Common Sense on the Envelope

1. Common Sense on the Envelope Praveen K. Paritosh Kenneth D. Forbus Qualitative Reasoning Group Department of Computer Science Northwestern University http://www.qrg.nwu.edu/

2. Outline Back of the Envelope Reasoning Common Sense QR Relevant Research A Similarity-Based Model Open Issues Conclusions

3. Real-world examples of BotE How muchoxygen is left? Is anyone still alive in there? Kursk Mir How longto repair it? QR-01 Talk USS Cole

4. Examples of Back of the Envelope Reasoning Q2.Estimate the drag force on a bicycle and rider traveling at 20 mph. Q3.Estimate the energy stored in a new 9-volt transistor battery. Q5.How long does it take to reach home from your office, or to get ready in the morning? Q6.How much money would you be spending on that vacation you have planned? Q7.You know a recipe that you made for yourself some time back – now you have to make it for eight people, and you want it less spicy and you ran out of one of the ingredients. Claim: Same processes underlie Q2, Q3 and Q5, Q6, Q7. QR-01 Talk

5. Back of the Envelope (BotE) Reasoning • Accurate data/models unavailable • Available data incomplete and/or inconsistent. • Need answers fast • Need a quantitative answer! Qualitative  Not Quantitative QR-01 Talk

6. Where do we do BotE – 1 • Real-world problems (nearly always incompletely specified) • Engineering/Design/Experimental Science • Evaluating Feasibility of an Idea • Planning Experiments and sizing components • Setting up and double-checking detailed analysis • Domains where BotE is the best one can do • Environmental Science (Consider a Spherical Cow, Harte, 1988) • Biophysics (O’Connor and Spotila, 1992) QR-01 Talk

7. Where do we do BotE – 2 • Everyday physical situations • How long will it take to get there? • Do I have enough money with me? • How much of the load can I carry at once? • A world of quantitative dimensions: Need to make quick, quantitative estimates to – • Interact (How much salt do I add while cooking some recipe?) • Understand (Is $7,000 too expensive for this laptop?) QR-01 Talk

8. resources specificity The Specificity/Economy Tradeoff Specificity = Resolution and certainty in the answer Resources = Time, information, formalization and computation required to reach the answer. QR-01 Talk

9. What underlies BotE • Qualitative Reasoning • Provides analytic framework • Facilitates comparison • Similarity – from similar scenarios: • Borrow modeling assumptions • Supply default, pre-computed information, parameter values • Reality check • Generalization along quantitative dimensions • What’s high, low or moderate? “1 Amp is too high a current for a walkman” QR-01 Talk

10. Constraints guiding Common Sense QR • 1. Incompleteness • Domain theories incomplete in coverage. • 2. Concreteness • Knowledge of concrete, specific situations (made use of by analogical reasoning) in addition to first-principles reasoning. • 3. Highly experiential: Experience improves • - ability to reason through similar scenarios. • - intuitions for what is reasonable, high, low in a domain. • 4. Focused reasoning • Tight reasoning, as opposed to maintaining ambiguity for completeness • 5. Pervasively quantitative • Real-world actions require that estimates manifest as exact values. QR-01 Talk

11. Psychological Research • Quantitative estimation • Intuitive statistics (Peterson and Beach, 1967) • Heuristics and biases (Tversky and Kahneman, 1974) • Metrics and mappings (Brown and Siegler, 1993) • Estimation in mechanical engineering curricula (Linder, 1999) • Models of similarity • Multidimensional scaling (Shepard, 1962; Torgerson, 1965) • Set-theoretic feature based account (Tversky, 1977) • Structure-mapping engine (Gentner, 1983) • Commonalities, alignable, and non-alignable differences (Markman and Gentner, 1993) • An account of generalization: SEQL QR-01 Talk

12. SME: Structure-Mapping Engine Inputs = propositional descriptions, w/ incremental updates Output = one or two mappings Base SME Operates in polynomial time, by exploiting graph labels & greedy algorithms Mappings = correspondences + structural evaluation + candidate inferences Target QR-01 Talk

13. SEQL: Category learning via progressive alignment • Produces abstractions incrementally, based on commonalties resulting from comparisons between exemplars. • Models conservative nature of concept learning • Skorstad, Gentner, & Medin (1988) showed SEQL could model sequence effects in learning QR-01 Talk

14. Similarity-Based Model of BotE Reasoning Two distinct processes - • Direct parameter estimation • Domain knowledge • Previous experience • Adapt from one or more similar scenarios • Building an estimation model • Parameter to be estimated not directly available • Estimation model relates it to parameters that can be directly estimated • Possibly recursive QR-01 Talk

