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Progress on the Structure-Mapping Architecture for Learning Dedre Gentner Kenneth D. Forbus Northwestern University
Symbolic modeling crucial for understanding cognition • Heavy use of conceptual knowledge is a signature phenomena of human cognition – People understand, make, compare, and learn from complex arguments – People learn conceptual knowledge from reading texts, and apply what they have learned to new situations – People reason and learn by analogy, applying precedents and prior experience to solve complex problems – People use symbolic systems (e. g. , language, maps, diagrams) • Symbolic models remain the best way to explore many conceptual knowledge issues
Overview • Structure-Mapping Architecture • Accelerating learning via analogical encoding – Brief review • Tacit analogical inference – Analogy on the sly • Similarity-based qualitative simulation • Transfer and outreach activities
Structure-Mapping Theory (Gentner, 1983) • Analogy and similarity involve – correspondences between structured descriptions – candidate inferences fill in missing structure in target Inference is selective. Not all base knowledge is imported Candidate Inference completes common structure • Constraints – Identicality: Match identical relations, attributes, functions. Map non-identical functions when suggested by higher-order matches – 1: 1 mappings: Each item can be matched with at most one other – Systematicity: Prefer mappings involving systems of relations, esp. including higher -order relations
Analyzing similarities and differences, reasoning from experience, applying relational knowledge Functional Overview US Israel Iraq Iran WMD Nuclear Reactor Invasion Similaritybased retrieval of relevant examples and knowledge Bombing SEQL Potentially relevant precedents SME Incrementally constructs generalizations, producing human-like relational abstractions within similar number of examples. MAC/FAC Long-term memory
Psychological Studies 1. Case-comparison method Previous work: Transfer New work: Learning of principles 2. Unaware analogical inference Previous work: Unaware inference New work: Attitude congeniality & unaware inferences New work: Unaware alignment-based decision making
Analogy • Core process in higher-order cognition • A general learning mechanism by which complex knowledge can be acquired • e. g. , causal structures & explanatory principles • Unique to humans (or nearly so): Similarity Species-general A A Analogy Species-restricted AA B Object match BB CD Relational match
Analogical Encoding in Learning Standard analogical learning: • Analogy can promote learning – Induces structural alignment – Generates candidate inferences Familiar Situation Inferences New Situation • But, memory retrieval of potential analogs is unreliable Inert knowledge: Learned material often fails to transfer to new situations Analogical encoding: • Solution: Analogical encoding Use comparison during learning to - highlight the common relational system - promote relational abstraction & transfer New Situation Compare New Situation Relational Schema New Situation
Case Comparison Method in Learning to Negotiate Studies of MBAs learning negotiation strategies Students study two analogous cases prior to negotiating Loewenstein, Thompson & Gentner, 1999 Thompson, Gentner & Loewenstein, 2000 Gentner, Loewenstein & Thompson, 2003 case 1 case 2 Separate Cases Condition Read each case, write principle and give advice. Comparison Condition Compare the two cases and write the commonalities Simulated Negotiation On a new analogous case
Negotiation transfer performance across three studies: Proportion using strategy exemplified in the cases Prop. Forming Contingent Contracts . 8. 7 . 58* . 6. 5. 4. 3. 2 . 19 . 24 . 1 0 No Cases Separate Cases Compare N=42 N=83 N=81
Better schemas Better transfer Prop. Forming Contingent Contracts . 8. 7 Compare . 6. 5. 4 Separate Cases . 3. 2. 1 0 0 0. 5 1. 0 1. 5 2. 0 Dyadic Schema Rating So, what happens if we just give them the principle?
