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# An immersive first-person navigation task for abstract knowledge acquisition

## Abstract

Advances in virtual reality (VR) technology have greatly benefited spatial navigation research. By presenting space in a controlled manner, changing aspects of the environment one at a time or manipulating the gain from different sensory inputs, the mechanisms underlying spatial behaviour can be investigated. In parallel, a growing body of evidence suggests that the processes involved in spatial navigation extend to non-spatial domains. Here, we leverage VR technology advances to test whether participants can navigate abstract knowledge. We designed a two-dimensional quantity space—presented using a head-mounted display—to test if participants can navigate abstract knowledge using a first-person perspective navigation paradigm. To investigate the effect of physical movement, we divided participants into two groups: one walking and rotating on a motion platform, the other group using a gamepad to move through the abstract space. We found that both groups learned to navigate using a first-person perspective and formed accurate representations of the abstract space. Interestingly, navigation in the quantity space resembled behavioural patterns observed in navigation studies using environments with natural visuospatial cues. Notably, both groups demonstrated similar patterns of learning. Taken together, these results imply that both self-movement and remote exploration can be used to learn the relational mapping between abstract stimuli.

## Introduction

Virtual reality (VR) technology has made great advances in recent years. The use of life-like graphics, head-mounted displays (HMD) as well as omnidirectional motion platforms helps to create interactive environments in which participants behave naturally and intuitively. Researchers on the other hand, remain in full control and can track participants’ behaviour in great detail with advanced sensors. Consequently, VR has increasingly been used to study spatial navigation1, social interaction2 and multisensory integration3. It has also been used in therapeutic settings such as stroke rehabilitation4, pain therapy5 or phobia treatment6. In all of these fields, researchers make use of the interactive nature of VR paradigms: participants are fully immersed in a highly-controlled representation and interact with their environment in a safe, yet realistic manner. They act from a first-person perspective, which creates a sense of presence and makes the environment relevant and stimulating.

Interactiveness, as facilitated by using VR, is also the central research topic in embodied learning. Research suggests that learning, even of abstract concepts or ideas, is aided by physically exploring them7,8,9. This has not only been shown in children, but also in adults10,11,12 and in a range of different tasks such as counting13,14, language15, letter recognition16,17, music concepts18 and understanding of physics19. Interacting with a concept increases its relevance, and makes it literally “graspable”. Interestingly, the effects of embodied learning have been shown on different levels of cognition that include passive viewing of images20, recognition memory21 and understanding of concepts18,19,22. The extent to which the way of learning shapes knowledge representation is still debated and may depend on the task at hand, but there are examples suggesting that the mode of acquisition has a lasting influence on knowledge representation23,24. For example, even in educated adults, who no longer use their fingers for counting, number representation is linked to their finger counting technique during early childhood14. Physically representing knowledge in an interactive manner could therefore serve as a transformative pedagogical approach that extends beyond initial learning to influence how knowledge is maintained and retrieved.

One skill we naturally learn in an embodied fashion is spatial navigation. During navigation, we use multiple senses such as vision, vestibular and motor feedback to find our way. However, the necessity of individual senses during navigation remains debated, with several studies showing no differences in behaviour with varying amount of inputs25,26,27 while others show an effect on behavioural performance28,29,30,31,32,33. The exact reason for the discrepancy in these studies is unclear, but the field generally agrees that inputs from different senses are integrated34,35,36. Especially in ambiguous situations, multi-sensory integration reduces error and thus is particularly important when we are trying to navigate difficult terrain while keeping track of our point of origin and destination.

This idea of keeping track of where we are, where we came from and where we are going entails the formation of a representation of the environment, the so-called cognitive map as it was first introduced by Tolman37. The cognitive map can be used to flexibly guide behaviour to find known paths as well as unknown short-cuts. 70 years of investigating Tolman’s claims has led to the identification of several core mechanisms for spatial navigation that span across species (for review38,39).

Beyond classical spatial navigation theories, a growing body of research suggests that spatial navigation mechanisms are involved in processing more than physical space40,41,42,43,44. This entails the formation of a cognitive map of abstract domains organized in n-dimensions41. For example, imagine a task for which you have to judge the similarity between animals. Given a giraffe, a rabbit and a seagull, depending on which dimensions you consider, you will probably reach different conclusions about their similarity: seagulls and rabbits are more similar when you consider size of the animal, population number and habitat range, while rabbits and giraffes are more similar when you consider diet and taxonomy. Flexibly accessing knowledge and making decisions about similarity can thus be seen as estimating distances between points (animals) in a high-dimensional space (animal features), similar to how you would judge the distance between the town hall and the train station when you want to figure out how much time you will need to walk between these places.

Evidence for abstract cognitive spaces have been observed in social45,46, semantic47, frequency48, odor49, abstract50,51,52 and musical53 domains. These studies had participants navigate in the space and form associations with selected positions. Critically, all studies used the abstract domains only and did not use an intuitive visuospatial representation of the domain. The tasks used in these studies resemble physical space navigation in their dimensionality, but are usually missing a first-person perspective. During navigation of these abstract spaces, single-cell recordings have demonstrated the recruitment of spatially tuned cells48,49. Consistently, modulations of BOLD-responses by inter-stimulus distances and grid-like hexadirectional signals have been observed in human fMRI50,51.

What many of these studies are lacking is the first-person perspective resembling physical space navigation and the interactiveness by which we typically explore our environment. When navigating physical space, it is proposed that egocentric representations gained through a first-person perspective may be transformed to an allocentric reference frame to support a viewpoint independent map-like representation54. By creating an immersive abstract space that can be navigated from a first-person perspective, we can measure similar transformations in an abstract environment. This would then afford a greater comparison to the processing principles underlying physical navigation.

By combining findings from embodied learning with well-established behavioural patterns as present in spatial navigation, we aim to explore the idea that abstract knowledge can be represented in a map-like fashion and that it can be navigated using physical movement. Such a representational format would eliminate the hurdle of an intuitive spatial representation, since it could be applied to any type of knowledge that can be organized along a number of dimensions. More concretely, we want to build on the idea that physical movement, via multi-sensory integration, limits ambiguity and error in a difficult environment, which could be beneficial in abstract space navigation. Further, physical exploration of a concept has been shown to improve performance in embodied learning research19. We propose that a physically navigable conceptual space should therefore help participants build up a mental representation, which in turn should lead to better distance and direction estimations.

To test the idea, we created a two-dimensional abstract quantity space (Fig. 1 and Supplementary video). This means participants saw quantities of two different geometric shapes: circles and rectangles. Each shape quantity defined the position along the respective axis of the space, so that each set of quantities encoded a position. Participants could manipulate the quantities to move between positions. To move in different directions, participants learned to connect a colour-code to their facing direction in the space. We trained participants extensively to use the colour and the quantities to navigate through this abstract space, and in the end, tested their ability to estimate distances and angles between positions, an ability supported by map-like representations of space. However, participants were never told that the abstract knowledge they were acquiring was spatialized by our task design. We deliberately refrained from explaining the relation between position and quantities as well as colour and direction and avoided spatial instructions as much as possible throughout the experiment. Critically, we were interested in two main questions: can participants learn to navigate such an abstract space using colour as directional and quantity as positional information? Second, are participants who navigate this space by physically exploring it more accurate in their distance and angle estimations than those who use a gamepad? Participants were randomly assigned to one of two groups. The movement group (MG) navigated the abstract space by walking and rotating on an omni-directional motion platform while visual feedback was provided in a head-mounted display. The gamepad group (GG) viewed the space also using the head-mounted display and standing up, but the motion sensors were disabled, so that participants used a gamepad for rotations and translations in the space. Both groups underwent extensive behavioural testing, with several different measures to identify similarities and differences to physical space navigation.

## Methods

### Participants

We used G*Power (3.1.9.2) to calculate the sample size per group aiming to achieve a power of 0.95 when using one-sided t-tests, with an assumed medium effect size of 0.5. Sample size was determined to be 45 participants per group. To account for drop-out, 103 participants were recruited from the Trondheim community to participate in the study. All experimental procedures were approved by the Regional Committee for Medical Health and Research Ethics (2018/1310). The study was carried out following all relevant guidelines and regulations in Norway. Prior to the study, participants provided their written informed consent and were screened for contraindications. Participants were compensated for their time at a rate of 100 NOK per hour. Of the initial 103 participants, five were unable to complete the task because of motion sickness or fatigue. An additional seven were excluded due to difficulty understanding the instructions and learning the task, and one participant was excluded due to participation in a related study. Therefore, 90 participants entered the analysis with 45 participants in each group (MG: 20 female, age range 20–34 years, mean age 25.2 years; GG: 23 female, age range 19–33 years, mean age 24.4 years). The MG was collected before the GG so that time point of sign-up for the study determined if a participant belonged to the MG or the GG. By collecting the MG prior to the GG we were able to match movement speed and exposure time between groups.

### Quantity space

Participants were asked to navigate a two-dimensional abstract quantity space. This space was defined by the amount of circles and the amount of rectangles. Thus, a position in the space is defined as a point in a coordinate system defined by the number of circles and the number of rectangles (see Fig. 1). Each shape axis spanned from 0 to 200 shape quantities which translates directly into conceptual meters: the area of the quantity space was $$200 \times 200$$ conceptual meters. To match each position in visual complexity, we added triangles as filler shapes. This way each position had 400 items. The specific positions of the shapes was randomized in the visual field. Hence, a location appeared slightly different each time it was visited.

