Half Orders of Magnitude

1. Half Orders of Magnitude Jerry R. Hobbs Artificial Intelligence Center SRI International Menlo Park, California

2. Some Multiple Choice Questions 1. About how many children are there in the average family? a) 1 b) 3 c) 10 d) 30 e) 100 2. About how many children are there in the average classroom? a) 1 b) 3 c) 10 d) 30 e) 100 3. About how many oranges are there in a basket full of oranges? a) 1 b) 3 c) 10 d) 30 e) 100 Often the most appropriate estimate of a quantity is to a half order of magnitude.

3. Basic Observation Often the most appropriate estimate of a quantity is to a half order of magnitude.

4. Levels of Structure on Scales not okay okay 0 -- + orders of magnitude half orders of magnitude integers reals

5. Outline of Talk 1. Shallow Defeasible HOM Arithmetic 2. “about” 3. Natural Half Orders of Magnitude 4. “where”

6. Shallow Defeasible HOM Arithmetic In order of magnitude reasoning, what happens at lower OM has no influence on events at higher OM. (Raiman, 1987; ...) But -- the Heap Paradox. In HOM reasoning, we want a way of concluding the HOM of the result from the HOMs of the constituents, with some degree of reliability. Several events at lower HOM can affect events at higher HOM.

7. What Is An HOM? Intuitively, HOM = 2 - 5, several; several2 = about 10 Formally, for arithmetic, 1 3.16 10 31.6 100 1.8 5.5 18 55 Assume entities in a given HOM are uniformly distributed throughout the HOM.

8. HOM Addition Given x of HOM h1 and y of HOM h2, h1< h2, what is probability that x+y is of HOM h2? 10h2+.25 10h2+.25 -10h1-.25 10h2+.25 -10h1+.25 10h2 10h2-.25 P = 1 - .96 * 10h1-h2 10h1-.25 10h1+.25 10h1 If h2=h1, then P = 4% If h2=h1+.5, then P= 68% If h2=h1+1, then P= 90% If h2=h1+1.5, then P= 97% several + several = 10 defeasible, if supporting evidence increasingly safe

9. HOM Multiplication Given x of HOM h1 and y of HOM h2, what is the probability that x * y is of HOM h1+h2? 10h2+.25 xy = 10h1+h2+.25 10h2 10h2-.25 xy = 10h1+h2-.25 P = 70% 10h1-.25 10h1+.25 10h1 Defeasible, especially if supporting evidence

10. “about” How is this word used in a corpus of business news, scientific articles, fiction, poetry, song lyrics, transcripts of conversation? Examined 86 examples. Topic: 52 Perimeter: 6 Spatial extent: 8 Approximately: 20

11. Implicit Precision There were 920 people at the meeting. Are the following true or false? a) There were about 1000 people at the meeting. -- TRUE b) There were about 900 people at the meeting. -- TRUE c) There were about 980 people at the meeting. -- FALSE a) Implicit precision = 200, 250, or 500. b) Implicit precision = 100 c) Implicit precision = 10

12. Data: What Counts as “about N” We have strong, coarse-grained intuitions about what range counts as “about N” when the speaker knows the right number. “About 80,000 people lost their long-distance service.” Real number probably lies between 77,000 and 84,000. Certainly not 87,000 and probably not 75,000.

13. What “about” Means X is about N: N = n * g, for some integer n and some HOM g. g is the implicit precision. N - .5g < X < N + .5g The HOM between 1 and 10 is usually 5 or 2. (3 lacks good divisibility properties.) The HOM between 10 and 100 is often 25, because it is close to 101.5 and has good divisibility properties.