Prop. Forming Contingent Contracts Aligning case and principle improves ability to use principle in transfer. 7 Separate Principle Plus case . 6. 5 . 44* Case 1 ________ Case 2 . 4 ___________ Compare Principle and Case 1 Case 2 ____________ _____ . 3. 2 Test: Face-to-face negotiation . 19 . 1 0 Separate Cases N=26 dyads Compare N=27 dyads Error bars assume binomial with prop=. 19 (baseline)
Comparison promotes transfer even when the principle is given - Why? Principles utilize abstract relational language • Relational language—verbs, prepositions, relational nouns— is contextually mutable interpretation difficulties – e. g. , force in physics =/= force in commonsense language • Assembling a complex relational structure is errorful • So, beginning learners don’t understand principles when presented solo Case provides a firm relational structure that is correct but overly specific – learning is context-bound – strongly situated – So unlikely to transfer Comparing a principle and a case – grounds the principle in a firm structure – invites abstracting the specific relations in the case
Learning Negotiation Principles- Experiment 1 Training: • participants read two passages – a negotiation principle (Contingent Contract) – an analogous case • Separate condition: Participants consider each passage separately. • Compare condition: Participants consider how the case and principle are alike. • Two orders: case ppl and ppl case • All participants answer the question "How could this be informative for negotiating? " 20 -minute delay Test: Recall task: subjects write out the principle they learned
Principle: Contingent Contract A contingent contract is a contract to do or not to do something depending on whether or not some future event occurs. At least two kinds of situations exist in which contingent agreements add potential for joint gains – when disagreeing over probabilities and when both parties try to influence an uncertain outcome. When the uncertain event itself is of interest, there are familiar economic contingent contracts with “betting” based on the probability of differences. Parties are dealing with uncertain quantities and actually or apparently differ in their assessment, and here contingent arrangements offer gains. When the parties feel capable of influencing an uncertain event, making the negotiated outcome dependent on its resolution may be a good idea. In both cases of course, contingent arrangements based on underlying differences are not a panacea. Crafting them effectively can be a high art. And once the outcome of the uncertain event is known, one party may have “won” and the other “lost. ” Whether the outcome will then be considered fair, wise, or even sustainable is an important question to be planned for in advance.
Training Case Two fairly poor brothers, Ben and Jerry, had just inherited a working farm whose main crop has a volatile price. Ben wanted to sell rights to the farm’s output under a long-term contract for a fixed amount rather than depend upon shares of an uncertain revenue stream. In short, Ben was risk-averse. Jerry, on the other hand, was confident that the next season would be spectacular and revenues would be high. In short, Jerry was risk-seeking. The two argued for days and nights. Ben wanted to sell immediately because he believed the price of the crop would fall; Jerry wanted to keep the farm because he believed the price of the crop would increase. Finally, Jerry proposed a possible agreement to his brother: They would keep the farm for another year. If the price of the crop fell below a certain price (as Ben thought it would), then they would sell the farm and Ben would get 50% of the farm’s current value, adjusted for inflation; Jerry would get the rest. However, if the price of the crop were to rise (as Jerry thought it would), Jerry would buy Ben out for 50% of the farm’s current value, adjusted for inflation, and would get to keep all of the additional profits for himself. Jerry was delighted when his brother told him he could agree to this arrangement, thereby avoiding further conflict.
Recall Scores (Max. = 8) Gentner & Colhoun Mean Recall Score 3. 4 Two blind raters Agreement: 94% 2. 5 (26) t(50) = 2. 10, p = 0. 041
Quotes from the Compare Group • P 18: "Contingent contract Principle: if there is an uncertain event occurring in the future which two parties disagree on, the outcome of this event becomes the determining factor in the outcome of the negotiation. " • P 30: "The contingency contract is created as an agreement to do/not do something in the future in the event of a situation. As the future is unknown, the CC is created on the probability that something will occur…" Quotes from the Separate Group • P 50: "It is important to consider how much you will lose or win when betting on an uncertain event. Negotiating in this situation is more complicated than just predicting the outcome. " (this was the entire answer) • P 51: "We read about the two poor brothers on the farm. One was risk-seeking and the other was risk-seeking, so they couldn't decide on whether or not to sell the farm…" (no mention of the principle)
Delayed Recall Read Principle & Case (20 mins) Immediate Recall Test case: Asian Merchant N=14; 7 sep, 7 comp (4 days) Long term Recall New test case N=14; 7 sep, 7 comp
Immediate Recall Scores Mean Recall Score Both Orders 4. 4 3. 1 (11) (12) T(23) = -2. 44, p = 0. 023
Delayed Recall Scores Mean Recall Score Both Order 4. 3 3. 1 (11) (12) T(21) = -1. 91, p = 0. 07 Combines two groups with slightly different procedures
Conclusions Comparison group > Separate group Case-first groups > Principle-first groups Comparing case and principle greatly benefits comprehension of principle The case provides firm relational structure and a clear (though overly specific) interpretation of the relational terms Comparing case with principle prompts rerepresentation and abstraction of the relational structure
Practical Implications • Case-based training is heavily used in professional schools (business, medicine, law) – intensive analyses of single cases – Our results suggest that learning could be greatly increased by changing to a comparison-based instructional strategy • Based on our findings, some institutions are revising their instructional methods – Medical School of Mc. Master University • Developing a new curriculum relying heavily on comparisonbased instruction – Harvard Business School • Exploring comparison-based method – CMU – discussions with Marsha Lovett
Unaware Analogical Inference Analogy as generally conceived: Current Studies: • Conscious • Non-aware • Discerning • Oblivious • Deliberate • Non-deliberate • Effortful • Accidental Suggestive evidence: Blanchette & Dunbar, 2002; Moreau, Markman & Lehman, 2001
New thrust: Study of “unwitting analogy” • Can analogical inferences occur without awareness of making the inferences? • Can analogical inferences occur without awareness of the analogy itself? • Can the highlighting effect of analogical alignment Influence future decision-making?
