The impact of problem structure on planning: insights from the Tower of London task

Abstract

Despite the large number of behavioral and functional neuroimaging studies employing the Tower of London (ToL), the task's structural parameters and particularly their impact on planning have not been addressed so far. In this paper, we highlight the structural properties of ToL problems and provide evidence for their systematic and substantial effects on the cognitive processes involved in planning. In a problem set with three-move problems, the following structural parameters were experimentally manipulated: the ambiguity of goal hierarchy, the demand for subgoal generation, and the existence of suboptimal alternatives. Analysis of preplanning time as an indicator for the planning process revealed highly significant effects for all three parameters which seems to reflect differences in cognitive processing due to structural task properties. Therefore, we suggest that the common consideration of ToL problem difficulty solely in terms of the minimum number of moves is not sufficient. Moreover, the applied problem sets should be more carefully selected and their structural parameters explicitly noted.

Introduction

Planning is a fundamental and ubiquitous cognitive function. As a form of human problem solving, it denotes the mental execution of a goal-directed behavior in order to predict and assess the resulting consequences. Therefore, prior to action, the mental representation of a given situation has to be transformed into a desired goal state via the generation of multiple hypothetical events. Apart from this look-ahead mechanism, planning also involves other cognitive processes such as recognition of goal attainment, execution-linked anticipation of future events, and storage of representations that can guide movement from the initial to the goal state [7]. In everyday life, planning problems are often ill-structured with incomplete specifications of the start and goal states as well as of the applicable transformations. For example, planning a dinner for a guest or designing a lab space are ill-structured tasks [12].

In contrast, most experimental planning tasks and clinical assessment tools are largely well-defined. Well-defined problems have definite initial states, as well as goals and transformations that are known. A great deal of past research on well-defined planning problems has focused on disc-transfer tasks such as the Tower of London (ToL). The ToL—a derivate of the classic Tower of Hanoi (ToH)—was originally developed by Tim Shallice [35] with the intention to provide a planning test with more graded levels of difficulty than the ToH. The ToL consists of three balls of differing colors that may be placed on three pegs of differing heights holding either one, two, or three balls, respectively. Solving a ToL problem requires the transformation of an initial state in order to match a predetermined goal state, with the restrictions that only one ball can be moved at a time and must not be placed outside the pegs. Levels of difficulty have usually been defined as the minimum number of moves to solve a given problem [35].

The neural basis of planning has long been associated with the involvement of the frontal lobes. Hence, the ToL has been widely used in clinical populations in order to assess the interrelation of planning abilities and frontal lobe functioning, for example in patients with frontal lesions (e.g., [3], [7], [27], [35]), in Parkinson's and Huntington's disease patients (e.g., [15], [21], [22], [23], [28], [44]), in children with Attention-Deficit/Hyperactivity Disorder (e.g., [26], [36]) and in patients with psychiatric disorders (e.g., [30], [38]). Moreover, the ToL was particularly used in functional neuroimaging studies in patients and normal controls revealing the importance of fronto-parietal and fronto-striatal networks in planning (e.g., [9], [10], [29], [33], [41]).

Despite its popularity and the extensive use of the ToL paradigm, there have been relatively few detailed behavioral investigations of the cognitive processes involved in the planning of ToL problems. Instead, a wide variety of problem selections and variants of the ToL with different cognitive demands were applied across studies (for review, see Ref. [5]). Thus, it might be questioned to which extent the different studies are actually comparable. Furthermore, due to the use of varying problem sets in identical ToL variants and given that for instance the original Shallice ToL configuration provides a 36-node problem space resulting in a total of 1260 problems (210 structurally unique problems with six isoforms each; cf. Ref. [5]), it might be assumed that different cognitive aspects of planning were assessed across studies. In addition, psychometric investigations of ToL problem sets found only a low internal consistency [16], [18] indicating the need for a more careful selection of problems. Furthermore, this clearly suggests that individual ToL problems involve a variety of cognitive processes which are not systematically controlled for. Although reliable problem sets were empirically developed [34], the structure of the problems has not been investigated so far. The study by Ward and Allport [43] is a notable exception, but it considers a five-disc tower task with an entirely different problem space and significant changes to the cognitive requirements (see also Ref. [5]). Therefore, the aim of the present study was to address the structural parameters of ToL problems and to provide evidence for their impact on planning.

From our point of view, the structure of ToL problems can be classified by global and specific parameters. Global parameters are associated with the general appearance of a given problem and relate to the entire solution or at least a portion of the solution, whereas specific parameters relate to specific moves. Global problem parameters are (1) the minimum number of moves, (2) the number of paths for achieving an optimal solution, (3) the goal hierarchy, (4) different patterns of subgoaling and (5) the number of suboptimal alternatives to the optimal solution.

