Half Orders of Magnitude

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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.
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Basic Observation
Often the most appropriate estimate of a
quantity is to a half order of magnitude.
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Levels of Structure on Scales
not okay
okay
0
--

orders of magnitude
half orders of magnitude
integers
reals
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Outline of Talk
1. Shallow Defeasible HOM Arithmetic 2.
about 3. Natural Half Orders of Magnitude 4.
where
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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.
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What Is An HOM?
Intuitively, HOM 2 - 5, several several2
about 10 Formally, for arithmetic,
1
3.16
10
31.6
100
1.8
18
5.5
55
Assume entities in a given HOM are uniformly
distributed throughout the HOM.
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HOM Addition
Given x of HOM h1 and y of HOM h2, h1 what is probability that xy 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 h2h1, then P 4 If h2h1.5, then
P 68 If h2h11, then P 90 If h2h11.5,
then P 97
several several 10
defeasible, if supporting evidence
increasingly safe
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HOM Multiplication
Given x of HOM h1 and y of HOM h2, what
is the probability that x y is of HOM
h1h2?
10h2.25
xy 10h1h2.25
10h2
10h2-.25
xy 10h1h2-.25
P 70
10h1-.25
10h1.25
10h1
Defeasible, especially if supporting
evidence
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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
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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
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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.
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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 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.
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The Examples Explained
b) about 900 n 9, g 100, 850 950 c) about 980 n 98, g 10, 975 985 a) about 1000 n 2, g
500, 750 875 X 15
Some Complications
g 5 Even multiples grab larger regions.
X is about 35 33 about 40 37 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
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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 MM, thumb tack,
diamond
handled with care between two fingers
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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
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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
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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 dont know
where to put these examples.
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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
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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
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"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.
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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.
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Future Questions
at vs. on vs. in Characterization of
near Characterizations of shapes without rough
radial symmetry
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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.