14. The Examples Explained b) “about 900”: n = 9, g = 100, 850 < X < 950 c) “about 980”: n = 98, g = 10, 975 < X < 985 a) “about 1000”: n = 2, g = 500, 750 < X < 1250 n = 4, g = 250, 875 < X < 1125 n = 5, g = 200, 900 < X < 1100 n = 10, g = 100, 950 < X < 1050

15. Some Complications g = 5: Even multiples grab larger regions. “X is about 35”: 33 < X < 37 “X is about 40”: 37 < X < 43 X often gets rounded down. 86,000 is about 80,000. 74,000 is not about 80,000. Given all this, the characterization works for: "about": 20/20 "approximately": 8/10 (2 were math. OM) "nearly": 13/13

16. Natural HOMs Linear extent: Examples: 6 feet person, door, chair, table, desk can be moved by one person, can accommodate one person 2 feet TV set, dog, basket, watermelon, sack can be held in two arms 8 inches book, football, cantelope can be held in one hand, manipulated with difficulty in one hand 3 inches pen, mouse, hamburger,orange, cup can be held with the fingers 1 inch french fry, eraser, peppermint candy can be bitten, can be manipulated easily with two fingers and thumb 1/4 inch M&M, thumb tack, diamond handled with care between two fingers

17. Natural HOMs Linear extent: Examples: 6 feet person, door, chair, table, desk can be moved by one person, can accommodate one person 18 feet office, room one person can move around can accommodate several people 20 yards house, restaurant, small yard, class 60 yards commercial building, large yard 200 yards small factory, field 600 yards large factory, large bridge, dam 1 mile town, airport 3 miles small city 10 miles large city, small county 30 miles large county 100 miles small state 300 miles large state, small nation 1000 miles typical large European nation 3000 miles the United States, China

18. Natural HOMs There are natural HOMs, anchored on persons, and characterized by distinctive ways of interacting with them. The natural HOM characteristic of a type of entity is part of what we know about the entity. Number of oranges in a basket? HOM: 3 inches HOM: 2 feet 2 HOMs difference About 10

19. “where” How is this word used in the same corpus? farms where corn is grown Where corn is grown, farmers prosper. The Midwest is where corn is grown. Where is corn grown? Examined 74 examples. Figure at Ground: PhysObj at Phys Loc: 7 Where are you? Prop of PhysObj at Prop of PhysLoc: 61 Where corn is grown, farmers prosper Abstraction at Abstraction: 6 I don’t know where to put these examples.

20. Relative Size of Figure and Ground Ground is same HOM as Figure: 36 Right here beside me is where you belong. Ground is one HOM larger than Figure: 13 the counter where slabs of meat were kept The front room was where Marvin stayed. Ground is two HOMs larger than Figure: 5 the houses where the workers live In 54 of 68 cases, HOM(Figure)  HOM(Ground)  HOM(Figure) + 2

21. Relative Sizes of Figure and Ground 11 cases where Ground is more than 2 HOMs larger than Figure: 10 cases: long-term activities of mobile entities the laws of New York, where the business is based 1 case: a treasure chest where a jewel is hidden 3 cases where Figure is larger than Ground: poetic My heart is where you are

22. "Several" Does "several" mean 1 HOM? several Ns: Ns  S If |S|  10, then |Ns|  2-5. If |S|  30-100, then |Ns|  3-8. 2-5: 13 of 25 Several women walked into the cafe. 3-8: 11 of 25 About 80,000 people lost their long-distance service and several communities lost their 911 emergency phone. 3-12: 1 of 25 ... criminal investigation of GE and several of its employees.

23. Opposing Tensions We want a rough logarithmic categorization scheme for sizes in which the categories are large enough that Aggregation operations have reasonably predictable results, Normal variation does not cross category boundaries But small enough that Our interactions with objects is predictable from their category. HOMs optimizes these criteria and is such a categorization scheme.

24. Future Questions “at” vs. “on” vs. “in” Characterization of “near” Characterizations of shapes without rough radial symmetry

25. Summary Half orders of magnitude provide a useful intermediate level of structure for scales. There are natural HOMs, centered on persons, with distinctive modes of interaction. Much of our knowledge about the typical sizes of entities is knowledge about their characteristic natural HOM. We can do limited defeasible arithmetic with HOMs. The meanings and uses of some words depends crucially on HOMs.