The ability to generate novel hypotheses is an important problem-solving capacity of humans. This ability is vital for making sense of the complex and unfamiliar world we live in. Often, this capacity is characterized as an inference to the best explanation - selecting the "best" explanation from a given set of candidate hypotheses. However, it remains unclear where these candidate hypotheses originate from. In this paper we contribute to computationally explaining these origins by providing the contours of the computational problem solved when humans generate hypotheses. The origin of hypotheses, otherwise known as abduction proper, is hallmarked by seven properties: (1) isotropy, (2) open-endedness, (3) novelty, (4) groundedness, (5) sensibility, (6) psychological realism, and (7) computational tractability. In this paper we provide a computational-level theory of abduction proper that unifies the first six of these properties and lays the groundwork for the seventh property of computational tractability. We conjecture that abduction proper is best seen as a process of deep analogical inference.
The role that the mental, or internal, representation plays when people are solving hard computational problems has largely been overlooked to date, despite the reality that this internal representation drives problem solving. In this work we investigate how performance on versions of two hard computational problems differs based on what internal representations can be generated. Our findings suggest that problem solving performance depends not only on the objective difficulty of the problem, and of course the particular problem instance at hand, but also on how feasible it is to encode the goal of the given problem. A further implication of these findings is that previous human performance studies using NP-hard problems may have, surprisingly, underestimated human performance on instances of problems of this class. We suggest ways to meaningfully frame human performance results on instances of computationally hard problems in terms of these problems’ computational complexity, and present a novel framework for interpreting results on problems of this type. The framework takes into account the limitations of the human cognitive system, in particular as it applies to the generation of internal representations of problems of this class.
Gestalt psychologists pointed out about 100 years ago that a key to solving difficult insight problems is to change the mental representation of the problem, as is the case, for example, with solving the six matches problem in 2D vs. 3D space. In this study we ask a different question, namely what representation is used when subjects solve search, rather than insight problems. Some search problems, such as the traveling salesman problem (TSP), are defined in the Euclidean plane on the computer monitor or on a piece of paper, and it seems natural to assume that subjects who solve a Euclidean TSP do so using a Euclidean representation. It is natural to make this assumption because the TSP task is defined in that space. We provide evidence that, on the contrary, subjects may produce TSP tours in the complex-log representation of the TSP city map. The complex-log map is a reasonable assumption here, because there is evidence suggesting that the retinal image is represented in the primary visual cortex as a complex-log transformation of the retina. It follows that the subject’s brain may be “solving” the TSP using complex-log maps. We conclude by pointing out that solving a Euclidean problem in a complex-log representation may be acceptable, even desirable, if the subject is looking for near-optimal, rather than optimal solutions.
The article reports three experiments designed to explore heuristics used in comparing the lengths of completed Euclidean Traveling Salesman Problem (E-TSP) tours. The experiments used paired comparisons in which participants judged which of two completed tours of the same point set was shorter. The first experiment manipulated two factors, the presence/absence of crossed arcs, and the relative areas of the enclosed polygons. Both factors significantly influenced judgments, with the absence of crossings and smaller areas being associated with shorter tours. The second experiment examined the effects of crossings only, and compared stimulus pairs using all possible combinations of no, one, and more than one crossing. The results showed a significant tendency for tours with one or more crossings to be judged longer than tours with none, while tours with more crossings were not judged to be longer than tours with only one. Apparently the mere presence of a crossing is sufficient to cause a tour to be judged as longer. The third experiment examined the effects of area only, and consisted of two parts. In the first part, participants judged which of two tours that differed in area was shorter. The results supported those of the first experiment, by finding that tours with smaller areas tended to be judged as shorter. In the second part of the experiment, participants judged the relative areas of each pair, to determine whether people can reliably differentiate the areas of such complex polygons. The results confirmed that they can, thereby supporting the feasibility of using differences in area as a heuristic to judge relative lengths. The results were discussed in terms of Carruthers’s (2015) proposal of goal modification and the suggestion is made that applying heuristics of the type identified may represent a specific form of goal modification.
In human problem solving, there is a wide variation between individuals in problem solution time and success rate, regardless of whether or not this problem solving involves insight. In this paper, we apply computational and parameterized analysis to a plausible formalization of extended representation change theory (eRCT), an integration of problem solving by problem space search and insight as problem restructuring which proposes that this variation may be explainable by individuals having different problem representations and search heuristic choices. Our analyses establish not only the intractability of eRCT in general, but also sets of restrictions under which eRCT-based problem solving can and cannot be done quickly. As such, our analyses both prove that several conjectures about what makes problem solving under eRCT possible in practice are incomplete, in the sense that not all factors in the model whose restriction is responsible for efficient solvability are part of the explanation, and provide several new explanations that are complete.
The paper concerns an important but underappreciated genre of algorithmic puzzles, explaining what these puzzles are, reviewing milestones in their long history, and giving two different ways to classify them. Also covered are major applications of algorithmic puzzles in cognitive science research, with an emphasis on insight problem solving, and the advantages of algorithmic puzzles over some other classes of problems used in insight research. The author proposes adding algorithmic puzzles as a separate category of insight problems, suggests 12 specific puzzles that could be useful for research in insight problem solving, and outlines several experiments dealing with other cognitive aspects of solving algorithmic puzzles.