Analogical insertion effect: believing that the analogical inference from B T actually occurred in T • Evidence for analogical insertion Blanchette & Dunbar 1999 Analogy: Anti-marijuana laws are like Prohibition Participants misrecognized parallel inferences as having occurred in marijuana passage • But, these pro-marijuana inferences were likely to be congenial to college students • Will analogical insertion occur if the inference is not so congenial?
Attitudes towards gayness assessed (Mass testing) 3 -4 weeks (unrelated context) Read paragraph “Is it OK to be gay” Analogy group Second paragraph analogizing gayness to left-handedness Control group No further text 15 -min filled delay Old-New recognition test Rate soundness of analogy Perrott, Gentner & Bodenhausen, 2005
Proportion “old” responses Perrott, Gentner & Bodenhausen, 2005 * Condition(2) X Item type(4) F(3, 228) = 4. 97, p=. 002, MSE =. 048
Results within analogy group Attitudes towards gays within predicted the rated soundness of the analogy But Likelihood of analogical insertion was not predicted by rated soundness of the analogy Even more surprisingly, Likelihood of analogical insertion was not predicted by attitude towards gays – No “attitude congeniality effect” Attitudes measured on 15 -item questionnaire composite scale from 1 (very negative) to 7 (very positive). Range: 1. 8 -6. 8 (M = 4. 7) Cutoffs for lower and upper quartiles = 3. 3 and 5. 8
Can analogical insertion occur without awareness of the analogy • Participants read a series of passages • Told that they would be asked questions about content of passages • We observed extent to which analogous passages early in the set influenced the interpretation of later passages • No goal other than comprehension • Inferences support understanding the input
Day & Gentner; 2003, in prep Current Studies • Participants read a series of passages • Some early passages are relationally similar to later passages • Will participants use structure-mapping in interpreting the later passages? TEST: • Participants answer TF questions about passages • Dependent measure: Answering True to questions that are inferences from earlier analogous passages.
Experiment 1 Two versions of each base passage • If participants use analogical inference from the earlier similar base passage, they will understand the target differently, depending on which base version they got. Target has some ambiguous portions
Example Source Passages Base 1: Base 2: Wealthy elderly woman dies mysteriously Her niece respectfully flies into town for the funeral Wealthy elderly woman dies mysteriously Her niece suspiciously leaves town when the death is announced People are surprised when the will leaves everything to the niece Target Passage: Wealthy elderly man dies mysteriously As soon as the death is announced, the man’s nephew immediately buys a ticket and flies to Rio de Janeiro People are surprised when the will leaves everything to the nephew
Expt. 1 Results: More false recognitions for baseconsistent statements Percentage ‘yes’ responses 100 73% 50 25% 0 Base-consistent Base-inconsistent Using base consistency as a within-subjects factor Day & Gentner, 2003 t (19) = 4. 79, p <. 001
E 1 Results: Analogical insertion • P’s interpreted the ambiguous portion of the target in a manner consistent with structurally matching information in the base. • The same target passage was interpreted differently, as a function of which base P’s had read • Evidence suggests that analogical inference influences the interpretation of new material • Not due to deliberate strategies: 90% noticed similarities between passages But, 80% said all passages were understandable on their own. • Not due to simple priming: further study showed inferences are specific to the structural role of the inserted information
Experiment 3 Is the analogical insertion effect occurring during online comprehension of target, or is it a later memory error?
