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Modelling alternative strategies for
mental rotation
David Peebles
July 22, 2019
Motivation for this work
• Several formal computational accounts of mental
imagery (e.g., Glasgow & Papadias, 1992; Just &
Carpenter, 1985; Kunda, McGreggor, & Goel, 2013;
Tabachneck-Schijf, Leonardo, & Simon, 1997)
• Long standing issue of whether imagery requires
• Some form of array based representation
• Abstract, amodal representations and processes
• All of the above employ an array representation
• Recent attempts using cognitive architectures
• Sigma (Rosenbloom, 2012)
• Soar (Lathrop, Wintermute, & Laird, 2011; Wintermute,
2012)
Motivation for this work
Questions
• How can mental imagery be modelled in ACT-R?
• What representations and processes can support it?
• Does it need an array based representation?
• What are the minimal changes necessary to do this?
Representations in ACT-R
• Many spatial imagery phenomena involve mental
representations of the shape, location, orientation and
spatial extent of imagined objects
• ACT-R has discrete symbolic representations in visual
module (e.g., shape = ‘square’)
• Only one x-y coordinate location for each object
Modifications made
(1,2)
(4,3)
(6,1)
(7,4)
(5,5)
(1,6)
(6,8)
(1)
VISUAL: POLYGON0-0 [POLYGON0]
POLYGON0-0
SCREEN-POS POLYGON-FEATURE8-0
VALUE “poly4”
COLOR WHITE
HEIGHT 8
WIDTH 8
POINTS ((1 2) (4 3) (6 1) (7 4) (5 5) (6 8) (1 6))
CENTRE-X 4
CENTRE-Y 4
REGULAR FALSE
SIDES 7
POLYGON T
(2)


x
y
1

 =


x
y
1

 ·


cos θ − sin θ 0
sin θ cos θ 0
0 0 1


(3)
1. Explicit representation of vertex coordinate locations in
stimulus objects
2. Encoding of vertex coordinates in the visual buffer
3. Affine and Boolean operations on spatial objects using
computational geometry and matrix multiplication
4. Use of imaginal action function of the imaginal buffer
Testing the approach
Initial application (Peebles, 2019) CogSci Saturday
• Mental scanning (Kosslyn, Ball, & Reiser, 1978)
• Mental rotation (Shepard & Metzler, 1971).
This application
• Different rotation strategies (Khooshabeh, Hegarty, &
Shipley, 2013).
Mental rotation
Standard task
• Pairs of similar images, one rotated around its centre.
Decide whether the images are identical or not.
• RT increases monotonically with the degree of angular
rotation between the images.
Strategies
• Holistic. Rotated figure manipulated as a single, whole
unit (Cooper, 1975; Shepard & Metzler, 1971).
• Piecemeal. Rotated figure subdivided, component pieces
manipulated separately (Just & Carpenter, 1976, 1985).
Holistic rotation strategy
Requires greater capacity to build and maintain complete
images in working memory (Bethell-Fox & Shepard, 1988;
Mumaw, Pellegrino, Kail, & Carter, 1984).
1. Search. Look for corresponding regions in the figures.
2. Confirm. Determine that the figures have related
features.
3. Transform and compare. Re-rotate whole figure
towards target.
Piecemeal rotation strategy
Favoured by lower spatial ability individuals
1. Search. Look for corresponding regions in the figures.
2. Transform and compare. Re-rotate selected piece
towards its corresponding target piece.
3. Confirm. Repeat to see if same rotation applies to other
corresponding pieces (Just & Carpenter, 1976, 1985)
Khooshabeh et al. (2013)
• Forced people to use holistic or
piecemeal strategy by using
fragmented versions of Shepard
and Metzler (1971) stimuli.
• Categorised people into high
and low spatial ability based on
performance.
• Assume that low ability
people use piecemeal strategy
most of the time.
• Compared high ability
people’s performance on
whole (holistic) and
fragmented (piecemeal)
stimuli.