To indicate heading direction, we applied a unique colour code to each shape dimension and the placeholder shapes. The colour code ranged from red to blue, and all colours were matched in luminance. The colour of the dimension 1 and dimension 2 shapes scaled with the cosine and sine of the heading angle, respectively. It was balanced whether the circles or rectangles (and their corresponding sets of placeholder shapes) were assigned dimension 1 or dimension 2 (see Fig. 1b for an illustration). Participants learned that if shapes were red (positive cosine or sine) and they moved forward, the quantity of this shape would increase. If the shapes were blue (negative cosine or sine), the quantity would decrease. If the shapes were purple (cosine or sine at zero), no change in the given shape quantity would occur. For an example of the colour code, see the quantity space Video S1 in the Supplementary information (also available on the OSF, see Additional information).

### Experimental set-up

The virtual environment was programmed in Vizard (version 5.7, WorldViz LLC), a python-based VR software. To allow participants to physically navigate the conceptual space, we employed a novel VR set-up consisting of an omnidirectional motion platform (Cyberith Virtualizer) and a head mounted display (Oculus Rift CV1). The HMD had a resolution of $$1080 \times 1200$$ pixels and a refresh rate of 90 Hz. In the motion platform, participants were secured in a harness. The harness was attached to a rotatable ring, affording the ability to freely rotate. Participants wore low-friction overshoes which allowed them to slide their feet across the baseplate as they stepped forward. The virtualizer generated a speed value from the sliding motion across the baseplate. Orientation and speed were measured at 90 Hz.

In the MG, the motion platform was used to navigate the space. The speed and ring orientation readings controlled the participant’s position, and the HMD orientation readings were used to update the angular colour code. In the GG, an Xbox One controller was used to navigate the space. The left stick on the controller determined the angular velocity while the right trigger determined the translational velocity. The translational velocity for both the GG and MG was smoothed by four frames (0.044 s).

### Tasks and analysis

Participants completed three tasks including navigation training, distance and direction estimation as well as a two alternative forced choice task. All tasks are described in detail in the following sections. See Fig. 2 for an illustration. The instructions for all tasks avoided spatial language as much as possible. For example, participants were told to “create patterns” rather than to “navigate”. We instructed participants to change colour and add shapes to create the pattern. All analyses were calculated in Matlab R2019a or RStudio 1.2.5.0.33 with R-3.6.355. For all analyses the alpha-level was set to 0.05 unless otherwise stated. We further used Bayes factor analysis and will follow the convention for Bayes factor cut-offs and interpretation given in56,57.

Participants performed a navigation training task in which they navigated between a start and a goal position (Fig. 2b). They were cued with an image of the goal location at the beginning of each trial, and they were able to retrieve this image at any point during the task. The shapes defining the goal were presented in black, as to not prime the participants with a colour code. Participants first completed a series of training trials, and later a set of testing trials. For the initial five training trials, participants were notified when they were in range of 10% of the goal location along both dimensions (100% being the total length of a dimension). For the subsequent three training trials, participants indicated when they reached the goal location with a button press. An error percentage was calculated by dividing the euclidean distance from the response location to the goal location by the maximum distance in the space. The error percentages were assigned into 5% increments from 0 to 100%, with each error range corresponding to a point score. Feedback was provided out of a maximum of 20 points, and displayed for 5 s. Along with the point value, a feedback message was shown to the participant depending on the error percentage.

For all training trials, participants were given unlimited use of a hint button. The hint indicated the distance to the goal in each dimension, and what colour the participant should set the shapes. For the subsequent testing trials, the hint function was disabled. Participants had to complete a minimum of 14 testing trials. If this minimum number was met, participants in the MG continued until the time in the task reached 54 min. If the participants met the minimum number of testing trials in the GG, they continued until they completed 28 trials. This was done in order to keep the mean number of trials consistent between groups. All testing trials were self-paced and there was no time limit. Goal locations were distributed across the entire space and were at least 30 conceptual meters from the start position. The navigation was continuous, meaning participants used the indicated goal location of the previous trial as the new starting position.

Mean movement speed for the MG was 9.004, $${\hbox {se}} = 0.458$$, for the GG’s mean movement speed was 9.099, $${\hbox {se}} = 0.039$$ . The speed was calculated using all testing trials of the task. Each trial was broken into segments of continuous movement. We excluded stationary periods and segments when the participant was viewing the goal or hint. If a distance of at least 10 conceptual meters was travelled within a segment, the mean velocity for the segment was calculated. This mean velocity was then calculated across all valid segments for each participant, and finally among participants. One subject in the GG never travelled a continuous 10 conceptual meters without pausing. This participant was therefore excluded from the calculation. We did not exclude this participant from the study, as the participant was not an outlier in other analyses. The movement speed corresponds to the number of items added/deleted per second during movement. Further, we added a slight fading function to the appearance of the individual shapes to ensure participants perceived the space as continuous. Additional tasks such as indicating responses and viewing messages were enabled by an Oculus remote for the MG, and buttons on the Xbox One controller for the GG.

To assess performance in the navigation task, we looked at three main variables: accuracy, path length and navigation time. Accuracy was calculated by creating a random distribution of 1000 possible end-positions and calculating the proportion of positions that are further away from the true goal than the indicated end-position. This was done for each trial and each participant. An accuracy of 0.95 thus means that 95% of the random positions are further away and thus corresponds to very high accuracy. Excess path length was calculated for each trial by subtracting the optimal path length from the taken path length. This difference was normalized by dividing it by the optimal path length. Navigation time was the time participants spent navigating, excluding goal retrievals and breaks from the trial time. General performance was assessed using t-tests against chance for each group individually as well as non-paired two-sided t-tests to compare groups as implemented in R (function t.test). As supporting evidence for group comparisons, Bayes factor analyses were conducted to reveal more detailed information and hidden trends. Bayes factor analysis allows us to compare two models quantifying how much more evidence we have for one model over the other. Here we compare the null model (there is no difference between groups) with the alternative model (the groups differ). Thus we can also find evidence for equality of the groups, not just the absence of an effect. We used the BayesFactor package in R with the ttestbf function. We adhere to standard Bayes factor interpretation and cut-offs as given in56,57.

We hypothesised that, similarly to observations of positional memory in physical space41,58, accuracy in close proximity to boundaries would be higher than for positions in the center. For this, we correlated distance to the border with accuracy for each participant and then tested the correlation values on a group level against zero.

We assessed potential dimensional biases in goal perception by calculating the mean error in goal location for each participant for each of the two dimensions: circles (C) and rectangles (R), and compared them on a group level with a paired t-test. Additionally, we compared the two groups with a non-paired t-test on the difference between the circle and rectangle dimension.

We looked at learning during the navigation training by fitting general linear models (GLM) for each participant individually. We then extracted the t-statistic for the regressor of interest for each participant. Since the t-statistic does not follow a normal distribution, we used a sign-flipping permutation method to create a null-distribution. This entails randomly sign-flipping the t-statistics and calculating the mean of the new distribution. This procedure was repeated 10,000 times. The true mean of the t-statisic distribution was then compared to the sign-flipped mean-distribution. We could then calculate the proportion of means that are larger (or smaller) than our true mean, resulting in a p-value that we assessed using an $$\alpha$$-level of 0.05. Z-values are created by subtracting the p-value from 1 and computing the inverse of the normal cumulative distribution function.

#### Distance and direction estimation task

After participants had completed the navigation training, we tested whether they accurately represented distance and directional relations within the abstract space (Fig. 2c). To this end, we asked participants to estimate the angle and distance between two cued locations. The goal location was presented in the same manner as in navigation training task, and could be viewed a maximum of two times with each view limited to 10 s. The first view occurred at the beginning of each trial, and the second view could be used by the participant at any point. After the first view of the goal, participants were given the starting location. They had to first estimate the direction from that location to the goal location. Participants indicated what colour the shapes should be by turning their head in the correct direction (MG) or using the controls on the gamepad (GG). They indicated their response with a button press, and the colour of the shapes was locked for the remainder of the trial. Participants then estimated how long it would take to create the target pattern from the start pattern by using a virtual slider. We used temporal distance instead of spatial distance since spatial language was avoided in the instruction of the task. The slider was set on a scale from 0 s to the time it took the participant to traverse the maximum distance in the space. The slider contained 29 increments from position 0 to position 28. For each trial, the initial slider position was always set to the center. Participants completed three practice trials using randomly generated locations. They then completed 36 testing trials. Trials evenly sampled locations separated by one of four distance bins (20–180 conceptual meters) and nine direction bins ($$40^{\circ }$$ spacing). All test positions were at least 10 conceptual meters apart from each other as well as from the axis limits in order to span the entire space. Testing trials were organized into four blocks of nine trials. No feedback was given in this task.