Experiment 3: Self-paced Reading Task • Base passage and target passage same as in Expts 1 and 2, except: • Target contains a later key sentence that is consistent with one base’s inference and inconsistent with the other’s: “George's absence from the service was conspicuous, especially since he had been seen around his uncle's estate prior to his death, and the police soon found out about his flight to Rio. ” If P’s insert the seeded inference into the target story, they will take longer to read the key test sentence when it is inconsistent with that inference
Experiment 3: Self-paced Reading Task • Base passage and target passage same as in Expts 1 and 2, except: • Target contains a later key sentence that is consistent with one base’s inference and inconsistent with the other’s: Results If P’s insert the seeded inference into the target story, they will take longer to read the key test sentence when it is inconsistent with that inference F (1, 19) = 6. 81, p <. 05 10 8. 88 9 Reading time (sec) “George's absence from the service was conspicuous, especially since he had been seen around his uncle's estate prior to his death, and the police soon found out about his flight to Rio. ” 8 7 6. 40 6 5 4 Baseconsistent inconsistent
Tacit analogical inferences Day & Gentner; 2003, in prep • People interpolated analogical inferences from a prior similar passage due to shared representational structure, not simply to general priming • Implication: Structure-mapping can operate in nonaware, nondeliberative processing • But –what about large number of analogy studies that show failure to transfer ? Current studies • Vary delay: 20 minute vs. 4 days later • Vary surface similarity between the passages • Future work: Progressive alignment effect? Does an obvious alignment potentiate more analogical creep?
Day & Bartels (2005) Unaware effects of analogy: Decision-making Structure mapping theory proposes that comparison involves the alignment of representational structures (Gentner, 1983; Gentner & Markman, 1997) This implies two kinds of differences: alignable differences: different values on same predicate or dimension; related to common structure non-alignable differences: none of the above Alignable differences are weighted more heavily in perceived similarity (Markman & Gentner, 1996) difference detection (Gentner & Markman, 1994) recall (Markman & Gentner, 1997) preference (e. g. , Roehm & Sternthal, 2001) Hypotheses: Alignment along a dimension renders that dimension more salient in immediate use Repeated alignment & use renders the dimension more salient in future encodings
Method: P’s choose among portable digital video players 1. First, participants gave preference ratings for models that 2. varied on only one alignable dimension: Firewire and USB connectivity: Battery life: Voice recorder: Hard drive capacity: Built-in FM radio: Wireless projection range : Support for WMV and MP 2 formats: Screen size: Weight: Strongly prefer Model A Yes 4 hr No 7 Gb Yes 12 ft No 2. 5 in 10 oz Model B Yes 4 hr No 4 Gb Yes 12 ft No 2. 5 in 10 oz Strongly prefer Model B
Method 1. First, participants gave preference ratings for models that 2. varied on only one alignable dimension: Firewire and USB connectivity: Battery life: Voice recorder: Hard drive capacity: Built-in FM radio: Wireless projection range : Support for WMV and MP 2 formats: Screen size: Weight: Strongly prefer Model A Yes 4 hr No 7 Gb Yes 12 ft No 2. 5 in 10 oz Model B Yes 4 hr No 4 Gb Yes 12 ft No 2. 5 in 10 oz Strongly prefer Model B
Method 2. Eventually, they make judgments between models varying on two dimensions, each favoring a different alternative Firewire and USB connectivity: Battery life: Voice recorder: Hard drive capacity: Built-in FM radio: Wireless projection range : Support for WMV and MP 2 formats: Screen size: Weight: Strongly prefer Model A Yes 4 hr No 10 Gb Yes 12 ft No 1. 5 in 10 oz Model B Yes 4 hr No 7 Gb Yes 12 ft No 2. 5 in 10 oz Strongly prefer Model B
Experiments Experiment 1 • Are more recently used dimensions weighted more in future decisions? That is, does aligning a dimension make it more salient for some period of time? Experiment 2 • Are dimension that have been used more frequently weighted more in future decisions? That is, does repeated alignment along a dimension render that dimension more salient in future encodings?
Types of item series 1 back: 1 v. 2 back: Diagnostic dimension A B C D E - - - - - 1 v. 3 back: 2 v. 3 back: Diagnostic dimension A B C D E - - - - - -
Results Day & Bartels (2005) Experiment 1 • Each response was coded as a value between 0 and 1 • . 5 would be chance; averages closer to 1 indicate a preference for the more recently diagnostic dimension Average response was. 62 (p <. 001) 18 out of 20 participant had average ratings greater than. 5 Participants weighted a dimension more if it had been used in a more recent decision
Results Day & Bartels (2005) Experiment 2 • Found correlation between preference ratings and number of prior uses of a dimension for each participant • Individual correlations transformed into Fisher’s Z for use in analysis Average transformed correlation was. 20 (p <. 01) Participants weight a dimension more if it had been used more frequently in prior decisions
Conclusions Day & Bartels (2005) • Finding an alignable difference along a dimension makes that dimension more salient for a period of time more recently aligned dimensions play a larger role in future decisions • Repeated alignment of a dimension increases its salience in future encodings higher numbers of repetitions greater dimension weights in decisions • These effects of comparison may go unnoticed, but may have pervasive effects on the mental landscape
Resistance is futile • Analogical insertion—interpolation of inferences into the target situation—can occur • when an analogy is given explicitly • when an alignable analog has been presented recently • Online comparisons increase the salience of aligned dimensions for future encodings • Hypothesis: Continual subtle learning occurs via structural matching and inference • Fits with MAC/FAC assumption of continual unbidden retrieval • • Challenges & Future work: • How recent? • How similar and in what ways? • Effects of intervening items?