Results – high spatial ability participants
Angular Difference in Orientation (degrees)
ResponseTime(s)
0 20 40 60 80 100 120 140 160 180
12345678
y = 0.026x + 2.276
q
q
q
q
q
q
q q
q
q
y = 0.019x + 1.703
q
Fragmented
Complete
Creating the models
Structure
• Two models instantiating the two strategies
• Holistic (7 productions)
• Piecemeal (8 productions)
• Rotation process (Just & Carpenter, 1976, 1985)
• Not a single ballistic rotation
• Series of discrete “rotate and compare” steps until images
are sufficiently congruent to stop.
Key parameters
• Rotation distance at each step
• Threshold distance to stop
• Imaginal delay time determines completion time for
imaginal buffer modification. Set to .1s (default =.2s).
Creating the models
Target image Rotated 60◦ clockwise
Stimuli
• Four component pieces of five random points – complex
irregular polygons (Cooper, 1975; Cooper & Podgorny,
1976)
• Model can attend to compound image or individual
components
Piecemeal strategy
1. Search. Look for corresponding
regions of the two figures.
Start
Look for
rotated figure
Attend to
rotated figure
Store rotated
figure; look to
normal figure
Attend to
normal figure
Pieces
match?
Different piece;
look again
no
Figures
aligned?
yes
Figures not
aligned; rotate
no
Enough
pieces
matched?
Figures aligned;
respond
yes
yes
Stop
no
Piecemeal strategy
1. Search. Look for corresponding
regions of the two figures.
2. Transform and compare.
Re-rotate selected piece towards
its corresponding target piece.
Start
Look for
rotated figure
Attend to
rotated figure
Store rotated
figure; look to
normal figure
Attend to
normal figure
Pieces
match?
Different piece;
look again
no
Figures
aligned?
yes
Figures not
aligned; rotate
no
Enough
pieces
matched?
Figures aligned;
respond
yes
yes
Stop
no
Piecemeal strategy
1. Search. Look for corresponding
regions of the two figures.
2. Transform and compare.
Re-rotate selected piece towards
its corresponding target piece.
3. Confirm. Repeat to see if the
same rotation will work for
other corresponding pieces.
Start
Look for
rotated figure
Attend to
rotated figure
Store rotated
figure; look to
normal figure
Attend to
normal figure
Pieces
match?
Different piece;
look again
no
Figures
aligned?
yes
Figures not
aligned; rotate
no
Enough
pieces
matched?
Figures aligned;
respond
yes
yes
Stop
no
Holistic strategy
1. Search. Look for corresponding
regions of the two figures.
Start
Look for
rotated figure
Attend to
rotated figure
Store rotated
figure; look to
normal figure
Attend to
normal figure
Pieces
match?
Different piece;
look again
no
Enough
pieces
matched?
yes
Do another
match
no
Figures
aligned?
yes
Figures not
aligned; rotate
no
Figures aligned;
respond
yes
Stop
Holistic strategy
1. Search. Look for corresponding
regions of the two figures.
2. Confirm. Determine that the
figures have related features.
Start
Look for
rotated figure
Attend to
rotated figure
Store rotated
figure; look to
normal figure
Attend to
normal figure
Pieces
match?
Different piece;
look again
no
Enough
pieces
matched?
yes
Do another
match
no
Figures
aligned?
yes
Figures not
aligned; rotate
no
Figures aligned;
respond
yes
Stop
Holistic strategy
1. Search. Look for corresponding
regions of the two figures.
2. Confirm. Determine that the
figures have related features.
3. Transform and compare.
Re-rotate whole figure towards
target.
Start
Look for
rotated figure
Attend to
rotated figure
Store rotated
figure; look to
normal figure
Attend to
normal figure
Pieces
match?
Different piece;
look again
no
Enough
pieces
matched?
yes
Do another
match
no
Figures
aligned?
yes
Figures not
aligned; rotate
no
Figures aligned;
respond
yes
Stop
Model performance
Angular Difference in Orientation (degrees)
ResponseTime(s)
0 20 40 60 80 100 120 140 160 180
12345678
y = 0.026x + 2.276
q
q
q
q
q
q
q q
q
q
y = 0.019x + 1.703
q
Fragmented
Complete
Human data
Angular Difference in Orientation (degrees)
ResponseTime(s)
0 20 40 60 80 100 120 140 160 180
12345678
y = 0.027x + 2.294
q
q
q
q
q
q
q
q
q
q
y = 0.019x + 1.995
q
Piecemeal
Holistic
Models
• Piecemeal slower because it requires more piece rotations
• To fit the data, rotation distance for additional piecemeal
rotations was larger than the initial rotation.
