, interaction. Lecture Notes in Computer Science (pp. 191–209). Berlin: Springer. Sereno, I. M., Pitzalis, S., & Martinez, A. (2001). Mapping of contralateral space in retinotopic coordinates by a parietal cortical area in humans. Science, 294, 1350–1354. Shimojima, A. (1996). Operational constraints in diagrammatic reasoning. In G. Allwein & J. Barwise (Eds.), Logical reasoning with diagrams (pp. 27–48). New York: Oxford University Press. Sloman, A. (1971). Interactions between philosophy and artificial intelligence: The role of intuition and nonlogical reasoning in intelligence. Artificial Intelligence, 2, 209–225. Sloman, A. (1975). Afterthoughts on analogical representations. In B. L. Nash-Webber & R. Schank (Eds.), Proceedings of theoretical issues in natural language processing (pp. 164–168). Cambridge, MA: Association for Computer Linguistics. Stevens, A., & Coupe, P. (1978). Distortions in judged spatial relations. Cognitive Psychology, 10, 422– 437. Tideman, T. N. (1979). A generalized v² for the significance of differences in repeated, related measures applied to different samples. Educational and Psychological Measurement, 39, 333–336. Tversky, B. (1991). Spatial mental models. The Psychology of Learning and Motivation, 27, 109–145.

1560


Tversky, B., & Schiano, D. J. (1989). Perceptual and conceptual factors in distortions in memory for graphs and maps. Journal of Experimental Psychology: General, 118(4), 387–398. Vieu, L. (1997). Spatial representation and reasoning in artificial intelligence. In O. Stock (Ed.), Spatial and temporal reasoning (pp. 5–41). Dordrecht, the Netherlands: Kluwer Academic Publishers. Zimmermann, K., & Freksa, C. (1996). Qualitative spatial reasoning using orientation, distance, and path knowledge. Applied Intelligence, 6, 49–58.

Appendix A To illustrate in more detail how the check for triangularity is performed in SARFCD, we will consider the reasoning situation shown in Fig. 14. For ease of exposition, we will display the direction lists of the involved entities in a linear fashion. However, keep in mind that direction lists are circular in SARFCD. Each list position is annotated with an abbreviation for the direction it is representing: N, NW, W, SW, S, SE, E, NE for north, northwest, west, southwest, south, southeast, east, northeast, respectively. Initially, the reasoning situation comprises the three entities Paris, Prague, and Algiers with the following direction lists (see Fig. 14, panel 1): Paris: N W S E Algiers Prague

Prague: E Paris

W

Algiers: S

N

Paris

The first attempt to establish a triangular conclusion leads to the situation shown in Fig. 14, panel 2, which corresponds to the following direction lists: Paris: N W S E Algiers Prague

Prague:

E W Paris Algiers

Algiers:

S N Paris Prague

To check for the triangularity of this solution, we first count the number of transitions from one entity to the other in each direction list. For Paris, one transition is needed to get from the first entity in Paris’s direction list (Algiers) to the other (Prague). This num-


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ber of transitions is divided by the number of directions that are distinguished in the direction list of Paris. As the direction list of Paris distinguishes four directions, the resulting value is ¼. For Prague, one transition is needed to get from the first entity in the list (Paris) to the other (Algiers). As the direction list of Prague distinguishes two directions, the fraction computed for Prague is ½. Following a similar procedure, the fraction associated with Algiers is computed to be also ½. A solution is considered to be triangular only if the sum of the fractions associated with the involved entities is not greater than ½. In the current situation, the sum of fractions associated with Paris, Prague, and Algiers is ¼ + ½ + ½ = 1¼, which is greater than ½ and, accordingly, the solution is not considered to be triangular. As a result, the direction system of Prague is refined and a new solution is attempted employing the refined direction system (see Fig. 14, panel 3): Paris: N W S E Algiers Prague

Prague:

E N W S Paris Algiers

Algiers:

S N Paris Prague

Computation of fractions as described above yields fractions of ¼, ¼, ½ for Paris, Prague, and Algiers, respectively. The sum of these fractions (¼ + ¼ + ½ = 1) is again greater than ½ indicating a lack of triangularity of the attempted solution. Accordingly, further steps of refinement take place (see Fig. 14, panels 4–6) until a feasible solution is found (panel 6): Paris: N W S E Algiers Prague

Prague:

E NE N NW W SW S SE Paris Algiers

Algiers:

SW S SE E NE N NW W Paris Prague

For this solution, the fractions associated with Paris, Prague, and Algiers are ¼, ⅛, ⅛, respectively, yielding a sum of ½. Consequently, the criterion for triangularity is met and the solution is kept as the solution to the reasoning problem.

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