How do people do common sense reasoning? • Today’s methods of qualitative reasoning are very useful – Many successful applications in engineering, education, supporting scientific reasoning • Are they also good models of how people common sense reasoning? – Yes, but similarity plays major role in reasoning • Important question for cognitive science – Central to understanding mental models
The standard Qualitative Reasoning community answer Scenario model Situation description input F G Model Builder H F Qualitative simulation F G H 1 st principles Domain Theory G H Qualitative Simulator F G H i F G H
First-principles qualitative simulation Useful properties Problematic properties • Handles incomplete and inexact data • Supports simple inferences • Explicit representation of causal theories • Exclusive use of 1 stprinciples domain theory – To prevent melting, remove kettle from stove • Representation of ambiguity – We easily imagine multiple alternatives in daily reasoning – inconsistent with psychological evidence of strong role for experience-based reasoning • Exponential behavior – inconsistent with rapidity & flexibility of human reasoning • Generates more complex predictions than people report – logically possible, but physically implausible
Working hypotheses about human common sense reasoning and learning (Forbus & Gentner, 1997) • Common sense = Combination of analogical reasoning from experience and first-principles reasoning • Within-domain analogies provide robustness, rapid predictions – Human learning requires accumulating lots of concrete examples – Structured, relational descriptions essential – feature vectors inadequate • First-principles reasoning emerges slowly as generalizations from examples – Human learning tends to be conservative – But human learning also tends to be faster than pure statistical learning • Qualitative representations are central – Appropriate level of understanding for communication, action, and generalization
An alternative: Hybrid qualitative simulation • Most predictions, explanations generated via within-domain analogies – Provides rapidity and robustness in common cases – Multiple retrieved behaviors leads to multiple predictions. – Logically possible behaviors that are rarely observed aren’t predicted. • 1 st principles reasoning relatively rare – 1 st principles domain theories fragmentary, partial • Some 1 st principles knowledge created by generalization over examples • Much of it taught via language • We built a similarity-based qualitative simulator to explore this approach
A Prototype SQS System Situatio n MAC/FAC Experience Library Rerep Engine SEQL Candidate Behaviors Projector Predictions
Experience Library Contents • Current sources – Classic QR examples • Generated envisonments using Gizmo Mk 2 – Feedback systems • Generated descriptions of behavior by hand • Test of whether system can operate without a complete 1 stprinciples domain theory • Each case consists of a qualitative state – Individuals, ordinal relations, model fragments – Concrete information about entities (stand-in for perceptual properties) – Description of transitions to other states
Example: Two Containers Liquid Flow State 0 State 1 ↓(Amount. Of Water Liquid F) ↑(Amount. Of Water Liquid G) ↓(Pressure Wf) ↑(Pressure Wg) (> (Pressure Wf) (pressure Wg)) (active. MF Liquid. Flow) State 2 ↑(Amount. Of Water Liquid F) ↓(Amount. Of Water Liquid G) ↑(Pressure Wf) ↓(Pressure Wg) (< (Pressure Wf) (pressure Wg)) (active. MF Liquid. Flow) →(Amount. Of Water Liquid F) →(Amount. Of Water Liquid G) →(Pressure Wf) →(Pressure Wg) (= (Pressure Wf) (pressure Wg)) (not (active. MF Liquid. Flow))
Input Scenario ↓(Amount. Of Water Liquid Beaker) ↑(Amount. Of Water Liquid Vial) ↓(Pressure Wb) ↑(Pressure Wv) (> (Pressure Wb) (pressure Wv)) →(Amount. Of Water Liquid Beaker) →(Amount. Of Water Liquid Vial) →(Pressure Wb) →(Pressure Wv) (= (Pressure Wb) (pressure Wv)) Behavior Prediction