For the analysis, we correlated distance estimates with the true distances for each participant and tested the resulting correlation values on a group level against 0. We then compared the two groups against each other, first with a traditional t-test and second with Bayes factor analysis to reveal possible trends in the data. Since the abstract space allows us to look at distance and visual similarity between positions separately, we created linear regression models for each participant modelling the estimated distance with the true distance and the visual similarity. Visual similarity was expressed by the normalized number of shared shapes between two positions. We standardized visual similarity by calculating visual similarity between the center position and each position on a radius of 100 conceptual meters away from the center, then normalizing all values. A detailed explanation of visual similarity can be found in Fig. 3. Colour did not factor into visual similarity here, since all tasks took measures to control for colour in the comparisons, either by only comparing stimuli with the same colour code or by presenting the stimuli in all black. We thus created a “look-up” table for visual similarity as a function of angle, independent of distance. With the same permutation method as described above, we compared the t-statistics for the two regressors.

We analysed the angles by calculating the mean absolute angular error over all 36 trials for each participant followed by a t-test against chance ($$90^{\circ }$$) on a group level. We further compared groups with a t-test and Bayes Factor analysis to elaborate on the results. Next we wanted to establish if the distance and/or the visual similarity between positions had an effect on the angular error. Again we fitted a linear regression model for angular error for each participant, adding distance and visual similarity as regressors.

#### Forced choice task

To determine whether perception of closeness were affected by visual similarity beyond Euclidean distance between positions, we sampled choices that were at directions maximally similar ($$135^{\circ }$$ and $$315^{\circ }$$) and maximally dissimilar ($$45^{\circ }$$ and $$225^{\circ }$$) in terms of visual similarity (Fig. 2d). This is possible since the number of items shared between two positions is not only dependent on the distance but also on the angle between positions (see Fig. 3). The goal was placed in the center of the space 90–100 conceptual meters away from the axis limits in each dimension. One choice was distorted by 10 conceptual meters in one similarity condition (similar or dissimilar) while the other choice was distorted by 60, 90, or 120 conceptual meters in the alternative similarity condition. Locations were presented in a colour code corresponding to $$45^{\circ }$$ or $$225^{\circ }$$. In the visual similarity sub-task, we had 96 trials with 16 repetitions per each sampled distance condition.

Along with testing the effect of visual similarity, we tested for a bias of angular alignment and colour priming. We aimed to determine whether the directional colour code that participants learned could prime them to choose a certain location. To test this, we presented half of the trials in the dissimilar condition with a colour code that was aligned with the correct answer’s true direction, and the other half in a colour code that was misaligned. This was only possible in the dissimilar condition because we used colour codes corresponding to $$45^{\circ }$$ and $$225^{\circ }$$. Note that all three positions A, B, and C were always presented in the same colour code, to make it easier to focus on the shapes only.

We also tested if there was a dimensional bias among the quality dimension shapes. To this end, we sampled positions in the lower left and upper right hand corner of the space where there was a dominance of placeholder and quality dimension shapes, respectively (Fig. 2e). This dominance stems from the logic of our quantity space: in the upper right corner, both quantity dimensions are near their upper limit, meaning participants saw a lot of circles and rectangles but only very few triangles. In the lower left corner, both quantity dimensions were near their lower limit, with only very few circles and rectangles but a lot of triangles present on the screen. Visually, differences in very low quantities are more salient. We therefore hypothesised that a dimensional bias would be accessible in these extreme cases. In this task, the goal was placed at a random location between 4 and 10 conceptual meters away from each axis limit for either corner. The choices were each distorted from the goal in a different dimension. One location was always distorted from A by 5 conceptual meters, while the other location was distorted from A by either 15, 30, or 55 conceptual meters in the alternate dimension. This created a layout in which one choice was always 10, 25, or 50 conceptual meters closer to A than the alternate choice. The dimensional bias sub-task consisted of 120 trials with 60 repetitions for both the upper and lower corner, and 10 repetitions per each of the 6 distance difference scenarios. The locations in this task were presented in a colour code corresponding to $$135^{\circ }$$ and $$315^{\circ }$$, with a random jitter of $$10^{\circ }$$ in either direction.

Analysis was run in R using the anova_test function from the rstatix toolbox. The data were collapsed across the different distances tested to maximize the number of trials. Further, performance or accuracy here is defined as percentage correct. To investigate the effect of visual similarity we used the data from the first sub-task and ran a three-way mixed ANOVA with one between factor of group (MG or GG) and two within factors for time point (pre and post) and visual-similarity (high and low). We checked ANOVA assumptions for outliers, normality and homogeneity, correcting for sphericity in our analysis. The exclusion of two potential outliers identified in these assumption checks did not change the interpretation of the statistical analysis and we thus report statistics based on data from all participants.

To investigate the effect of colour priming, we looked only at trials in which the low-visual similarity option was the correct choice. We again used a three-way mixed ANOVA with one between factor (group) and two within factors for time point and alignment (aligned or misaligned with correct choice). Excluding outliers from the analysis did not affect the results.

For the second sub-task (the one investigating the possibility of a dimensional bias), we calculated a four-way mixed ANOVA with one between factor of group (MG or GG) and three within factors of time point (pre- or post-training), shape (circles or rectangles) and corner (lower left or upper right). Again, we identified extreme outliers that opposed ANOVA assumptions of normality. Since excluding the outliers did not change the results, we decided not to exclude them from this analysis.

## Results

#### General performance

Here, we assessed the general performance of the participants in the navigation training to ensure participants in both groups understood the task and navigated in the instructed way. Firstly, we looked at accuracy, which is a measure of how close an indicated goal location is to the true goal location in comparison to a random distribution of possible locations. Both groups performed significantly above chance in indicating the goal location ($${\hbox {chance-level}} = 0.5$$, $${\hbox {mean}}_{\mathrm{MG}} = 0.8623$$, $${\hbox {t}}_{\mathrm{MG}} = 33.6084$$, $$\hbox {df}_{\mathrm{MG}} = 44$$, $$\hbox {p} < 0.001$$, $$\hbox {mean}_{\mathrm{GG}} = 0.8641$$, $$\hbox {t}_{\mathrm{GG}} = 38.6638$$, $$\hbox {df}_{\mathrm{GG}} = 44$$, $$\hbox {p} < 0.001$$) but did not differ from one another ($$\hbox {t} = 0.1266$$, $$\hbox {df} = 86.4404$$, $$\hbox {p} = 0.8995$$). Bayes factor analysis revealed moderate evidence for the equality of the groups ($$\hbox {BF}_{\mathrm{10}} = 0.2223$$). The mean trial duration for the MG was 116 s and differed significantly from trial durations of the GG with a mean of 89 s ($$\hbox {t} = -3.3006$$, $$\hbox {df} = 81.792$$, $$\hbox {p} = 0.0014$$). The difference between groups is also strongly supported by Bayes factor analysis ($$\hbox {BF}_{\mathrm{10}} = 22.5455$$). Excess path lengths did not differ significantly between groups ($$\hbox {t} = -1.9457$$, $$\hbox {df} = 82.107$$, $$\hbox {mean}_{\mathrm{MG}} = 1.2594$$, $$\hbox {mean}_{\mathrm{GG}} = 0.9517$$, $$\hbox {p} = 0.0551$$) with Bayes Factor analysis providing only anecdotal evidence against the null hypothesis ($$\hbox {BF}_{\mathrm{10}} = 1.1476$$).

As has been found in physical space navigation58,59, we expected to find an effect of distance to the boundary, i.e. accuracy should increase closer to the boundary. For both the MG and GG, we found significant negative correlations for distance to border and accuracy ($$\hbox {mean}_{\mathrm{MG}} = -0.1929$$, $$\hbox {t}_{\mathrm{MG}} = -5.646$$, $$\hbox {df}_{\mathrm{MG}} = 44$$, $$\hbox {p}_{\mathrm{MG}} = 1.115{\mathrm{e}}^{-06}$$, $$\hbox {mean}_{\mathrm{GG }}= -0.2043$$, $$\hbox {t}_{\mathrm{GG}} = -7.2935$$, $$\hbox {df}_{\mathrm{GG}} = 44$$, $$\hbox {p}_{\mathrm{GG}} = 4.245{\mathrm{e}}^{-09}$$) but no difference between groups ($$\hbox {t} = 0.2586$$, $$\hbox {df} = 84.747$$, $$\hbox {p} = 0.7965$$). Participants in both groups thus showed higher accuracy for goal positions at the borders than in the center, matching findings in physical space navigation.

Finally, we tested for a dimensional bias in the indicated goal location. A dimensional bias would mean participants perceive differences in quantities at different thresholds for the circle and the rectangle dimension. We found no significant dimensional bias in the MG ($$\hbox {mean}_{\mathrm{C}} = 24.5346$$, $$\hbox {std}_{\mathrm{C}} = 8.7333$$, $$\hbox {mean}_{\mathrm{R}} = 23.4573$$, $$\hbox {std}_{\mathrm{R}} = 8.6842$$, $$\hbox {t} = -1.1111$$, $$\hbox {df} = 44$$, $$\hbox {p} = 0.2726$$) but found a significant bias in the GG, with lower error in the rectangle dimension than in the circle dimension ($$\hbox {mean}_{\mathrm{C}} = 25.6617$$, $$\hbox {std}_{\mathrm{C}} = 10.5541$$, $$\hbox {mean}_{\mathrm{R}} = 22.2035$$, $$\hbox {std}_{\mathrm{R}} = 7.3661$$, $$\hbox {t} = -29786$$, $$\hbox {df} = 44$$, $$\hbox {p} = 0.0047$$). When comparing the two groups against each other with a non-paired t-test of the difference between error in the circle dimension and in the rectangle dimension we found no significant difference ($$\hbox {t} = -1.5741$$, $$\hbox {df} = 85.2889$$, $$\hbox {p} = 0.1192$$). The dimensional bias in the GG but not in the MG will be discussed below.