Insightful problem solving is a vital part of human thinking, yet very difficult to grasp. Traditionally, insight has been investigated by using a set of established "insight tasks," assuming that insight has taken place if these problems are solved. Instead of assuming that insight takes place during every solution of the 9 Dot, 8 Coin, and Matchstick Arithmetic Problems, this study explored the likelihood that solutions evoked the "Aha! experience," which is often regarded as the defining characteristic of insight. It was predicted that the rates of self-reported Aha! experiences might vary based on the necessary degree of constraint relaxation. The main assumption was that the likelihood of experiencing an Aha! would decrease with increasing numbers of constraints that must be relaxed, because several steps are needed to achieve a representational change and solve the problem, and thus, the main feature of suddenness of a solution might be lacking. The results supported this prediction, and demonstrated that in many cases participants do solve these classical insight problems without any Aha! experience. These results show the importance of obtaining insight ratings from participants to determine whether any given problem is solved with insight or not.
Recent evidence suggests that solving problems through insight can enhance long-term memory for the problem and its solution. Previous findings have shown that generation of the solution as well as experiencing a feeling of Aha! can have a beneficial relationship to later memory. These findings lead to the question of how learning in problem-solving tasks in which a novel solution needs to be generated-such as in tasks used to study insight-differs from the classical generation effect. Because previous studies on learning from insight on one hand and the generation effect on the other hand have measured different types of memory, the present study examined two kinds of memory measures: indirect (solving old and new problems at test) and direct (recognition memory). At encoding, we manipulated whether participants had the chance to solve Compound Remote Associates Task items and compared later memory for generated solutions (generate condition) to solutions that were presented after failing to generate one (fail-to-generate condition), and to solutions that were presented without a chance at generation (read condition). Participants also reported if they had an Aha! experience for each problem. While both Aha! experiences and generated solutions were associated with more positive emotional responses, only the generation variable was associated with differences in later memory performance. While attempts to generate had an advantage over the read condition in recognition memory performance (generate >fail-to-generate > read), only when generation was successful did it enhance the solution rate of old items during testing (generate > read > fail-to-generate). Contrary to generation effects with other verbal stimuli, these results suggest that the generation effect in problem-solving tasks in which a novel solution needs to be found differs from the classical generation effect. Seeing a correct solution for a longer time (read) seems in the current case to be more helpful for solving the same problem later on, compared to being presented with the solution after a failed attempt at problem solving.
People often use spatial metaphors (e.g., think "laterally," "outside the box") to describe exploration of the problem space during creative problem solving. In this paper, we probe the potential cognitive underpinnings of these spatial metaphors. Drawing on theories of situative cognition, semantic foraging theory, and environmental psychology, we formulate and test the hypothesis that larger physical spaces can facilitate divergent (but not convergent) processes in problem space exploration. Across two experiments, participants worked on a battery of problem solving tasks intended to represent divergent (alternative uses, shape invention) and convergent (remote associates, letter extrapolation) problem solving processes in either a large or a small room. In Experiment 1, participants in the larger room produced more novel alternative uses for everyday objects, and created more novel shape inventions, but generated less practical alternative uses, than participants in the smaller room. In Experiment 2, participants in the larger room (including a variant larger room) also produced more novel alternative uses for everyday objects, and less practical alternative uses, than participants in a small room, but did not create more novel shape inventions. These results suggest that spatial metaphors for problem space exploration may reflect meaningful cognitive phenomena: People may be able to search more broadly in a problem space if they are in an environment where broad physical search is a salient affordance; however, this effect appears to be relatively small and may depend on having sufficiently motivated participants.
Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation format might affect solution success, and whether the efficiency of solution varies depending on the nature of the problem solving task. This study addressed the question of whether symbolic or graphical representation format provides different affordances with respect to two different types of problems: computation and interpretation. Graphical representation was found to facilitate problem solving among college students, and problems that required the comparison of slopes were more difficult when presented in a symbolic format than in graphical format.
Students benefit from learning multiple procedures for solving the same or related problems. However, past research on comparison instruction has focused on comparing multiple formal procedures. This study investigated whether the benefits of comparing procedures extend to comparisons that involve informal and formal procedures. We also examined how learner characteristics, including prior knowledge and attitudes toward mathematics, affect learning from comparing procedures. We addressed these issues in college students' learning procedures for solving systems of equations problems in algebra. Learners who liked mathematics learned equally well whether they received comparison or sequential instruction. However, among learners who did not like mathematics, instruction that included support for comparisons between the formal and informal procedures led to greater gains in conceptual knowledge than did sequential instruction of the procedures.
