1. Back of the Envelope Reasoning Praveen Paritosh Ken Forbus QR04 Practice Talk Thursday, July 8, 2004

2. Back of the Envelope Reasoning Numeric answers Specificity-Resources tradeoff Only thing you can do in many domains –Environmental science, Biophysics, etc. –New, unfamiliar domains. Useful –Engineers, Scientists, Policymakers, everyone who reads a newspaper Problem solving + feel for numbers

3. 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. Constraints guiding Common Sense QR

4. Some examples How many K-8 elementary school teachers are in the USA? How much money is spent on newspapers in USA per year? How much time would be saved per year nationwide by increasing the speed limit from 55 to 65 mph? What is the annual cost of healthcare in USA? How much tea (weight) is there in China? Last summer, the US Army bought Microsoft Windows/Office/Server software for 500,000 computers. The deal included the software and six years of support. How much did the army pay for this?

5. How many K-8 elementary school teachers are in the USA? Number of teachers = number of students / students per teacher Number of students = population * fraction in the age range of K- 8 students * fraction of kids who go to school Number of students = 300 mil * 9/75 * 1 Number of teachers = 40 mil / 25 = 1.6 mil 1.9 million, source: Statistical Abstracts.

6. How much money is spent on newspapers in USA per year? Total money spent = Money spent per buyer * number of buyers Annual expense per buyer = Units bought per year * cost per unit Annual expense per buyer = 365 * 0.75 = 250 Number of buyers = 300 mil * ¼ = 75 mil Total money spent = 75 mil * 250 = 20 billion 26 billion, source: Statistical abstracts

7. What is the total annual gasoline consumption by cars in US? Total consumption = Total miles driven/ miles per gallon Total miles driven = Number of cars in the US * Miles driven per car per year Miles driven per car per year = Miles driven per day * 365 If we say that every household owns a car, since some don’t and some might have more than one, then Number of cars in the US = number of households = population / average size of American household. Now we have a model, and using the following numbers, Population ~ 300 million, Average size of household ~ 3, Daily miles driven ~ 20, Miles per gallon for a car ~ 20. We get an estimate of 36.5 billion gallons.

8. Formalizing BoTE reasoning Estimate parameter directly Create estimation model Find modeling strategy Find values for parameters in model Use known value if available Estimate based on similar situation 1 2 3

9. Problem Solving Representation –Of problems, strategies and domain knowledge. Retrieval –Of relevant knowledge and strategies Reasoning –Workspace: keeping track of progress made. –Agenda: figuring out what to do next.

10. BotE-Solver Representation –Domain knowledge: 1.2 million fact subset of Cycorp’s CYC knowledge base. –Strategies: Suggestions. Retrieval –Pattern matching and backward chaining. Reasoning –Built on top of FIRE reasoning engine –AND/OR tree as workspace. –Difficulty ordered agenda.

11. SOLVE Suggestions based problem solver Represent problem solving progress in an and/or tree structure Incremental solution generation, get- next-solution Ordered subgoals

12. An example (defSuggestion VolumeFormulaForSphere (volumeOfObject ?obj ?vol) :test (shapeOfObject ?obj SphereShape) :subgoals ((radius ?obj ?radius)) :result-step (evaluate ?vol (TimesFn 4.187 ;;4/3*Pi (ExponentFn ?radius 3))))

13. (annualSales NewspaperCopy UnitedStatesOfAmerica (YearFn 2003) ?money) (defSuggestion PerBuyerStrategy (annualSales ?obj ?place ?time ?money) :subgoals ((annualExpensePerBuyer ?obj ?place ?time ?money-per-buyer) (numberOfBuyers ?obj ?place ?time ?number)) :result-step (evaluate ?money (TimesFn ?money-per-buyer ?number))) (defSuggestion UnitaryStrategyForCost (annualExpensePerBuyer ?obj ?place ?time ?money-per-buyer) :subgoals ((annualUnitsBoughtPerBuyer ?obj ?place ?time ?units) (cost ?obj ?unit-cost)) :result-step (evaluate ?money-per-buyer (TimesFn ?units ?unit-cost))) (defSuggestion FractionOfTotalStrategy (numberOfBuyers ?obj ?place ?time ?number) :subgoals ((populationDuring ?place ?any-time ?total) (percentOfBuyers ?obj ?place ?percent)) :result-step (evaluate ?number (QuotientFn (TimesFn ?total ?percent) 100)))