• Confirmatory action faster because distance known.
Conclusions
Representations and processes
• Not pixel arrays nor discrete symbols – intermediate
numerical level that abstracts from pixel level.
• Quantitative, subsymbolic processes assumed to be at a
level closer to the visual system but controlled and
monitored by higher level actions.
• Approach works well within the constraints of the
architecture with minimal adaptations
• May allow ACT-R to interact with other standard
vector-based images (e.g., SVG)
Conclusions
Future work
• Mental scanning and rotation are relatively simple —
repeated actions producing linear RT functions.
• Just use translation, rotation and Euclidean distance
measuring processes.
• More stringent test by modelling more challenging tasks.
• Raven’s Progressive Matrices (c.f. Kunda et al., 2013)
• “Pedestal blocks world” or “Nonholonomic car motion
planning” task (Wintermute, 2012)
• https://github.com/djpeebles/act-r-mental-rotation-models
References I
Bethell-Fox, C. E., & Shepard, R. N. (1988). Mental rotation:
Effects of stimulus complexity and familiarity.. Journal
of Experimental Psychology: Human Perception and
Performance, 14(1), 12–23.
Cooper, L. A. (1975). Mental rotation of random
two-dimensional shapes. Cognitive Psychology, 7(1),
20–43.
Cooper, L. A., & Podgorny, P. (1976). Mental transformations
and visual comparison processes: Effects of complexity
and similarity. Journal of Experimental Psychology:
Human Perception and Performance, 2(4), 503–514.
Glasgow, J., & Papadias, D. (1992). Computational imagery.
Cognitive Science, 16(3), 355–394.
Just, M. A., & Carpenter, P. A. (1976). Eye fixations and
cognitive processes. Cognitive psychology, 8(4), 441–480.
References II
Just, M. A., & Carpenter, P. A. (1985). Cognitive coordinate
systems: Accounts of mental rotation and individual
differences in spatial ability. Psychological Review, 92(2),
137–172.
Khooshabeh, P., Hegarty, M., & Shipley, T. F. (2013). Individual
differences in mental rotation. Experimental Psychology,
60(3), 164–171.
Kosslyn, S. M., Ball, T. M., & Reiser, B. J. (1978). Visual images
preserve metric spatial information: Evidence from
studies of image scanning.. Journal of Experimental
Psychology: Human Perception and Performance, 4(1),
47–60.
References III
Kunda, M., McGreggor, K., & Goel, A. K. (2013). A
computational model for solving problems from the
Raven’s Progressive Matrices intelligence test using
iconic visual representations. Cognitive Systems
Research, 22, 47–66.
Lathrop, S. D., Wintermute, S., & Laird, J. E. (2011). Exploring
the functional advantages of spatial and visual
cognition from an architectural perspective. Topics in
Cognitive Science, 3(4), 796–818.
Mumaw, R. J., Pellegrino, J. W., Kail, R. V., & Carter, P. (1984).
Different slopes for different folks: Process analysis of
spatial aptitude. Memory & Cognition, 12(5), 515–521.
References IV
Peebles, D. (2019). Modelling mental imagery in the ACT-R
cognitive architecture. In A. Goel, C. Seifert, & C. Freksa
(Eds.), Proceedings of the 41st annual conference of the
cognitive science society, Montreal, Canada: Cognitive
Science Society.
Rosenbloom, P. S. (2012). Extending mental imagery in Sigma.
In J. Bach, B. Goertzel, & M. Iklé (Eds.), International
conference on artificial general intelligence (pp. 272–281).
Berlin, Heidelberg: Springer.
Shepard, R. N., & Metzler, J. (1971). Mental rotation of
three-dimensional objects. Science, 171(3972), 701–703.