#### Learning

As a second step we included a separate analysis on the navigation training data, aiming to quantify participants’ learning behaviour. Specifically, we looked for improvements in navigation time, path lengths and accuracy. Navigation time and path lengths can be viewed as measures for efficiency and certainty, accuracy as a measure of precision.

To assess learning over the course of the training periods, we predicted navigation times from trial numbers in participant-specific linear regression models (Fig. 4a,b). Trial number negatively predicted navigation time in the MG ($$\hbox {p} = 0.0001$$, $$\hbox {z} = 3.719$$) and the GG ($$\hbox {p} = 0.0001$$, $$\hbox {z} = 3.719$$). This indicates that participants became faster with more experience, thereby demonstrating learning. When comparing both groups using a non-paired, two-sided t-test we found no significant difference ($$\hbox {t} = -0.0377$$, $$\hbox {df} = 87.919$$, $$\hbox {p} = 0.97$$). Bayes Factor analysis provides moderate evidence for the equality of groups ($$\hbox {BF}_{\mathrm{10}} = 0.2209$$). Taken together, these results suggest both groups show an equal amount of improvement in their navigation durations.

Next, we predicted path lengths from optimal path lengths and trial numbers (Fig. 4c,d). We found that trial number negatively explained path lengths for the MG ($$\hbox {p} = 0.0001$$, $$\hbox {z} = 3.719$$) but not for the GG ($$\hbox {p} = 0.0674$$, $$\hbox {z} = 1.495$$), suggesting that the MG improved in finding the most direct path, while the improvement in the GG did not reach statistical significance. Testing for group differences in learning using a non-paired two-sided t-test, we failed to see a difference ($$\hbox {t} = -1.9764$$, $$\hbox {df} = 87.865$$, $$\hbox {p} = 0.0512$$). In addition, using Bayes factor analysis to elaborate on this trend we found only anecdotal evidence for a difference between groups ($$\hbox {BF}_{\mathrm{10}} = 1.2086$$). As shown in the general performance section, mean excess path lengths did not differ between groups. This suggests that the absence of a learning effect in the GG was due to more accurate navigation in early trials, making it difficult to compare learning effects.

Lastly, we looked at accuracy scores, quantifying the precision with which participants navigated to target positions (see “Methods”, (Fig. 4e,f)). Here, we modeled accuracy from trial numbers and observed no significant effect for either the MG ($$\hbox {p} = 0.6831$$, $$\hbox {z} = -0.4764$$) or the GG ($$\hbox {p} = 0.2926$$, $$\hbox {z} = 0.5458$$). We next compared learning between groups, first using a non-paired, two-sided t-test finding no significant difference ($$\hbox {t} = -0.7247$$, $$\hbox {df} = 87.314$$, $$\hbox {p} = 0.4706$$) and second using Bayes factor analysis ($$\hbox {BF}_{\mathrm{10}} = 0.2782$$), revealing moderate evidence for the equality of groups. Accuracy thus remained stable over the experiment, which was likely due to participants being allowed to look at the goal as often as they liked, where participants would match their current position to the goal position.

### Distance and direction estimation task

In the estimation task, participants judged the angle and distance between pairs of positions (36 trials), which allowed us to probe their ability to plan goal-directed trajectories through the quantity space. The estimation task was conducted right after the navigation training (see Fig. 2a).

To test if participants could accurately estimate distances between a start and a goal location, we compared true distances with their distance estimates and found significant positive correlations in both the MG ($$\hbox {t} = 16.083$$, $$\hbox {df} = 44$$, $$\hbox {mean Pearson r} = 0.4968$$, $$\hbox {p} < 0.001$$) and the GG ($$\hbox {t} = 19.288$$, $$\hbox {df} = 44$$, $$\hbox {mean Pearson r} = 0.5353$$, $$\hbox {p} = 0.001$$). We then compared the two groups against each other, finding no significant effect ($$\hbox {t} = 0.9269$$, $$\hbox {df} = 87.01$$, $$\hbox {p} = 0.3565$$). These results provide evidence that both groups performed the distance estimation task successfully, but didn’t differ from one another. Furthermore, a Bayes Factor analysis supports the absence of a difference between groups ($$\hbox {BF}_{\mathrm{10}} = 0.3221$$). Next, we wanted to assess factors that could drive distance estimates. We used visual similarity and true distance to predict distance estimates and found that both distance ($$\hbox {p}_{\mathrm{MG}} = 0.0001$$, $$\hbox {z}_{\mathrm{MG}} = 3.719$$, $$\hbox {p}_{\mathrm{GG}} = 0.0001$$, $$\hbox {z}_{\mathrm{GG}} = 3.719$$) and visual similarity ($$\hbox {p}_{\mathrm{MG}} = 0.0001$$, $$\hbox {z}_{\mathrm{MG}} = 3.719$$, $$\hbox {p}_{\mathrm{GG}} = 0.0001$$, $$\hbox {z}_{\mathrm{GG}} = 3.719$$) were significantly contributing to participants’ estimates (Fig. 5). This illustrates that both true distances as well as visual similarity affect judgement in this task.

In addition to estimating the distance of a given start position to a goal location, we asked participants to estimate the direction to the goal location. By regarding both components we could assess the accuracy of the mental representation of the abstract space. Both groups showed absolute angular errors smaller than expected from chance level behaviour ($$\hbox {t}_{\mathrm{MG}} = 17.599$$, $$\hbox {df}_{\mathrm{MG}} = 44$$, $$\hbox {mean}_{\mathrm{MG}} = 46.9936$$, $$\hbox {p}_{\mathrm{MG}} < 0.001$$, $$\hbox {t}_{\mathrm{GG}} = 16.64$$, $$\hbox {df}_{\mathrm{GG}} = 44$$, $$\hbox {mean}_{\mathrm{GG}} = 48.9885$$, $$\hbox {p}_{\mathrm{GG}} < 0.001$$), showing they could estimate directions with good accuracy (Fig. 5). We further compared groups, but found no significant difference ($$\hbox {t} = 0.5019$$, $$\hbox {df} = 87.175$$, $$\hbox {p} = 0.617$$). Using Bayes Factor analysis to elaborate on these results ($$\hbox {BF}_{\mathrm{01}}= 0.2466$$), we found slightly lower angular errors for the MG than for the GG. Next we tried to predict absolute angular error from distance and visual similarity and found distance to be a significant predictor ($$\hbox {p}_{\mathrm{MG}} = 0.0001$$, $$\hbox {z}_{\mathrm{MG}} = 3.719$$, $$\hbox {p}_{\mathrm{GG}} = 0.0001$$, $$\hbox {z}_{\mathrm{GG}} = 3.719$$), while visual similarity was not a significant predictor ($$\hbox {p}_{\mathrm{MG}} = 0.7814$$, $$\hbox {z}_{\mathrm{MG}} = -0.7769$$, $$\hbox {p}_{\mathrm{GG}} = 0.1637$$, $$\hbox {z}_{\mathrm{GG}} = 0.9794$$). See Fig. 5 for illustrations of these results. Direction unlike distance perception was thus independent from visual similarity, but absolute angular error did scale with distance; suggesting an integration of the two features.

### Forced choice task

The forced choice task was performed before as well as after the navigation training and estimation task. We therefore have two time points: “pre” and “post”. We further designed trials in two different ways, to test for effects of visual similarity and colour priming (sub-task one) and dimensional bias and symmetry of space (sub-task two, see “Methods”).

#### Effect of visual similarity

In this part of the forced choice paradigm, we chose positions B and C to be either aligned or misaligned with the visual similarity distribution, to test if participants showed a bias towards choosing the visually similar choice even if this was not the choice closer to the probe. Further, we aimed to investigate if a possible bias was reduced with training, i.e. participants learned to base their choice more on distance than on visual similarity. We performed a three-way mixed ANOVA with one between factor of group (MG or GG) and two within factors for time point (Pre or Post) and visual similarity (high or low). We found a main effect of time point ($$\hbox {F} = 19.974$$, $$\hbox {df} = 88$$, $$\hbox {p} = 2.33{\mathrm{e}}^{-05}$$) and of similarity ($$\hbox {F} = 356.811$$, $$\hbox {df} = 88$$, $$\hbox {p} = 1.03{\mathrm{e}}^{-32}$$) providing evidence that performance increased after training and that high-visual similarity affects distance judgements. We observed no effect of group ($$\hbox {F} = 0.015$$, $$\hbox {df} = 88$$, $$\hbox {p} = 0.902$$) and no interactions. See Fig. 6 for an illustration and Supplementary Table S1 for the full ANOVA table. Participants thus showed a visual similarity bias, which was not reduced after training.