Example: Heat Flow State 0 ↓(Temperature Coffee) ↑(Temperature Ice. Cube) (> (Temperature Coffee) (Temperature Ice. Cube)) (active. MF Heat. Flow) Retrieved analogue Input Scenario ↓(Temperature Brick) ↑(Temperature Water) (> (Temperature Brick) (Temperature Water)) (active. MF Heat. Flow) State 1 →(Temperature Coffee) →(Temperature Ice. Cube) (= (Temperature Coffee) (Temperature Ice. Cube)) (not (active. MF Heat. Flow)) Predicted Behavior →(Temperature Brick) →(Temperature Water) (= (Temperature Brick) (Temperature Water)) (not (active. MF Heat. Flow))
Example: Discrete action feedback system
Mappings for Feedback Example Feedback Control System Water Level Regulation System Sensor Floating ball Comparator Ball Stick Controller String + Pulleys Actuator Valve Temperature set point Proper water level Room air Tank water Room Water tank Oven Water supply Heat flow process Liquid flow process Furnace on process Valve open process
Stored Feedback System Behavior Quantities S 1 S 3 S 4 S 5 S 6 < (Temperature Room)vs. Set. Point S 2 = > > = < (Ds (temperature Room)) 1 -1 (active. MF Furnace. On) Yes No (active. MF Heat. Flow) Yes Retrieved Behavior S 1 S 6 S 2 S 5 S 3 S 4
Mapped Feedback System Behavior Retrieved Behavior Quantities S 1 (Temperature Room)vs. Set. Point S 2 S 3 S 4 S 5 S 6 < = > > = < (Ds (temperature Room)) 1 Yes (active. MF Furnace. On) (active. MF Heat. Flow) Quantities (Level Tank. Water) vs. Proper. Water. Level (Ds (Level Tank. Water)) -1 No Yes S 1 S 2 < = S 3 S 4 > > Yes S 5 S 6 = 1 -1 (active. MF Valve. Open) Yes No (active. MF Liquid. Flow) Yes Predicted Behavior <
Example: Proportional action control system • Amount of correction applied is proportional to the error signal • SQS prototype with current library makes incorrect prediction – Retrieves discrete-action controller behavior – Currently has no means of detecting inconsistencies • Possible solutions – Include some first-principles reasoning for reality checks – When failure detected, add new behavior to Experience Library to improve future performance
Current Issue: Combining Behaviors P(Wg) P(Wf) F Two mappings, how to combine? A G B Level(Wf) Aof(Wg) I- Q+ Aof(Wf) A G Level(Wg) FR(F G) Q+ H F Q+ I+ H B
Pastiche Mappings • Retrieve behaviors for unexplained parts of system • Combine by re-evaluating closed-world assumptions Perform influence resolution to combine influences across cases F G H
Next steps: Hybrid qualitative simulation • Significantly expand Experience Library – Plan: Use EA NLU system to describe qualitative states in QRG Controlled English • Test skolem resolution strategies – Identify hypothesized entities with unmapped current situation entities when possible. • Formulate criteria for using multiple remindings – When to generate alternate predicted behaviors? • Develop more selective rerepresentation strategies – Currently performed exhaustively • Explore learning strategies – Store rerepresented results and new behaviors – Use SEQL to construct generalizations
Geometric Analogy Problems • Evans classic 1968 work ANALOGY – Miller Analogies Test geometric problems – Non-trivial human intelligence task • Goal of our simulation: – Show that general-purpose simulations can handle this task – Another source of data for tuning visual representations in our sketching system
Sketching the Geometric Analogy Problems A 1 B 2 C 3 4 5
Finding the Answer: Evans A is to B as C is to 1, 2, 3, 4 or 5? • Compute all transformations Ag. B, Cg 1, Cg 2, … • Search for best match between transformation for Ag. B with all of the transformations for Cg 1, Cg 2, …
Finding the Answer: Our simulation A is to B as C is to 1, 2, 3, 4 or 5? Differences compared at second level Two-stage structure mapping
Results (Based on Evans’ answer key) MAT Problems ANALOGY s. KEA/SME 1 -9, 11, 13 -18, 20 Correct 10 Incorrect Correct 12 Correct (prefers reflection) Correct (prefers rotation) Incorrect (prefers rotation) Correct (prefers rotation) 19
Problem case 12
Summary: Geometric Analogies Simulation • SME + qualitative spatial representations provide a basis for solving geometric analogy problems • Two-stage structure mapping provides an elegant model for this task – Explicit transformation rules unnecessary – Applicable to other analogy tasks?