Research on problem solving typically does not address tasks that involve following detailed and/or illustrated step-by-step instructions. Such tasks are not seen as cognitively challenging problems to be solved. In this paper, we challenge this assumption by analyzing verbal protocols collected during an Origami folding task. Participants verbalised thoughts well beyond reading or reformulating task instructions, or commenting on actions. In particular, they compared the task status to pictures in the instruction, evaluated the progress so far, referred to previous experience, expressed problems and confusions, and-crucially-added complex thoughts and ideas about the current instructional step. The last two categories highlight the fact that participants conceptualised this spatial task as a problem to be solved, and used creativity to achieve this aim. Procedurally, the verbalisations reflect a typical order of steps: reading-reformulating-reconceptualising-evaluating. During reconceptualisation, the creative range of spatial concepts represented in language highlights the complex mental operations involved when transferring the two-dimensional representation into the real world. We discuss the implications of our findings in terms of problem solving as a multilayered process involving diverse types of cognitive effort, consider parallels to known conceptual challenges involved in interpreting spatial descriptions, and reflect on the benefit of reconceptualisation for cognitive processes.
The traveling salesman problem (TSP) is a combinatorial optimization problem that requires finding the shortest path through a set of points (“cities”) that returns to the starting point. Because humans provide heuristic near-optimal solutions to Euclidean versions of the problem, it has sometimes been used to investigate human visual problem solving ability. The TSP is also similar to a number of tasks commonly used for neuropsychological assessment (such as the trail-making test), and so its utility in assessing reliable individual differences in problem solving has sometimes been examined. Nevertheless, the task has seen little widespread use in clinical and assessment domains, in part because no standard software implementation or item set is widely available with known psychometric properties. In this paper, we describe a computerized version of TSP running in the free and open source Psychology Experiment Building Language (PEBL). The PEBL TSP task is designed to be suitable for use within a larger battery of tests, and to examine both standard and custom TSP node configurations (i.e., problems). We report the results of a series of experiments that help establish the test’s reliability and validity. The first experiment examines test-retest reliability, establishes that the quality of solutions in the TSP are not impacted by mild physiological strain, and demonstrates how solution quality obtained by individuals in a physical version is highly correlated with solution quality obtained in the PEBL version. The second experiment evaluates a larger set of problems, and uses the data to identify a small subset of tests that have maximal coherence. A third experiment examines test-retest reliability of this smaller set that can be administered in about five minutes, and establishes that these problems produce composite scores with moderately high (R = .75) test-retest reliability, making it suitable for use in many assessment situations, including evaluations of individual differences, personality, and intelligence testing.
While both insight and divergent thinking tasks are used to study creativity, there are reasons to believe that the two may call upon very different mechanisms. To explore this hypothesis, we administered a verbal insight task (riddles) and a divergent thinking task (verbal fluency) to 16 native English speakers and 16 non-native English speakers after they underwent Transcranial Direct Current Stimulation (tDCS) of the left middle temporal gyrus and right temporo-parietal junction. We found that, in the case of the insight task the depolarization of right temporo-parietal junction and hyperpolarization of left middle temporal gyrus resulted in increased performance, relative to both the control condition and the reverse stimulation condition in both groups (non-native > native speakers). However, in the case of the divergent thinking task, the same pattern of stimulation resulted in a decrease in performance, compared to the reverse stimulation condition, in the non-native speakers. We explain this dissociation in terms of differing task demands of divergent thinking and insight tasks and speculate that the greater sensitivity of non-native speakers to tDCS stimulation may be a function of less entrenched neural networks for non-native languages.
Large-Scale Multimedia Retrieval (LSMR) is the task to fast analyze a large amount of multimedia data like images or videos and accurately find the ones relevant to a certain semantic meaning. Although LSMR has been investigated for more than two decades in the fields of multimedia processing and computer vision, a more interdisciplinary approach is necessary to develop an LSMR system that is really meaningful for humans. To this end, this paper aims to stimulate attention to the LSMR problem from diverse research fields. By explaining basic terminologies in LSMR, we first survey several representative methods in chronological order. This reveals that due to prioritizing the generality and scalability for large-scale data, recent methods interpret semantic meanings with a completely different mechanism from humans, though such humanlike mechanisms were used in classical heuristic-based methods. Based on this, we discuss human-machine cooperation , which incorporates knowledge about human interpretation into LSMR without sacrificing the generality and scalability. In particular, we present three approaches to human-machine cooperation ( cognitive , ontological , and adaptive ), which are attributed to cognitive science, ontology engineering, and metacognition, respectively. We hope that this paper will create a bridge to enable researchers in different fields to communicate about the LSMR problem and lead to a ground-break-ing next generation of LSMR systems.
Two experiments on performance on the traveling salesman problem (TSP) are reported. The TSP consists of finding the shortest path through a set of points, returning to the origin. It appears to be an intransigent mathematical problem, and heuristics have been developed to find approximate solutions. The first experiment used 10-point, the second, 20-point problems. The experiments tested the hypothesis that complexity of TSPs is a function of number of nonboundary points, not total number of points. Both experiments supported the hypothesis. The experiments provided information on the quality of subjects' solutions. Their solutions clustered close to the best known solutions, were an order of magnitude better than solutions produced by three well-known heuristics, and on average fell beyond the 99.9th percentile in the distribution of random solutions. The solution process appeared to be perceptually based.
Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students' problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content, are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed.