Next we used only half of the trials (those in the low-visual similarity condition) to assess an effect for color priming or angle priming. We hypothesised that if the applied colour (colour encodes direction) is aligned with the angle between AB or AC, the aligned position may be a more likely choice. We again used a three-way mixed ANOVA with one between factor (group) and two within factors for time point and alignment (aligned or misaligned with correct choice). We found no main effect of alignment ($$\hbox {F} = 0.125$$, $$\hbox {df} = 88$$, $$\hbox {p} = 0.725$$), group ($$\hbox {F} = 0.067$$, $$\hbox {df} = 88$$, $$\hbox {p} = 0.796$$) nor any interaction effects (Supplementary Table S2). We again found the main effect of time point ($$\hbox {F} = 10.1369$$, $$\hbox {df} = 88$$, $$\hbox {p} = 0.002$$). We could therefore not find any evidence for colour priming induced by learning.

#### Effect of dimensional bias

The second sub-task of the forced choice task compared biases participants may have for the two dimensions defined by the rectangles and circles. A dimensional bias would entail, that differences in quantities may be more accurately perceived in either the circle or the rectangle dimension. Therefore, we chose position B to differ from A only in one dimension, and position C to differ from A in the other dimension. Further, we tested positions in the upper right and lower left corner of the space to determine if participants’ performance changed when they had to rely on the filler items instead of the circle and rectangle dimensions and if there was a systematic difference in performance in these corners of the space. Positions in the upper right corner were defined by very high amounts of circles and rectangles (low amounts of fillers). Positions in the lower left were defined by very low amounts of circles and rectangles (high amounts of fillers). We calculated a four-way mixed ANOVA with one between factor of group (MG or GG) and three within factors of time point (pre or post), shape (circles or rectangles) and corner (lower left or upper right). We found a main effect of time point ($$\hbox {F} = 59.989$$, $$\hbox {df} = 88$$, $$\hbox {p} = 1.53{\mathrm{e}}^{-11}$$), which suggest learning, and a main effect of shape ($$\hbox {F} = 7.69$$, $$\hbox {df} = 88$$, $$\hbox {p} = 7.00{\mathrm{e}}^{-03}$$) which suggests a dimensional bias. We found no other main or interaction effects (see Supplementary Table S3). In post-hoc analyses, we computed pair-wise comparisons between all subgroups looking for an effect of shape to understand the dimensional bias. We found a significant effect of shape only in the upper-right corner in both pre ($$\hbox {df} = 44$$, $$\hbox {p} = 0.017$$) and post ($$\hbox {df} = 44$$, $$\hbox {p} = 0.034$$) for the GG; however, this does not survive correction for multiple post-hoc tests (Supplementary Table S4). Since we find the subtle dimensional bias in pre in the GG but not in the MG, our experimental manipulation cannot explain the dimensional bias. See Fig. 7 for an illustration.

## Discussion

In this study, we created a first-person view abstract space navigation task. We closely matched aspects of physical space, such as the number of dimensions, mode of navigation and informational content. Our aim was to establish a new paradigm that combines both theories of cognitive spaces and embodied learning: an abstract space that uses first-person navigation, contains both positional as well as directional information, can be learned by physical exploration and can easily be extended. We used the paradigm to investigate a key question of the field: If we build up cognitive maps of any type of knowledge, and if physical interaction enhances learning, can we navigate an abstract space in the same manner we navigate the real world? Furthermore, does physical movement improve learning?

Our results suggest that participants can learn the representation of an abstract environment both through physical navigation and through the use of a gamepad. We find trial durations shortening over time in the navigation task as well as accurate performance in angle and distance judgements. Participants demonstrate learning in the forced choice task with increased performance after navigation training. We did not, however, find strong differences between groups, suggesting that our embodied learning manipulation did not aid in the facilitation of an accurate representation of the space.

Along with observing learning in the three sub-tasks, we found certain biases that resemble navigational biases during spatial exploration of the physical world. For example, in the navigation training we found that the estimation of positions was more accurate near the borders than in the center of the space. This effect becomes also evident when comparing the distance judgements in the forced choice test: during piloting we found that for center positions we needed larger differences of distances (50, 80 and 110) than for positions in the corners (10, 25, and 50). Yet, performance in the corner positions was still higher than in the center positions (see ”Supplementary information” text as well as Tables S5 and S6). This is consistent with findings from studies testing positional memory in navigation paradigms58,59.

Additionally, in the angle estimation task, visual similarity did not explain angular error. Instead, we found that the larger the distance between two positions the smaller the angular error. Participants thus made smaller errors in the angle estimation when the distances between positions were large. This is not surprising since the general direction from one position to the other, i.e. the differences in the quantities are more apparent when positions are far apart. For example, a participant is given a starting position A and has to estimate the angle to position B. When B is far away, the participant is able to reduce the possible angles considerably by judging if more or fewer circles as well as more or fewer rectangles are needed. Thus the $$360^{\circ }$$ decision space is reduced to a $$90^{\circ }$$ decision space. If position B on the other hand is close to A and potentially within the perceptual threshold, the decision space cannot be reduced as easily and the estimation is more difficult62. In typical spatial navigation research, this can be easily explained when researchers consider the visual angle also known as the angular size of a goal location: an object that is far away from you has a smaller visual angle on your retina than an object that is close to you. In our task, the same rules apply, when we take into account that participants do not have a dot-like representation of the individual positions but rather a field-like representation. This is evident in the perceptual thresholds of a position in the space (see forced choice paradigm). Estimating the exact angle to the goal thus becomes more difficult if the goal is close to you, compared to when it is far away.

In this study, we show that participants can navigate an abstract space in the same manner they navigate physical space, even showing similar behavioural biases. But, what are the implications of such a paradigm? For one, navigation is a well-trained, intuitive behaviour. Here we show that it can be applied to the acquisition of knowledge beyond the layout of a physical space. This opens up a new avenue of learning techniques: providing abstract information in a spatial manner, that in turn can be physically navigated. By mapping relationships between abstract stimuli onto a spatial representational format, learners could acquire the knowledge in an intuitive manner. Imagine for example, learning about politics or deciding whom to vote for. Politicians (or parties) can be represented as individual points (or areas) in the large n-dimensional space of opinion. We could for example, represent politicians along three dimensions: the degree to which they like to invest in science, the degree to which they see global warming as a problem and the degree to which they support capitalism. Using a spatial layout makes it very easy to estimate which two politicians have the most similar opinions (are the closest in space) and which differ the most. You could also navigate to your own position in this high dimensional space and then see which politicians are close to you, by potentially viewing the dimensions that are most important to you or using dimensional reduction techniques. Large amounts of information, i.e. all the different positions of the different politicians would therefore become spatially intuitive and easily comparable.

Spatial learning strategies are already established in the field of memorization: the method of loci uses an imagined mind palace, in which the to be remembered information is placed in different rooms. When the knowledge needs to be recollected, participants imagine navigating through their mind palace visiting the positions at which they previously placed the items. This memorization strategy has shown to be very effective63 not only in imagination64 but also in combination with a virtual environment65.

Language also uses several spatial metaphors to describe abstract relationships. “You couldn’t be further from the truth”, “A comes before B”, “It is close to midnight”, are examples of spatial language that describe non-spatial content. These types of metaphors have been shown to shape the way we think66 pointing at the inherent link of how we learn and how knowledge is represented in our brain67.

We constantly learn new, complicated concepts and making use of spatial representations with advanced VR techniques provides an enticing opportunity. Beyond the possibilities for learning, our paradigm could be useful for investigations of knowledge representations: do we represent knowledge in a spatial manner and do we use mechanisms from spatial navigation to access and manipulate non-spatial information? One interesting part of spatial navigation is the flexibility of switching between allocentric and egocentric reference frames. Relations or distances between landmarks and other positions in the space are preserved in both allocentric and egocentric maps67. In our study, we find evidence for such a preservation of relations between positions well as the integration of directional information and distances. This suggests that participants do acquire a cognitive map of the quantity space, even when they only use a first-person perspective to navigate the space. The paradigm itself is easily extendable with another dimension or additional “rooms” (by adding a new kind of shape, once one dimension is at its maximum). We could also include landmarks, obstacles, or distortions to test the limits of abstract cognitive maps.

Why was there no clear difference between groups? We had hypothesised that participants in MG would show clear differences in performance to GG, since they used physical movement to navigate and thus had multiple senses providing information that could be integrated and potentially reduce error.

One possibility is that our task provided too much information: path integration and multi-sensory integration was not necessary, since all information of position and direction was constantly available in the current view. Visual accessibility has been pointed out as a factor that renders physical movement for accurate formation of a cognitive map30 unnecessary. Our space was therefore easily solvable. Still we found perceptual (visual) biases in performance that could have been reduced after embodied learning. Here it is possible that effects of our group manipulation are only detectable with more training distributed over multiple days.

Further differences in performance may only be evident on a longer time-scale: Johnson-Glenberg et al.68, trained students in physics concepts by varying the degree of embodied learning in the knowledge acquisition. They found that performance in a physics test right after training did not differ between groups but when they tested the students again one week later they found a significant effect of embodiedness: the higher the level of embodiment at encoding the better the retention of the knowledge. For future studies that use embodied learning approaches we would therefore recommend post-tests to assess retention of knowledge gained.