Tabachneck-Schijf, H. J. M., Leonardo, A. M., & Simon, H. A.
(1997). CaMeRa: A computational model of multiple
representations. 21, 305–350.
References V
Wintermute, S. (2012). Imagery in cognitive architecture:
Representation and control at multiple levels of
abstraction. Cognitive Systems Research, 19, 1–29.

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Modelling alternative strategies for mental rotation

  • 1. Modelling alternative strategies for mental rotation David Peebles July 22, 2019
  • 2. Motivation for this work • Several formal computational accounts of mental imagery (e.g., Glasgow & Papadias, 1992; Just & Carpenter, 1985; Kunda, McGreggor, & Goel, 2013; Tabachneck-Schijf, Leonardo, & Simon, 1997) • Long standing issue of whether imagery requires • Some form of array based representation • Abstract, amodal representations and processes • All of the above employ an array representation • Recent attempts using cognitive architectures • Sigma (Rosenbloom, 2012) • Soar (Lathrop, Wintermute, & Laird, 2011; Wintermute, 2012)
  • 3. Motivation for this work Questions • How can mental imagery be modelled in ACT-R? • What representations and processes can support it? • Does it need an array based representation? • What are the minimal changes necessary to do this?
  • 4. Representations in ACT-R • Many spatial imagery phenomena involve mental representations of the shape, location, orientation and spatial extent of imagined objects • ACT-R has discrete symbolic representations in visual module (e.g., shape = ‘square’) • Only one x-y coordinate location for each object
  • 5. Modifications made (1,2) (4,3) (6,1) (7,4) (5,5) (1,6) (6,8) (1) VISUAL: POLYGON0-0 [POLYGON0] POLYGON0-0 SCREEN-POS POLYGON-FEATURE8-0 VALUE “poly4” COLOR WHITE HEIGHT 8 WIDTH 8 POINTS ((1 2) (4 3) (6 1) (7 4) (5 5) (6 8) (1 6)) CENTRE-X 4 CENTRE-Y 4 REGULAR FALSE SIDES 7 POLYGON T (2)   x y 1   =   x y 1   ·   cos θ − sin θ 0 sin θ cos θ 0 0 0 1   (3) 1. Explicit representation of vertex coordinate locations in stimulus objects 2. Encoding of vertex coordinates in the visual buffer 3. Affine and Boolean operations on spatial objects using computational geometry and matrix multiplication 4. Use of imaginal action function of the imaginal buffer
  • 6. Testing the approach Initial application (Peebles, 2019) CogSci Saturday • Mental scanning (Kosslyn, Ball, & Reiser, 1978) • Mental rotation (Shepard & Metzler, 1971). This application • Different rotation strategies (Khooshabeh, Hegarty, & Shipley, 2013).
  • 7. Mental rotation Standard task • Pairs of similar images, one rotated around its centre. Decide whether the images are identical or not. • RT increases monotonically with the degree of angular rotation between the images. Strategies • Holistic. Rotated figure manipulated as a single, whole unit (Cooper, 1975; Shepard & Metzler, 1971). • Piecemeal. Rotated figure subdivided, component pieces manipulated separately (Just & Carpenter, 1976, 1985).
  • 8. Holistic rotation strategy Requires greater capacity to build and maintain complete images in working memory (Bethell-Fox & Shepard, 1988; Mumaw, Pellegrino, Kail, & Carter, 1984). 1. Search. Look for corresponding regions in the figures. 2. Confirm. Determine that the figures have related features. 3. Transform and compare. Re-rotate whole figure towards target.
  • 9. Piecemeal rotation strategy Favoured by lower spatial ability individuals 1. Search. Look for corresponding regions in the figures. 2. Transform and compare. Re-rotate selected piece towards its corresponding target piece. 3. Confirm. Repeat to see if same rotation applies to other corresponding pieces (Just & Carpenter, 1976, 1985)
  • 10. Khooshabeh et al. (2013) • Forced people to use holistic or piecemeal strategy by using fragmented versions of Shepard and Metzler (1971) stimuli. • Categorised people into high and low spatial ability based on performance. • Assume that low ability people use piecemeal strategy most of the time. • Compared high ability people’s performance on whole (holistic) and fragmented (piecemeal) stimuli.