On the other hand, specific problem parameters concern the existence of certain types of goal moves and subgoal moves on the optimal solution path, for example counterintuitive subgoal moves and alternative goal moves. Generally, solving ToL problems involves goal moves and in most of the problems also a certain number of subgoal moves. A goal move places a ball into its goal position. In ToL problems, there cannot be more than one goal move available at any given time. Subgoal moves are defined as moves that are essential to the optimal solution, but which do not place a ball into its goal position (cf. Ref. [43]). As opposed to regular subgoal moves, counterintuitive subgoal moves specifically require to move a ball that is already placed in its goal position, thereby representing a conflict between the global goal and a local subgoal. In the ToH, patients with lesions to the prefrontal cortex have been reported to fail at recognizing and/or resolving similar goal–subgoal conflicts [11]. In addition, research on the Orc-and-Hobbit paradigm [17] and the water jug problem [4] has also shown that counterintuitive subgoal moves enhance problem difficulty by increasing the difference between the current state and the goal state (see also Ref. [8]). Similarly, in ToL problems with an alternative goal move,¹ the optimal solution requires one to inhibit the prepotent response, i.e. moving a ball into its goal position, in order to avoid blocking subsequent moves [3], [14], [31].

However, given that specific parameters such as counterintuitive subgoal moves and alternative goal moves are present only in about 4.3% and 7.1% of all ToL problems, respectively, the focus of this paper will be on global parameters. Further restrictions were made concerning the minimum number of moves and the number of paths to achieve an optimal solution: As the planning process in the ToL relies on working memory involvement [29], [32], [45], confoundations with planning performance are likely. Consequently, the influence of problem structure on planning was assessed in three-move problems in order to keep working memory contributions at a minimum. In addition, since parameters such as subgoaling patterns may differ between solution paths, only problems with one optimal solution path were selected for further investigation.

Thus, the global parameters experimentally manipulated in the present study were goal hierarchy, subgoaling, and suboptimal alternatives. Goal hierarchy concerns the obviousness of goal priority gathered from the structure of the goal state (Fig. 1a). Problems with “tower” goal states, where all three balls are stacked on the tallest peg, provide an unambiguous goal hierarchy as the ball at the bottom has to be in its goal position before the second from the bottom and so on. In contrast, in problems having “flat” goal states, where there is one ball on each peg, the prioritization of goal moves is completely ambiguous. Consequently, goal states in which two balls are placed on one peg and one ball on another peg constitute a partially ambiguous sequence of goal moves. Past research has demonstrated that the grade of ambiguity in goal hierarchy strongly influences planning performance in the ToH in children [6], [20] as well as in the Ward and Allport tower task in adults [43].

With regard to different patterns of subgoaling,² in three-move ToL problems one can differentiate problems that require subgoal generation and problems that do not, i.e. problems with a subgoal move preceding two goal moves vs. problems with three successive goal moves for an optimal solution (Fig. 1b). Hence, one can also distinguish different heuristics in solving a given problem (for overview, see Ref. [42]). While problems with three successive goal moves can be solved by simple difference reduction (also referred to as ‘hill-climbing’), this heuristic will not be sufficient in problems requiring the generation of subgoals for optimal solution. Depending on the obviousness of goal hierarchy [19], means–end analysis can be directly applied to this kind of problems when prioritization is at least partially clear. In contrast, when the goal state provides conflicting information for sequencing the goal moves, it is necessary to search ahead and evaluate the available moves for their ‘goodness’ [43].

In the ToL, searching ahead can result in a suboptimal alternative which allows one to solve the problem within one to two additional subgoal moves (Fig. 1c). Competing with the optimal solution, a suboptimal alternative may thus impact planning by requiring one to inhibit its realization and to proceed in the search for an optimal solution. ToL problems vary in this regard, with differences concerning the existence and number of suboptimal alternatives. Three-move problems either have one suboptimal alternative that achieves the goal state within four or five moves, or they do not.

To summarize, the ToL is widely used as a planning tool in a variety of domains and particularly in clinical research and neuroimaging. However, the impact of the task's structural properties on the cognitive processes involved in planning has not been taken into account so far. In contrast, research on the ToL as well as related tasks clearly indicates the need for addressing the structural parameters of problems. Therefore, the intention of the present study was to focus on the issue of problem structure and to provide empirical evidence for its importance in investigating planning processes.