Our finding that an abstract space can be navigated both with and in the absence of physical movement introduces future questions. One potential direction is to investigate how training over a longer duration could elicit an effect of embodied learning. Furthermore, it could be of interest to adapt this into an imaging study in an effort to understand the neuronal correlates of our behavioural measures. The use of a map-like representation of knowledge could also be interesting for research in generalization and transfer of knowledge41,42. For example, do we transfer relationships between items made in one space into another? Both the method of loci, as well as spatial language of metaphors point to space as a useful tool to understanding and remembering complicated relations40,66. We suggest that intuitive spatial behaviour paired with constantly improving VR technology could be a useful tool for learning. Future research should also aim to test if participants inherently organize abstract knowledge in a spatial manner, or if applying a spatial format to abstract knowledge through experimental manipulation is necessary for such an organization.

In conclusion, our paradigm shows that first-person navigation of abstract spaces is possible, using not only button presses but also physical movement. Participants are able to flexibly use colour and quantities as directional and positional information. Such a paradigm can be used to assess the usefulness of multi-sensory integration in the form of embodied learning (potentially on a longer time-scale)8, landmark- and boundary-based navigation in abstract spaces46 (i.e. prototyping), short-cuts, inferences and transferability of knowledge between different cognitive spaces. Physical exploration of an environment is an intuitive and already well-trained task. Presenting knowledge in a map-like fashion that can be explored thus poses an interesting opportunity for teaching as a new “learning strategy”: walking and creating connections between objects in an abstract space.

## Data availability

Accession codes Data and code is available at: https://osf.io/nvu4t/.

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## Acknowledgements

This work is supported by the European Research Council (ERC-CoG GEOCOG 724836). C.D.’s research is further supported by the Max Planck Society, the Kavli Foundation, the Centre of Excellence scheme of the Research Council of Norway—Centre for Neural Computation (223262/F50), The Egil and Pauline Braathen and Fred Kavli Centre for Cortical Microcircuits, the Jebsen Centre for Alzheimer’s Disease, and the National Infrastructure scheme of the Research Council of Norway—NORBRAIN (197467/F50).

## Author information

Authors

### Contributions

D.K., J.B. and C.D. originally designed the study and its hypothesis. N.S. programmed the task and collected the data, under the supervision of D.K.. D.K. analysed the data and wrote the first draft of the manuscript. R.K., J.B. and C.D. supervised the work. All authors contributed to the final version of the manuscript.

### Corresponding authors

Correspondence to Doerte Kuhrt or Christian F. Doeller.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

Supplementary Video S1.

## Rights and permissions

Reprints and Permissions

Kuhrt, D., St. John, N.R., Bellmund, J.L.S. et al. An immersive first-person navigation task for abstract knowledge acquisition. Sci Rep 11, 5612 (2021). https://doi.org/10.1038/s41598-021-84599-7

• Accepted:

• Published:


text:
# Questions & Answers of Mathematical Logic

#### Topics of Mathematical Logic 25 Question(s)

Question No. 1

The statement $\style{font-family:'Times New Roman'}{\left(\neg p\right)\Rightarrow\left(\neg q\right)}$ is logically equivalent to which of the statments below?

I. $\style{font-family:'Times New Roman'}{p\Rightarrow q}$

II. $\style{font-family:'Times New Roman'}{q\Rightarrow p}$

III. $\style{font-family:'Times New Roman'}{\left(\neg q\right)\vee p}$

IV. $\style{font-family:'Times New Roman'}{\left(\neg p\right)\vee q}$

Question No. 2

Consider the first-order logic sentence $\style{font-family:'Times New Roman'}{F:\forall x\left(\exists yR\left(x,y\right)\right)}$. Assuming non-empty logical domains, which of the sentence below are implied by F?

$\style{font-family:'Times New Roman'}{\mathrm I.\;\exists y\left(\exists xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{II}.\;\exists y\left(\forall xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{III}.\;\forall y\left(\exists xR\left(x,y\right)\right)}$

$\style{font-family:'Times New Roman'}{\mathrm{IV}.\;\neg\exists y\left(\forall y\neg R\left(x,y\right)\right)}$

Question No. 29

Let p,q, and r be proposition and the expression $\style{font-family:'Times New Roman'}{\left(p\rightarrow q\right)\rightarrow r}$ be a contradiction. Then, the expression $\style{font-family:'Times New Roman'}{\left(r\rightarrow p\right)\rightarrow q}$ is

Question No. 111

Let p, q, r, denote the statements "It is raining". "It is cold", and "It is pleasant", respectivelyThen the statement "It is not raining and it is pleasant, and if it is raining only if it is raining and it is cold" is represented by

Question No. 111

Consider the following expressions:

(i) false

(ii) Q

(iii) true

(iv) PQ

(v) ¬QP

The number of expressions given above that are logically implied by P(P Q) is__________ .

Question No. 19

Which one of the following is NOT equivalent to $p↔q$?

Question No. 111

Consider the following two statements.
S1: If a candidate is known to be corrupt, then he will not be elected
S2: If a candidate is kind, he will be elected
Which one of the following statements follows from S1 and S2 as per sound inference rules of logic?

Question No. 11

Consider the statement
“Not all that glitters is gold”
Predicate glitter(x) is true if x glitters and predicate gold(x) is true if x is gold. Which one of the following logical formulae represents the above statement?

Question No. 63

Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?

Question No. 163

Which one of the following Boolean expressions is NOT a tautology?

Question No. 211

Consider the following statements:
P: Good mobile phones are not cheap
Q: Cheap mobile phones are not good
L: P implies Q
M: Q implies P
N: P is equivalent to Q
Which one of the following about L, M, and N is CORRECT?

Question No. 263

The CORRECT formula for the sentence, “not all rainy days are cold” is

Question No. 27

What is the logical translation of the following statement?
“None of my friends are perfect.”

Question No. 47

Which one of the following is NOT logically equivalent to $¬\exists x\left(\forall y\left(\alpha \right)\wedge \forall z\left(\beta \right)\right)$ ?

Question No. 1

Consider the following logical inferences.

I1: If it rains then the cricket match will not be played.
The cricket match was played.
Inference: There was no rain.
I2: If it rains then the cricket match will not be played.
It did not rain.
Inference: The cricket match was played.

Which of the following is TRUE?

Question No. 13

What is the correct translation of the following statement into mathematical logic?

“Some real numbers are rational”

Question No. 30

Which one of the following options is CORRECT given three positive integers x, y and z , and a predicate

$P\left(x\right)=¬\left(x=1\right)\wedge \forall y\left(\exists z\left(x={y}_{*}z\right)⇒\left(y=x\right)\vee \left(y=1\right)\right)$

Question No. 30

Suppose the predicate F(x, y, t) is used to represent the statement that person x can fool person y at time t. which one of the statements below expresses best the meaning of the formula $\forall x\exists y\exists t\left(¬F\left(x,y,t\right)\right)$?

Question No. 23

Which one of the following is the most appropriate logical formula to represent the statement:

“Gold and silver ornaments are precious”

The following notations are used:

G(x): x is a gold ornament.
S(x): x is a silver ornament.
P(x): x is precious.

Question No. 24

The binary operation $\tiny\Box$ is defined as follows

 P Q P $\tiny\Box$ Q T T T T F T F T F F F T

Which one of the following is equivalent to P $\vee$ Q?

Question No. 26

Consider the following well-formed formulae:

I. $¬\forall x\left(P\left(x\right)\right)$
II. $¬\exists x\left(P\left(x\right)\right)$
III. $¬\exists x\left(¬P\left(x\right)\right)$
IV. $\exists x\left(¬P\left(x\right)\right)$

Which of the above are equivalent?

Question No. 30

Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton, and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent (a, b) means a and b are equivalent. Which of the following first order logic statements represents the following:
Each finite state automaton has an equivalent pushdown automaton.

Question No. 31

P and Q are two propositions. Which of the following logical expressions are equivalent?
I. P $\vee$ $~$ Q
II.$~$ ($~$ P $\wedge$ Q)
III. (P $\wedge$ Q)$\vee$ (P $\wedge$ $~$ Q) $\vee$ ($~$ P $\wedge$ $~$ Q)
IV. (P $\wedge$ Q) $\vee$ (P $\wedge$ $~$ Q) $\vee$ ($~$ P $\wedge$ Q)

format: plain


text:
8,646 views
Consider the following language.

$L = \{{ x\in \{a,b\}^*\mid}$number of $a$’s in $x$ divisible by $2$ but not divisible by $3\}$

The minimum number of states in DFA that accepts $L$ is _________

### 1 comment

Number of $a’s$ in $x$ are divisible by $2$ but not divisible by $3$ means number of $a's$ can be $2,4,8,10,14,16,20,22,26,\ldots$

Terms which are divisible by both $2$ and $3$ like $6,12,18,24,\ldots$ are dropped from the above sequence at finite interval in $*$ position from above sequence $2,4,*,8,10,*,14,16,*,20,22,*,26,28,*,32,34,\ldots$

Here, sequence $\{6,12,18,24,\ldots\}$ is in AP. Now, to drop this sequence from our original sequence which represent the number of $a's$, we have to go through at least total $7$ states which says total $6$ $a's$ have seen so far and since terms of this sequence come at regular interval, So, we will make a cycle in the DFA as :

Now, observe that states $0$ and $6$ are equivalent because by definition, $2$ states $p$ and $q$ are equivalent i.e.