  • 11. Results – high spatial ability participants Angular Difference in Orientation (degrees) ResponseTime(s) 0 20 40 60 80 100 120 140 160 180 12345678 y = 0.026x + 2.276 q q q q q q q q q q y = 0.019x + 1.703 q Fragmented Complete
  • 12. Creating the models Structure • Two models instantiating the two strategies • Holistic (7 productions) • Piecemeal (8 productions) • Rotation process (Just & Carpenter, 1976, 1985) • Not a single ballistic rotation • Series of discrete “rotate and compare” steps until images are sufficiently congruent to stop. Key parameters • Rotation distance at each step • Threshold distance to stop • Imaginal delay time determines completion time for imaginal buffer modification. Set to .1s (default =.2s).
  • 13. Creating the models Target image Rotated 60◦ clockwise Stimuli • Four component pieces of five random points – complex irregular polygons (Cooper, 1975; Cooper & Podgorny, 1976) • Model can attend to compound image or individual components
  • 14. Piecemeal strategy 1. Search. Look for corresponding regions of the two figures. Start Look for rotated figure Attend to rotated figure Store rotated figure; look to normal figure Attend to normal figure Pieces match? Different piece; look again no Figures aligned? yes Figures not aligned; rotate no Enough pieces matched? Figures aligned; respond yes yes Stop no
  • 15. Piecemeal strategy 1. Search. Look for corresponding regions of the two figures. 2. Transform and compare. Re-rotate selected piece towards its corresponding target piece. Start Look for rotated figure Attend to rotated figure Store rotated figure; look to normal figure Attend to normal figure Pieces match? Different piece; look again no Figures aligned? yes Figures not aligned; rotate no Enough pieces matched? Figures aligned; respond yes yes Stop no
  • 16. Piecemeal strategy 1. Search. Look for corresponding regions of the two figures. 2. Transform and compare. Re-rotate selected piece towards its corresponding target piece. 3. Confirm. Repeat to see if the same rotation will work for other corresponding pieces. Start Look for rotated figure Attend to rotated figure Store rotated figure; look to normal figure Attend to normal figure Pieces match? Different piece; look again no Figures aligned? yes Figures not aligned; rotate no Enough pieces matched? Figures aligned; respond yes yes Stop no
  • 17. Holistic strategy 1. Search. Look for corresponding regions of the two figures. Start Look for rotated figure Attend to rotated figure Store rotated figure; look to normal figure Attend to normal figure Pieces match? Different piece; look again no Enough pieces matched? yes Do another match no Figures aligned? yes Figures not aligned; rotate no Figures aligned; respond yes Stop
  • 18. Holistic strategy 1. Search. Look for corresponding regions of the two figures. 2. Confirm. Determine that the figures have related features. Start Look for rotated figure Attend to rotated figure Store rotated figure; look to normal figure Attend to normal figure Pieces match? Different piece; look again no Enough pieces matched? yes Do another match no Figures aligned? yes Figures not aligned; rotate no Figures aligned; respond yes Stop
  • 19. Holistic strategy 1. Search. Look for corresponding regions of the two figures. 2. Confirm. Determine that the figures have related features. 3. Transform and compare. Re-rotate whole figure towards target. Start Look for rotated figure Attend to rotated figure Store rotated figure; look to normal figure Attend to normal figure Pieces match? Different piece; look again no Enough pieces matched? yes Do another match no Figures aligned? yes Figures not aligned; rotate no Figures aligned; respond yes Stop
  • 20. Model performance Angular Difference in Orientation (degrees) ResponseTime(s) 0 20 40 60 80 100 120 140 160 180 12345678 y = 0.026x + 2.276 q q q q q q q q q q y = 0.019x + 1.703 q Fragmented Complete Human data Angular Difference in Orientation (degrees) ResponseTime(s) 0 20 40 60 80 100 120 140 160 180 12345678 y = 0.027x + 2.294 q q q q q q q q q q y = 0.019x + 1.995 q Piecemeal Holistic Models • Piecemeal slower because it requires more piece rotations • To fit the data, rotation distance for additional piecemeal rotations was larger than the initial rotation. • Confirmatory action faster because distance known.
