Searching for Alternatives in Spatial Reasoning: Local Transformations and Beyond

Searching for alternative solutions of an indeterminate reasoning task is an important and necessary step in order to draw certain inferences as in the case of deduction. To elucidate the underlying mental representations and processes of the search for alternatives in spatial reasoning, an experiment was conducted that used specific material stemming from AI research of Qualitative Spatial Reasoning. The results showed that searching for alternative solutions can be best explained as a revision process starting with an initial mental model of the premises. Proceeding from one solution to an alternative is apparently achieved by local transformation. Interestingly, local transformations have a "logic of their own": They can lead to systematic errors of omission and to errors of commission. Spatial Reasoning and Mental Models Dealing with spatial problems is a frequent and important challenge in everyday as well as in professional life. It occurs across various fields like spatial navigation or spatial configuration and design. In this paper, we will concentrate on a special sort of spatial problem solving, namely reasoning based on spatial relational descriptions. This type of reasoning can be investigated with recourse to several background theories of thinking developed in cognitive psychology. According to previous research in spatial reasoning (Byrne & Johnson-Laird, 1989; Evans, Newstead, & Byrne, 1993) and according to our own previous findings (Knauff, Rauh, & Schlieder, 1995; Knauff, Rauh, Schlieder, & Strube, 1998; Rauh & Schlieder, 1997) the most promising and most successful framework is the theory of mental models. Mental Model Theory as Framework The core assumption of the mental model theory (JohnsonLaird, 1983; Johnson-Laird & Byrne, 1991) states that when we reason we build an integrated representation of the situation that the premises describe. This integrated representation—the mental model—is in certain aspects analogous to the state of affairs and, as a consequence, lacks the information whether relationships are explicitly mentioned in the premises and or are implicitly determined by the representational format. A further consequence of the assumption of integrated representation becomes evident when certain kinds of inferences have to be drawn. Take deductive inference for example: To test whether a contingent relationship in the initial mental model is necessarily true, the reasoner has to test all the alternative models of the premises. If a contradictory example is found, the putative conclusion will be rejected; if not it will be accepted as a valid conclusion. The search for alternative models takes place during what we call the phase of model variation. It seems to be a deliberate mental process so fragile that it causes many systematic reasoning errors. There are errors of omission, i.e. inferences that could have been validly drawn, and there are errors of commission, i.e. inferences that are not justified by the premises. Therefore, model variation has attracted much attention, but little is empirically known about how the mental search for alternative models is accomplished by the human process of reasoning. For a precise investigation of the model variation phase, there is the need for relational material with a rich inherent structure and unambiguous semantics. Spatial Reasoning with Interval Relations Traditional investigations of spatial reasoning used relations like left-of, right-of, in front of, and behind. As argued elsewhere (Knauff et al., 1998), these spatial relations have no clear semantics. Therefore, studies of reasoning using these spatial relations are problematic because it is unclear whether the results obtained can be attributed to the inference processes, or are due to the ambiguity of these relations. To remedy this situation, we use Allen’s (1983) set of 13 qualitative interval relations that enables one-dimensional spatial reasoning. These relations have clear geometric semantics based on the bounding points of the intervals, i.e. their starting points and ending points. They also have the property of being jointly exhaustive and pairwise disjoint (JEPD)—a property that also reduces the risk of misinterpretations. In Table 1, we shortly introduce these relations together with verbalizations that we use in our experiments. With these relations, reasoning tasks known as three-term series problems can be constructed. One example is "X overlaps Y from the left. Y surrounds Z." The example also shows that there are many three-term series problems generated from these relations that have more than one solution. To be precise, there are 42 three-term series problems that have three solutions, 24 that have five solutions, 3 that have nine, and another 3 that have thirteen solutions. We utilize this property in order to construct indeterminate three-term series problems to investigate precisely the phase of model variation. In the next section, we will present a more formal analysis of these tasks. From this analysis and the revealed properties of the different tasks, hypotheses can be derived that we will test in a model variation experiment. A Formal Framework for Model Variation In principle, there are two ways to construct alternative models of the premises. The first consists of repeating the complete construction of alternative models one after another (model iteration). We will examine the more plausible variaTable 1: The 13 qualitative interval relations, associated natural language expressions, and a graphical example (adapted and augmented according to Allen, 1983). Relation symbol Natural language description Graphical example X < Y X lies to the left of Y X m Y X touches Y at the left X o Y X overlaps Y from the left X s Y X lies left-justified in Y X d Y X is completely in Y X f Y X lies right-justified in Y