$p \approx q \overset{\textit{def}}\Leftrightarrow \forall x \in \Sigma^{*}(\hat{\delta}(p,x) \in F \Leftrightarrow \hat{\delta}(q,x) \in F)$

It says $2$ states are equivalent if for all strings, they both either go to final state or both go to non-final state means they have the same nature.

So, here, we can merge states $0$ and $6$. Other states can't be merged because they are in cycle and if we merge states which are in cycle, machine will not accept the desired number of $a's$ and reject $6,12,18,\ldots$ $a's.$

So, our final collapsing $\textsf{DFA}$ is :

@ Sir,

Do e(empty string, so number of a’s = 0) is divisible by 2 and not divisible by 3?

slpp95prashanth no it won't be the case

Number of $a$'s is divisible by $2$

 $a$ $b$ $A^*$ $B$ $A^*$ $B$ $A^*$ $B$

Number of $a$'s  is not divisible by $3$

 $a$ $b$ $D$ $E^*$ $D$ $E^*$ $F^*$ $E^*$ $F^*$ $D$ $F^*$

Now we can combine both these table to get the desired DFA

Number of $a$'s is divisible by $2$ but not divisible by $3$

 $a$ $b$ $AD$ $BE$ $AD$ $AE^*$ $BF$ $AE^*$ $AF^*$ $BD$ $AF^*$ $BD$ $AE^*$ $BD$ $BE$ $AF^*$ $BE$ $BF$ $AD$ $BF$

now we can use minimization technique to check whether we can merge states in DFA or not

$\{\ AD,\ BD\ ,BE\ ,BF\ \}$ $\{AE^*\ ,AF^*\ \}$

$\{\ AD, BF\ \}$ $\{BD\ ,BE\ \}$ $\{AE^*\ ,AF^*\ \}$

$\{\ AD\ \} \{ \ BF\ \}$ $\{BD\ ,BE\ \}$ $\{AE^*\ \} \{\ AF^*\ \}$

$\{\ AD\ \} \{ \ BF\ \}$ $\{BD\ \} \{\ BE\ \}$ $\{AE^*\ \} \{\ AF^*\ \}$

$\{\ AD\ \} \{ \ BF\ \}$ $\{BD\ \} \{\ BE\ \}$ $\{AE^*\ \} \{\ AF^*\ \}$

$\therefore$ There will be $6$ states in the given DFA

by

edited

In the same manner if given as “number of a’s divisible by 6 but not divisible by 12” , then minimal states would be 12 and if “number of a’s divisible by 2 or divisible by 3 ” would be 6 states right ?

@
Yes, you are correct.

## State A was final in divisible by 2 DFA, but in combined DFA it was not considered as final?

neither 2 is divisible by 3 nor 3 is divisible by 2.

So, Minimal DFA. contains LCM(2,3) = 6 states

### 1 comment

Thank you, these are very much required.

This is a question of product automata.

DFA for number of a’s in x divisible by 2:                                     DFA for number of a’s not divisible by 3:

Now applying product automata:

Therefore 6 states. 15 and 14 are final states.


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## The Annals of Mathematical Statistics

### On the Line-Graph of the Complete Bigraph

J. W. Moon

#### Abstract

The line-graph, $L(G)$, of any given simple graph, $G$, is that graph whose points can be put in a one-to-one correspondence with the two edges of $G$ in such a way that two points in $L(G)$ are adjacent if, and only if, their corresponding edges in $G$ are adjacent. The complete $m$ by $n$ bigraph, $B_{mn}$, contains $mn$ edges which join each point in one set of $m$ points to each point in a second set, disjoint from the first, containing $n$ points. As general references on terminology see, e.g., Harary [4] and Ore [8]. If we suppose that $m \geqq n \geqq 1$ it can be seen that $L(B_{mn})$ has the following three properties: (1) The graph has $mn$ points each of which is adjacent to $m + n - 2$ other points. (2) $n \binom{m}{2}$ of the pairs of adjacent joints are mutually adjacent to $m - 2$ other points and the remaining $m \binom{n}{2}$ pairs of adjacent points are mutually adjacent to $n - 2$ other points. (03) Any two distinct nonadjacent points are mutually adjacent to two points. The object of this note is to show that if any graph satisfies these three conditions then it is isomorphic to $L(B_{mn})$ except possibly when $(m, n) = (4, 4), (4, 3)$ or $(5, 4)$. This will generalize a result of Shrikhande [10] (see also Mesner [7]) who, using different terminology, has already shown this for the case that $m - n$. The corresponding problem for the line-graph of the ordinary complete graph of $n$ points has been treated by Connor [3], Shrikhande [9], Hoffman [5], and Chang [1], [2].

#### Article information

Source
Ann. Math. Statist. Volume 34, Number 2 (1963), 664-667.

Dates
First available in Project Euclid: 27 April 2007

http://projecteuclid.org/euclid.aoms/1177704179

Digital Object Identifier
doi:10.1214/aoms/1177704179

Mathematical Reviews number (MathSciNet)
MR148050

Zentralblatt MATH identifier
0138.19204

JSTOR

#### Citation

Moon, J. W. On the Line-Graph of the Complete Bigraph. Ann. Math. Statist. 34 (1963), no. 2, 664--667. doi:10.1214/aoms/1177704179. http://projecteuclid.org/euclid.aoms/1177704179.


text:
Maltese and Arabic/Work plan

This is a workplan for development efforts for the Maltese and Arabic translator in Google Summer of Code 2012.

• Bidix-trimmed coverage means the coverage of the morphological analyser after being trimmed according to the bilingual dictionary of the pair, that is, only containing stems which are also in the bilingual dictionary (but omissions leading to generation errors do not change the trimmed coverage)
• Testvoc for a category means that the category is testvoc clean, in both translation directions.
• Evaluation is taking ${\displaystyle n}$ words and performing an evaluation for post-edition word error rate (WER). The output for those ${\displaystyle n}$ words should be clean, ie. no #/@, and no (or as few as possible) *, for all WER tests.
• Note on the WER results: both tests use references that are post-edited from the respective MT outputs, but the texts were used in apertium development, so unrestricted texts would probably look quite different. Texts are in SVN in apertium-mt-ar/dev/story/

Week Dates B-trimmed Cov. Goal B-trimmed Cov. Reached (news, wp) Testvoc Evaluation WER Reached (Google WER) Notes
0 23/04—20/05 45% 68.6%, 68.1% `<sent>` `<cm>` `<ij>` 500 words 3.17 % (47.44 %) Preliminary evaluation. Translate the story total coverage and without diagnostics. Get a baseline WER. Aim to get the two mt verb scripts merged.
1 21/05—27/05 50% 74.2%, 73.8% `<num>` `<pr>` - - - Basic numerals, and prepositions should be testvoc clean. Maltese verb scripts merged.
2 28/05—03/06 53% 75.3%, 74.5% `<cnjcoo>` `<cnjadv>` `<cnjsub>` - - - Create ar verb script based on mt, add ar open-class words according to frequency
3 04/06—10/06 59% 75.3%, 74.4% `<adv>` 200 words 8.70 % (20.94 %) Continue adding open class words to ar (adding to bidix/mt as you go)
4 11/06—17/06 63% 76.0%, 75.1% `<prn>` `<det>` - - - Monodix completion; closed classes (apart from verb-attached pronouns) should be testvoc clean.
5 18/06—24/06 68% 79.0%, 78.3% `<adj>` - - - Bidix expansion; bidix adjectives should be testvoc clean.
6 25/06—01/07 70% 79.1%, 78.4% `<n>` `<np>` 500 words 23.11 % (33.53 %) Midterm evaluation. Bidix expansion, multiwords.
7 02/07—08/07 73% 79.4%, 78.6% - - - - Transfer
8 09/07—15/07 75% 79.5%, 78.8% - - - - Transfer, corpus testvoc
9 16/07—22/07 77% 80.5%, 80.0% - 200 words 17.28 % (35.89 %) Disambiguation, retrain tagger
10 23/07—29/07 80% 80.9%, 80.2% `<v>` - - - Testvoc
11 30/07—05/08 80% 81.2%, 80.5% - - - - Testvoc
12 06/08—12/08 80% 82.0%, 80.9% all categories clean 500 words 21.34 % (39.00 %) Final evaluation. Tidying up, releasing


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# CMS Detector Performance Summaries

Posledne pridané:
2016-04-04
08:49
 Initial Report of the Fast Timing Working Group /CMS Collaboration A preliminary set of results highlighting the unique capabilities of fast-timing for resolving information from individual collisions at the High-Luminosity LHC (HL- LHC) is presented. These results explore the possibilities made available by using fast timing to enhance the reconstruction and physics capabilities of the CMS detector in terms of pileup mitigation and searches for new physics. [...] CMS-DP-2016-008; CERN-CMS-DP-2016-008.- Geneva : CERN, 2016 - 14 p. Fulltext: PDF;