  • 21. Conclusions Representations and processes • Not pixel arrays nor discrete symbols – intermediate numerical level that abstracts from pixel level. • Quantitative, subsymbolic processes assumed to be at a level closer to the visual system but controlled and monitored by higher level actions. • Approach works well within the constraints of the architecture with minimal adaptations • May allow ACT-R to interact with other standard vector-based images (e.g., SVG)
  • 22. Conclusions Future work • Mental scanning and rotation are relatively simple — repeated actions producing linear RT functions. • Just use translation, rotation and Euclidean distance measuring processes. • More stringent test by modelling more challenging tasks. • Raven’s Progressive Matrices (c.f. Kunda et al., 2013) • “Pedestal blocks world” or “Nonholonomic car motion planning” task (Wintermute, 2012) • https://github.com/djpeebles/act-r-mental-rotation-models
  • 23. References I Bethell-Fox, C. E., & Shepard, R. N. (1988). Mental rotation: Effects of stimulus complexity and familiarity.. Journal of Experimental Psychology: Human Perception and Performance, 14(1), 12–23. Cooper, L. A. (1975). Mental rotation of random two-dimensional shapes. Cognitive Psychology, 7(1), 20–43. Cooper, L. A., & Podgorny, P. (1976). Mental transformations and visual comparison processes: Effects of complexity and similarity. Journal of Experimental Psychology: Human Perception and Performance, 2(4), 503–514. Glasgow, J., & Papadias, D. (1992). Computational imagery. Cognitive Science, 16(3), 355–394. Just, M. A., & Carpenter, P. A. (1976). Eye fixations and cognitive processes. Cognitive psychology, 8(4), 441–480.
  • 24. References II Just, M. A., & Carpenter, P. A. (1985). Cognitive coordinate systems: Accounts of mental rotation and individual differences in spatial ability. Psychological Review, 92(2), 137–172. Khooshabeh, P., Hegarty, M., & Shipley, T. F. (2013). Individual differences in mental rotation. Experimental Psychology, 60(3), 164–171. Kosslyn, S. M., Ball, T. M., & Reiser, B. J. (1978). Visual images preserve metric spatial information: Evidence from studies of image scanning.. Journal of Experimental Psychology: Human Perception and Performance, 4(1), 47–60.
  • 25. References III Kunda, M., McGreggor, K., & Goel, A. K. (2013). A computational model for solving problems from the Raven’s Progressive Matrices intelligence test using iconic visual representations. Cognitive Systems Research, 22, 47–66. Lathrop, S. D., Wintermute, S., & Laird, J. E. (2011). Exploring the functional advantages of spatial and visual cognition from an architectural perspective. Topics in Cognitive Science, 3(4), 796–818. Mumaw, R. J., Pellegrino, J. W., Kail, R. V., & Carter, P. (1984). Different slopes for different folks: Process analysis of spatial aptitude. Memory & Cognition, 12(5), 515–521.
  • 26. References IV Peebles, D. (2019). Modelling mental imagery in the ACT-R cognitive architecture. In A. Goel, C. Seifert, & C. Freksa (Eds.), Proceedings of the 41st annual conference of the cognitive science society, Montreal, Canada: Cognitive Science Society. Rosenbloom, P. S. (2012). Extending mental imagery in Sigma. In J. Bach, B. Goertzel, & M. Iklé (Eds.), International conference on artificial general intelligence (pp. 272–281). Berlin, Heidelberg: Springer. Shepard, R. N., & Metzler, J. (1971). Mental rotation of three-dimensional objects. Science, 171(3972), 701–703. Tabachneck-Schijf, H. J. M., Leonardo, A. M., & Simon, H. A. (1997). CaMeRa: A computational model of multiple representations. 21, 305–350.
  • 27. References V Wintermute, S. (2012). Imagery in cognitive architecture: Representation and control at multiple levels of abstraction. Cognitive Systems Research, 19, 1–29.