Using Strategies and AND/OR Decomposition for Back of the Envelope Reasoning

1. Using Strategies and AND/OR Decomposition for Back of the Envelope Reasoning Praveen Paritosh Ken Forbus Qualitative Reasoning 2004 2nd August 2004

2. Outline • Examples and motivation • Worked out examples • Implementation of BotE-Solver • Results and current status • What next

3. Some examples • How many K-8 school teachers are in the USA? • How much money is spent on newspapers in USA per year? • What is the total annual gasoline consumption by cars in US? • 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?

4. 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 [contrast first principles reasoning]

5. 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 quantity: 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.

6. Outline • Examples and motivation • Worked out examples • Implementation of BotE-Solver • Results and current status • What next

7. How many K-8 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, 2003.

8. 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, 2003.

9. 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. 35 billion gallons, Statistical Abstracts, 2003.

10. Outline • Examples and motivation • Worked out examples • Implementation of BotE-Solver • Results and current status • What next

11. Formalizing BoTE reasoning Estimate parameter directly 1 Use known valueif available Feel for numbers 2 Estimate basedon similar situation Create estimation model Problem solving Find modeling strategy 3 Find values forparametersin model

12. 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.

13. 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.

14. AND/OR Decomposition Total annual gasoline consumption MileageStrategy Suggestion/OR-node Total miles driven Miles per Gallon Goal/AND-node PerUnitStrategy Annual miles driven per car Number of cars HouseholdStrategyForCountingUnits PerDayForYearStrategy Number of cars per house Number of households Miles driven per day

15. An example suggestion (defSuggestion HouseholdStrategyForCountingUnits :trigger (unitsTotal ?obj ?place ?time ?total-units) :test (ownedBy ?obj FamilyCohabitationUnit) :subgoals ((numberOfHouseholds ?place ?time ?num-households) (unitsPerHousehold ?obj ?units-per-household)) :result-step (evaluate ?total-units (TimesFn ?num-households ?units-per-household)))

16. Axioms for HouseholdStrategyForCountingUnits (ist-Asserted SuggestionMT (suggestionResultStep HouseholdStrategyForCountingUnits (evaluate ?total-units (TimesFn ?num-households ?units-per-household)))) (ist-Asserted SuggestionMT (suggestionSubgoals HouseholdStrategyForCountingUnits (TheList (numberOfHouseholds ?place ?time ?num-households) (unitsPerHousehold ?obj ?units-per-household)))) (ist-Asserted SuggestionMT (suggestionGoalForm HouseholdStrategyForCountingUnits (unitsTotal ?obj ?place ?time ?total-units))) (ist-Asserted SuggestionMT (implies (ownedBy ?obj FamilyCohabitationUnit) (suggestFor (unitsTotal ?obj ?place ?time ?total-units) HouseholdStrategyForCountingUnits))) (ist-Asserted SuggestionMT (isa HouseholdStrategyForCountingUnits Suggestion)) (suggestionResultStep HouseholdStrategyForCountingUnits (evaluate ?total-units (TimesFn ?num-households ?units-per-household))) (suggestionSubgoals HouseholdStrategyForCountingUnits (TheList (numberOfHouseholds ?place ?time ?num-households) (unitsPerHousehold ?obj ?units-per-household))) (suggestionGoalForm HouseholdStrategyForCountingUnits (unitsTotal ?obj ?place ?time ?total-units)) (implies (ownedBy ?obj FamilyCohabitationUnit) (suggestFor (unitsTotal ?obj ?place ?time ?total-units) HouseholdStrategyForCountingUnits)) (isa HouseholdStrategyForCountingUnits Suggestion) Uniform representation of domain knowledge and strategies provides substrate for learning.

17. Control Status Flags SOLVED FAILED MOOT MOOT-VIA-FAILURE MOOT-VIA-SUCCESS Solution generation Incremental Two modes Agenda processing IN-PLAY processing Subgoals Fully ordered Workings of the Solver

18. Outline • Examples and motivation • Worked out examples • Implementation of BotE-Solver • Results and current status • What next

19. (annualSales NewspaperCopy UnitedStatesOfAmerica (YearFn 2003) ?money) How much money is spent on newspapers in USA per year? (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)))

20. (cardinality K-8SchoolTeacher ?numteachers) How many K-8 school teachers are in the USA? (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))))

21. Current Status • Can successfully solve 8 problems • Strategy library contains 23 suggestions

22. Problem Number Problem, and its predicate calculus representation Answer found by BotE-Solver, and comparison to a known answer if available #Axioms 1 How many popcorn kernels would it take to fill in the 1890 Maple big classroom? (CountContained CS381ClassRoom Popcorn ?number) (?number . 1.343444e+7) BotE-Solver: 13 million Correct answer: not available! 30 2 How much money is spent on newspapers in the US? (annualSales NewspaperCopy UnitedStatesOfAmerica (YearFn 2003) ?money) (?money . 2.1884363e+10) BotE-Solver: 21 billion Correct answer: 26 billion 30 3 How many K-8 teachers are there in the US? (cardinality K-8SchoolTeacher ?numteachers) (?numteachers . 1056454) BotE-Solver: 1.05 million Correct Answer: 1.9 million 20 4 What is the annual cost of healthcare in the US? (annualSales HealthCare UnitedStatesOfAmerica (YearFn 2003) ?money)) (?money . 799428834000) BotE-Solver: 0.8 trillion Correct Answer: 1.6 trillion 12 Solved Problems - I