2016-03-14
08:38
 ECAL trigger performance plots /CMS Collaboration ECAL trigger performance plots. Effect of the new ECAL response correction strategy, turn-on curves and resolution for 13 TeV run of 2015 and comparison with 8 TeV run of 2012 CMS-DP-2016-007; CERN-CMS-DP-2016-007.- Geneva : CERN, 2016 - 19 p. Fulltext: PDF;

2016-02-22
10:21
 Results on CASTOR Performance during LHC Run 2 /CMS Collaboration Results on various aspects of the performance of the CASTOR calorimeter in LHC Run 2 are presented. This includes the alignment, calibration and triggers of the detector. The intercalibration of the gains of the fine mesh PMT's using beam-halo muons is discussed, this in combination with results of a study on the noise and baseline. [...] CMS-DP-2016-006; CERN-CMS-DP-2016-006.- Geneva : CERN, 2016 - 12 p. Fulltext: PDF;

2016-02-18
17:13
 The BCM1F Detector: commissioning at LHC Run II and first results. /CMS Collaboration Talk at ADAMAS workshop: The BCM1F Detector: commissioning at LHC Run II and first results.. CMS-DP-2016-005; CERN-CMS-DP-2016-005.- Geneva : CERN, 2016 - 9 p. Fulltext: PDF;

2016-01-14
11:22
 Tracking and Luminosity Calibration of the PLT /CMS Collaboration The Pixel Luminosity Telescope (PLT) is one of the newest additions to the CMS detector for the LHC Run II data taking period. On each side of the CMS detector it consists of eight 3-layer telescopes based on silicon pixel detectors that are placed around the beam pipe viewing the interaction point at small angle. [...] CMS-DP-2016-004; CERN-CMS-DP-2016-004.- Geneva : CERN, 2015 - 5 p. Fulltext: PDF;

2016-01-14
11:22
 Machine Induced Background with BCM1F 2015 /CMS Collaboration The Fast Beam Condition Monitor BCM1F consists of 24 single-crystal CVD 5 mm x 5 mm diamonds positioned 1.8 m on either side of the interaction point at a radius of 6.5 cm from the beam pipe. The signal is read out, shaped by a frontend ASIC, and converted to an optical signal which is then transmitted to the backend electronics in USC55. [...] CMS-DP-2016-003; CERN-CMS-DP-2016-003.- Geneva : CERN, 2015 - 10 p. Fulltext: PDF;

2016-01-14
11:22
 The CMS Beam Halo Monitor Detector System /CMS Collaboration A new Beam Halo Monitor (BHM) detector system has been installed in the CMS cavern to measure the machine-induced background (MIB) from the LHC. This background originates from interactions of the LHC beam halo with the final set of collimators before the CMS experiment and from beam gas interactions. [...] CMS-DP-2016-002; CERN-CMS-DP-2016-002.- Geneva : CERN, 2015 - 4 p. Fulltext: PDF;

2016-01-14
11:21
 Performance of the CMS Beam Halo Monitor /CMS Collaboration The CMS Beam Halo Monitor has been successfully installed in the CMS cavern in LHC Long Shutdown 1 for measuring the machine induced background for LHC Run II. The system is based on 40 detector units composed of radiation hard synthetic quartz Cherenkov radiators coupled to fast photomultiplier tubes for a direction sensitive measurement. [...] CMS-DP-2016-001; CERN-CMS-DP-2016-001.- Geneva : CERN, 2015 - 6 p. Fulltext: PDF;

2015-12-30
11:40
 Electron and Photon performance using data collected by CMS at $\sqrt{s}$ = 13 TeV and 25ns This note describes preliminary performances of the electron and photon reconstruction with the CMS experiment using data collected at $\sqrt{s} = 13~\mathrm{TeV}$ and with a LHC bunch spacing of 25ns. Electron and photon key discriminating variables and their data to simulation agreement are shown. [...] CMS-DP-2015-067; CERN-CMS-DP-2015-067.- Geneva : CERN, 2015 - 36 p. Fulltext: PDF;

2015-12-17
11:21
 HIG PAG DPS plots for presentations before Moriond 2016 A few distributions related to Higgs analyses are shown in these slides CMS-DP-2015-066; CERN-CMS-DP-2015-066.- Geneva : CERN, 2015 - 64 p. Fulltext: PDF;


text: Nope, this is serious stuff; it’s about finding the slope of a line, finding the equation of a line... High School Math Solutions – Perpendicular & Parallel Lines Calculator. The slope-intercept form is , where is the slope and is the y-intercept. Related Symbolab blog posts. It also makes a right angle with the line segment. Required fields are marked *. Contact us for more details. Nope, this is serious stuff; it’s about finding the slope of a line, finding the equation of a line... Read More. The Perpendicular Line Calculator is a free to use digitally available tool that gives users the equation of the perpendicular line. Slope=rise/run=(y2-y1)(x2-x1) 2. Draw the lines whose equations are y = 2x , y = 2x + 1, y = 2x + 2, y = 2x + 3, y = 2x - 1, y = 2x - 2, and y = 2x - 3. A perpendicular bisector is a line which intersects or segments the given line into two equal parts. Ski Vacation? Perpendicular line equation calculator used to find the equation of perpendicular bisector. To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram|Alpha and will be in touch soon. perpendicular-line-calculator. Ski Vacation? Slope, Distance and More. Mensagem recebida. Find the Perpendicular Line y=-2 , (-5,7), Use the slope-intercept form to find the slope. Obrigado pelo feedback. The Perpendicular Line Calculator is a free to use digitally available tool that gives users the equation of the perpendicular line. Use our simple online Perpendicular bisector equation calculator to determine the bisector equation for the two given points. Sometimes, the line equation with the coordinates are given to us to solve the perpendicular lines. image/svg+xml. Save my name, email, and website in this browser for the next time I comment. The formula used as follows: The distance from Ax+By+C=0 to P(x 1,y 1) is $$d = \frac{(A\,x_1 + B\,y_1 + C)}{\sqrt{A^2 + B^2}}$$ To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: To add a widget to a MediaWiki site, the wiki must have the. The perpendicular line is vertical and the slope is undefined. This website uses cookies to ensure you get the best experience. Our below online calculator can be used to solve the perpendicular lines. In such a case, if the said two lines are perpendicular to the same line, then those two lines are going to be parallel to each other and still those lines will never intersect. Just fill out your information so we can prioritize what to build. Por favor tente novamente usando um diferente meio de pagamento. Enter (x 1 , y 1 ) = Step 2: Now, hit the calculate button to get the desired equation. Find more Mathematics widgets in Wolfram|Alpha. Your email address will not be published. By using this website, you agree to our Cookie Policy. Perpendicular Bisector is the division of something into two equal or congruent parts. Parallel and Perpendicular Lines. High School Math Solutions – Perpendicular & Parallel Lines Calculator. pt. Log InorSign Up. Perpendicular Line Through a Point calculator Formula: For: Line: ax + by = c Point: (x1,y1) Perpendicular Line: Y = (b/a)X - (b/a)x1 + y1 Using the slope-intercept form, the slope is . Objective Understand the fact that parallel lines have the same slope and that if two lines are perpendicular, then their slopes are negative reciprocals of each other.. Slope and Parallel Lines . Parallel and Perpendicular lines. Let us build one for your website. A free graphing calculator - graph function, examine intersection points, find maximum and minimum and much more This website uses cookies to ensure you get the best experience. parallel, perpendicular, slope, intersection, calculator-- Enter Line 1 Equation-- Enter Line 2 Equation (only if you are not pressing Slope) The procedure to use the perpendicular line calculator is as follows: Step 1: Begin with entering the line equation and the coordinate point passing through the line in the respective field. Formula to find the perpendicular distance from a point to a line. The calculator provided in this section can be used to find the perpendicular distance from a point to a line. All of our calculators are free to embed on your site with attribution. the equation of a perpendicular line through a point. Point-slope form: y-y1=m(x-x1) can write this using the slope and one point on the line. Your email address will not be published. In ancient mathematics, the perpendicular lines are defined as the lines which intersect at right angles (that is at 90 degrees). It is also known as angle bisector. For instance, the adjacent sides of a rectangle and a square always remain perpendicular to each other. Get the free "Equation of a Perpendicular Line" widget for your website, blog, Wordpress, Blogger, or iGoogle. Slope-intercept form: y=mx+b y-intercept is the b and m is slope (gradient) 1. The bisector can either cross the line segment it bisects, or can be a line segment or ray that ends at the line. In ancient mathematics, the perpendicular lines are defined as the lines which intersect at right angles (that is at 90 degrees). Parallel lines have the same slope, to find the parallel line at a given point you should simply calculate the... To create your new password, just click the link in the email we sent you. Perpendicular Line Calculator Perpendicular lines are the intersection of 2 different lines meeting at 90 degree to each other. In simpler words, any line is said to be a perpendicular line, when one line meets the other line at an angle of 90 degrees. Step 3: This will give you the perpendicular line equation in the output tab. To embed this widget in a post, install the Wolfram|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. Calculator finds the Perpendicular Length or Distance from a point to a line by putting the X and Y coordinates and the values for the given line (Ax+ By + C), very usefull tool for analytical geometry.