Analysis of strategic knowledge used in Back of

the Envelope Reasoning

Praveen ParitoshKen Forbus

QRG Meeting, 8th March 2005

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 power can a adult human generate? • What is Jason Kidd’s point per game for this

season?

Outline

• Motivation

• Examples

• BotE-Solver

• Analysis of Strategic Knowledge

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]

1. IncompletenessDomain theories incomplete in coverage.

2. ConcretenessKnowledge 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 reasoningTight reasoning, as opposed to maintaining ambiguity for completeness

5. Pervasively quantitativeReal-world actions require that estimates manifest as exact values.

Constraints guiding Common Sense QR

How many K-8 school teachers are in the USA?

Number of teachers = number of students / students per teacherNumber 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 * 1Number of teachers = 40 mil / 25 = 1.6 mil

1.9 million, source: Statistical Abstracts, 2003.

How much money is spent on newspapers in USA per year?

Total money spent = Money spent per buyer * number of buyersAnnual expense per buyer = Units bought per year * cost per unit

Annual expense per buyer = 365 days * $0.75 = $250Number of buyers = 300 mil * ¼ = 75 mil Total money spent = 75 mil * $250 = $20 billion

$26 billion, source: Statistical abstracts, 2003.

Formalizing BoTE reasoning

Estimate parameter directly

Create estimation model

Find modeling strategy

Find values forparameters

in model

Use known valueif available

Estimate basedon similar situation

1

2

3

Feel for numbers

Problem solving

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.

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.

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)))

HouseholdStrategyForCountingUnits

Total annual gasoline consumption

Total miles driven

Miles per Gallon

MileageStrategy

Number of cars

Miles driven per day

Annual miles driven per car

Number of households

Number of cars per house

PerUnitStrategy

PerDayForYearStrategy

Suggestion/OR-node

Goal/AND-node

AND/OR Decomposition

Analogical Estimation

• 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.

– Representations augmented with symbolic representation lead to more accurate estimates [Paritosh, submitted to CogSci05].

Analysis of Strategic Knowledge

• Claim: There is a core collection of strategic knowledge, specifically, seven strategies that capture most of back of the envelope reasoning.

• Source: – Strategies in Bote-Solver– Analysis of problems from Force and

Pressure, Rotation and Mechanics, Heat and Astronomy from Clifford Swartz’s Back-of-the-Envelope Physics.

Analysis of strategic knowledge

• Knowledge Level (Newell, 1982)– A description of what the system knows

• Heuristic Classification (Clancey, 1985)– “Forms of knowledge and reasoning patterns of familiar problem

solving situations and solutions, heuristically related.”– Applied Newell’s paradigm to ten expert systems of the era

(MYCIN, SACON, WINE, etc.)

Data

Heuristic matchSolution AbstractionsData abstractions

Solutions

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.

BotE Strategies

(Q O ?Value) → {(Qi Oi ?Vi)}

• Such that ?Vi are already known or easier to estimate.

• Each strategy also contains the answers to the following questions:– When does it apply– How do we combine ?Vi’s to find ?value– How sure are we of our estimate of ?value

Object based strategies

Given Q:O:?V, transform it into Q:O1:?V1, Q:O2:?V2,… such that

?V = f(?V1, ?V2,…)• 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.

• Similarity: – O is similar to O1 => ?V = ?V1– O is similar to {Oi} => ?V = average({?Vi})

• 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.

Kinds of strategies

• Object based: (Q O ?V) → {(Q Oi ?Vi)}

• Quantity based: (Q O ?V) → {(Qi O ?Vi)}

• System based: (Q O ?V) → {(Qi Oi ?Vi)}

1. Mereology

• Extensive: If Q is an extensive parameter,

• {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.

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 equation 2 has to meet dimensional constraints.

Extra Stuff

Pseudocode

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

get-next-solutionget-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

propagate-bindingspropagate-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)

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))

Examples

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 driversTime spent driving per person = Total distance driven per person /SpeedTotal distance driven per person = Distance per day * number of days driving

Number of drivers: assume one driver per household = 1/3 * population = 100 milTotal distance driven per person = 20 * 365 = 8000Time spent per person = 8000/55 = 150 hoursTime spent if it was 65 mph = 8000/65 = 120 hoursTime saved per person = 30 hrs

Total time saved = 3 billion hours = 300,000 years.

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 = 650Cost of desktop software = 500,000 * 650 = 325 mil Cost of upgrades = 500,000 * 200 = 100 mil

Assume 1 in 100 servers = 5000 serversExchange server = 600Windows 2003 server = 650SQL Server = 1000Cost of server software = 2250 * 5000 = 11.25 milCost of upgrades = 400 * 5000 = 2 mil

Total cost = 440 millionSoftmart, inc, PA got paid = 470 million

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, forthe 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?]

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 people5 cups a day5 grams a cupLets say they stock an years supply25 billion grams = 25 million kilos per dayIn an year 365*25 million ~ 10 billion kilos

CARVE, feel for numbers

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

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.

CARVE

Dimensional partitioning for each quantity

(isa Algeria (HighValueContextualizedFn Area AfricanCountries)...

Add these facts to original cases

Structural clustering using SEQL

S1 S2 S3Cj

Ci

Ci*

Quantity 1

L2L1

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.

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.

Numbers in News

• The cost of Mars Pathfinder mission was 270 million.

• 50 billion tax cut.

• 62,000 square feet.

Old slides

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))))

SOLVE

• Suggestions based problem solver

• Represent problem solving progress in an and/or tree structure

• Incremental solution generation, get-next-solution

• Ordered subgoals

For More

The BotE website:

www.cs.northwestern.edu/~paritosh/bote/

Library of problems, solutions, strategies.

Slides from QR04

What is the total annual gasoline consumption by cars in US?Total consumption = Total miles driven/ miles per gallonTotal miles driven = Number of cars in the US * Miles driven per car per yearMiles driven per car per year = Miles driven per day * 365If 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.

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.

Workings of the Solver

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

Outline

• Examples and motivation

• Worked out examples

• Implementation of BotE-Solver

• Results and current status• What next

Current Status

• Can successfully solve 8 problems

• Strategy library contains 23 suggestions

(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)))

How much money is spent on newspapers in USA per year?

(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))))

How many K-8 school teachers are in the USA?

ProblemNumber

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 billionCorrect 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 millionCorrect 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 trillionCorrect Answer: 1.6 trillion

12

Solved Problems - I

ProblemNumber

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 millionCorrect 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 poundsCorrect 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 millionCorrect 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

Outline

• Examples and motivation

• Worked out examples

• Implementation of BotE-Solver

• Results and current status

• What next

Object based strategies

Given Q:O:?V, transform it into Q:O1:?V1, Q:O2:?V2,… such that

?V = f(?V1, ?V2,…)• 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.

• Similarity: – O is similar to O1 => ?V = ?V1– O is similar to {Oi} => ?V = average({?Vi})

• 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.

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 equation 2 has to meet dimensional constraints.