14. (cardinality K-8SchoolTeacher ?numteachers) (defSuggestion StudentsPerTeacherStrategy (cardinality K-8SchoolTeacher ?numteachers) :subgoals ((cardinality K-8SchoolStudent ?numstudents) (studentsPerTeacher K-8School ?perteacher)) :result-step (evaluate ?numteachers (TimesFun ?numstudents ?perteacher))) (defSuggestion UniformAgeDistributionStrategy (cardinality K-8SchoolStudent ?numstudents) :subgoals ((populationDuring UnitedStatesOfAmerica (YearFn 1997) ?population) (minimumAge K-8SchoolStudent ?min) (maximumAge K-8SchoolStudent ?max) (lifeExpectancyForGroupInRegion UnitedStatesOfAmerica Person (YearsDuration ?life))) :result-step (evaluate ?numstudents (TimesFn (QuotientFn (DifferenceFn ?max ?min) ?life) ?population))))

15. What next Represent 20-30 problems, with the goal of re-usable representations, in strategies and theoretical knowledge added. Feel for numbers –Build symbolic representations for numbers Large, expensive, upper class, etc. –Analogical estimator: makes guesses for a numeric parameter based on analogy.

16. Numbers in symbolic knowledge representation Consider the Great black-bucked gull –Wingspan = Large –Wing-span = 0.272 sq. m. Numbers not handled right –Similarity: computing and making inferences –Retrieval –Generalization

17. Extra Stuff

18. Examples

19. How much time would be saved by increasing the speed limit from 55 to 65 mph? Total time spent driving = time spent driving per person * number of drivers Time spent driving per person = Total distance driven per person /Speed Total distance driven per person = Distance per day * number of days driving Number of drivers: assume one driver per household = 1/3 * population = 100 mil Total distance driven per person = 20 * 365 = 8000 Time spent per person = 8000/55 = 150 hours Time spent if it was 65 mph = 8000/65 = 120 hours Time saved per person = 30 hrs Total time saved = 3 billion hours = 300,000 years.

20. The Microsoft Army – How much does software for 500,000 computers cost? Total cost on the software = cost of desktop software + server software + upgrades Windows XP/Office CDW prices = 650 Cost of desktop software = 500,000 * 650 = 325 mil Cost of upgrades = 500,000 * 200 = 100 mil Assume 1 in 100 servers = 5000 servers Exchange server = 600 Windows 2003 server = 650 SQL Server = 1000 Cost of server software = 2250 * 5000 = 11.25 mil Cost of upgrades = 400 * 5000 = 2 mil Total cost = 440 million Softmart, inc, PA got paid = 470 million

21. What is the annual cost of healthcare in USA? Lets say everyone was insured. Total cost of healthcare has to be less than insurance premiums, for the insurance companies to stay in business. Cost of healthcare = average insurance premium * population = 3,000 * 300 million ~ 1 trillion [1.6 trillion last year. How does this work?]

22. How much tea is there in China? Total tea = amount of tea consumed per day * stockpile in time Tea consumed per day = tea consumed per person per cup * number of cups * population 1 billion people 5 cups a day 5 grams a cup Lets say they stock an years supply 25 billion grams = 25 million kilos per day In an year 365*25 million ~ 10 billion kilos

23. CARVE, feel for numbers

24. CARVE: Symbolizing Quantity Don’t do it –Sorites –Context/Utterer sensitivity Dimensional partitions: Large and Small, based on distributional properties of the quantity. (isa Algeria (HighValueContextualizedFn Area AfricanCountries)) Structural partitions: Boiling point and Poverty line, denote changes of quality.

25. CARVE Dimensional partitioning for each quantity (isa Algeria (HighValueContextualizedFn Area AfricanCountries). Add these facts to original cases Structural clustering using SEQL S1S1 S2S2 S3S3 CjCj CiCi Ci*Ci* Quantity 1 L2L2 L1L1

26. Analogical Estimator (GrossDomesticProduct Brazil ?x) The value is known. Find an analogous case for which value is known. Find anything in the KB which might help me make an estimate.

27. Thesis and Evaluation More powerful and flexible back of the envelope reasoning can be done using these symbolic representations of quantity. Evaluation: –Corpus of problems with/without representations generated by CARVE.

28. Numbers in News The cost of Mars Pathfinder mission was 270 million. 50 billion tax cut. 62,000 square feet.

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