15. A Simple Example How many pieces of popcorn would fit in this room? Possibly don’t have num-popcorn in memory, num-popcorn = volume-room/volume-popcorn (1) Approximating room to a cuboid, and popcorn to a cube (considering the voids left after packing in popcorn kernels this is a reasonable assumption), num-popcorn = l*b*h / a^3 (2) QR-01 Talk

16. An Extended Example Estimate the energy stored in a 9v transistor battery. (Linder, 1999) • Nobody used first-principles chemistry. • People recalled walkmans, clocks, flashlights and other scenarios where they came across 9v battery. • A lot of people adapted estimates from 1.5v battery, or car battery. QR-01 Talk

17. Estimation Model Suppose I did not know anything about the 9v battery except its size, but I knew examples of where 1.5v AA batteries were being used. If I make the assumption that these two batteries are fundamentally the same, and only the difference in volume should be responsible for difference in energies stored. Etransistor/EAA = Vtransistor/VAA …(1) In a small hand-held flashlight, all the power provided by the batteries is used up in lighting the bulb. N * EAA = Pbulb * Life …(2) Where Pbulb is power rating of the flashlight bulb, and Life is the time that a new set of batteries will take before they die out, and N is the number of batteries in a flashlight. Parameters and Calculations N = 2 (number of batteries) Pbulb = 1 Watts Life = 2 hours EAA = 1 * 2 * 3600 * 0.5 = 3600 J Vtransistor/VAA = 2 Etransistor = 7200 J An example solution QR-01 Talk

18. Open Issue #1 How do quantitative dimensions factor in our similarity judgments? • Quantitative dimensions effect similarity judgments. • Aligned versus non-aligned dimensions • Quantitative similarities/differences  Relational representations QR-01 Talk

19. Open Issue #2 What are the quantitative inferences that analogy sanctions? • Don’t need an overall match to make estimates along a certain dimension only • A good match does not mean that all the aligned dimensions in the base and the target are equally close. QR-01 Talk

20. Open Issue #3 How do we generalize along quantitative dimensions? • No formal/given qualitatively distinct regions • e.g., price of a computer • Experiential development of a sense of quantitativeness • Abstract central tendency and distributions • What does a mid-range server cost? • Managing multiple distributions of different quantities of the same dimension • A notion of expensiveness over prices of different things • Cars, computers, houses are expensive, but houses are the most expensive. QR-01 Talk

21. Conclusions • Understanding BotE is an interesting problem • Key component of commonsense qualitative reasoning. • Important real-world problem-solving strategy • Raises new questions about similarity. • Current efforts • Implementation in progress • Using IPSA (Pisan, 1998) as starting point • Exploring extensions to SME, SEQL • Taking account of numerical similarity in SME • Computing distribution information via alignable differences in SEQL QR-01 Talk

22. Extra slides QR-01 Talk

23. Model Power = Force * Velocity Parameters Power (produced by the human during cycling) = 200 Watts Velocity (given) = 9m/s Force (to be estimated) Solution Fdrag = 200/9 ≈ 22 N Q2 Estimate the drag force on a bicycle and rider traveling at 20 mph (9 m/s). QR-01 Talk

24. Model Fdrag = Cdrag (1/2 ρV2) A …(1) Or, Fdrag = KV2 for same sized objects in the same density fluid. …(2) Plugging the value of K back into (2) gives us Fdrag. Similar scenario: Free-fall, known terminal velocity, VT = 50 m/s Here, Fdrag_free_fall = Weight. …(3) K = Fdrag_free_fall/VT2 = Weight/ VT2 …(4) Parameters [A, Cdrag, ρ (density of air)] can be lumped into K, V (velocity), VT= 50 m/s Calculations K = 750/ 50^2 = 0.3 Fdrag = 0.3 * 9 *9 ≈ 25 N QR-01 Talk

25. Structure-Mapping Theory (Gentner, 1983) • Analogy involves • correspondences between structured descriptions • candidate inferences fill in missing structure in target • 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 • Growing body of evidence that same processes are used in perception, problem solving, conceptualchange(Goldstone, Medin, & Gentner, 1991; Markman & Gentner, 1993; Medin, Goldstone, & Gentner, 1993; Goldstone 1994; Gentner & Markman, 1995, 1997) QR-01 Talk

26. Correctness • Generate multiple solutions • Use the sense of quantity for a reality check • Is that too small? QR-01 Talk

27. Integrated Problem Solving Architecture (Pisan, 1998) QR-01 Talk