23. Problem Number Problem, and its predicate calculus representation Answer found by BotE-Solver, and comparison to a known answer if available # Axioms 5 How many cars are bought per year in the US? (unitsBoughtPerYear Automobile UnitedStatesOfAmerica (YearFn 2003) ?num) (?num . 8920000) BotE-Solver: 8.9 million Correct Answer: 8 million 30 6 What is the weight of garbage thrown away by American families each year? (annualProduction Garbage-Generic UnitedStatesOfAmerica (YearFn 2003) (Pound-UnitOfMass ?garbage-mass)) (?garbage-mass . 446000000) BotE-Solver: 446 million pounds Correct answer: not available! 10 7 How many hotdogs are sold in a baseball season in Wrigley Field? (unitsSold HotDogSandwich WrigleyField BaseballSeason ?num-dogs) (?num-dogs . 1600000) BotE-Solver: 1.6 million Correct answer: not available! 18 8 What is the total amount of gasoline consumption by cars in the US? (annualAutomobileGasConsumption UnitedStatesOfAmerica (YearFn 2003) (Gallon-US ?oil-consumption))) (?oil-consumption . 32558000000) BotE-Solver: 32.6 billion gallons Correct Answer: 35 billion gallons 20 Solved Problems - II

24. Outline • Examples and motivation • Worked out examples • Implementation of BotE-Solver • Results and current status • What next

25. What next • Represent about 50 problems • Show stable re-usable representations, in strategies and theoretical knowledge added. • Knowledge level analysis of the kinds of knowledge involved in BotE. [Preview in the next slides] • Feel for numbers • Build symbolic representations for numbers • Large, expensive, upper class, etc. [CARVE, Paritosh, 2003] • Analogical estimator: makes guesses for a numeric parameter based on analogy.

26. BotE Strategies • Abstract form: (Q O ?Value) • Some examples discussed • Money spent on : newspapers sold every year in the US • Number of : k-8 teachers in the US • Mapping from a question to Q:O not unique.

27. Object based strategies Given Q:O:?V, transform it into Q:O1:?V1, Q:O2:?V2,… such that ?V = f(?V1, ?V2,…) • Similarity: • O is similar to O1 => ?V = ?V1 • O is similar to {Oi} => ?V = average({?Vi}) • Mereology: • {Oi} is the complete set of objects that share the non-overlapping extensive parameter Q. • Weight of a basket of fruits = sum of weight of each fruit and the basket. • Ontology: • If O is an instance of a collection C for which we know the range/distribution of Q, or whose other instances have a known value for Q.

28. Quantity based strategies • Given Q:O:?V, transform it into Q1:O:?V1, Q2:O:?V2,… such that • ?V = f(?V1, ?V2,…) -- (2) • Density: Rates, densities, averages along a dimension multiplied by the extent along that dimension. • Physical laws: F = m * a. [Domain specific knowledge] • Scale-up: unit conversion. • The composition function, f, in eqn 2 has to meet dimensional constraints.

29. Extra Stuff

30. Pseudocode

31. get-solution get-solution(original-goal) if in-play?(original-goal) get-next-solution(original-goal) else if the agenda is empty then quit process-agenda

32. get-next-solution get-next-solution(ao-node) if there are cached solutions at this node ;; cur-bmarker points to the current solution increment cur-bmarker(ao-node) return cached solution if goal-node?(ao-node) if ao-node has a child suggestion that is in-play get-next-solution(in-play-suggestion(ao-node)) ;; No in-play suggestion, if this goal has a younger ;; sibling that can give us more new bindings, re-instantiate ;; this node with those bindings. if younger-siblings(ao-node) bindings = get-next-solution(younger-sibling(ao-node)) if bindings found enqueue-on-agenda(re-instantiate-node(ao-node,bindings)) if suggestion-node?(ao-node) ;; seek downward solution = get-next-solution(eldest-child,ao-node) add-bindings(solution,ao-node) increment cur-bmarker(ao-node) return solution

33. propagate-bindings propagate-bindings(goal-node) if elder-sibling?(goal-node) increment cur-bmarker(goal-node) next-node = instantiate-node(elder-sibling(goal-node), current-bindings(goal-node)) add-to-tree(next-node) enqueue-on-agenda(next-node) return ;; No elder sibling, so we can now see ;; if the parent node got solved if parent(goal-node) exists bindings = get-next-solution(goal-node) while we have bindings for goal-node result-bindings = do-result-step(parent(goal-node)) if result-bindings found add-bindings(result-bindings,parent(goal-node) propagate-bindings(parent(parent(goal-node))) bindings = get-next-solution(goal-node)

34. process-agenda process-agenda(ao-node) if goal-node?(ao-node) solutions = ask(goal-node) ;; primitive problem solving step if solutions found propagate-bindings(goal-node) else suggestions = gather-suggestions(goal-node) if suggestions found enqueue-on-agenda(suggestions) else ;; no suggestions found, see if we can ;; re-instiate this node if younger-siblings(goal-node) bindings = get-next-solution(younger-sibling(goal-node) if bindings found enqueue-on-agenda(re-instantiate-node(ao-node,bindings)) else update-failed (goal-node) if suggestion-node?(ao-node) ;; since the subgoals are ordered, add the first one to agenda enqueue-on-agenda(youngest-child(ao-node))

35. Examples

36. 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.

37. 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

38. 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?]

39. 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

40. CARVE, feel for numbers

41. 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

42. 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.

43. Dimensional partitioning for each quantity (isa Algeria (HighValueContextualizedFn Area AfricanCountries) . . . Ci Add these facts to original cases Quantity 1 S2 Structural clustering using SEQL Cj S1 S3 Ci* L1 L2 CARVE

44. 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.

45. 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.

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

47. Old slides

48. 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))))

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

50. For More The BotE website: www.cs.northwestern.edu/~paritosh/bote/ Library of problems, solutions, strategies.