1 © amit mitra & amar gupta formats, symbols & units of measure continuation of our...

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© Amit Mitra & Amar Gupta

FORMATS, SYMBOLS & UNITS OF MEASURE

Continuation of our discussion of Pattern and its semantics

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Object

May b

e contain

ed in

0 to man

y[con

tain 1 to m

any]

May be used by 0 to many[involve values in 1]

RULE MEANING Expressed by 1 or more[expression of 1]

Rule ExpressionObject SetMap to 1

[result of 0 or more]

REPRESENT/ENCRYPT(REPRESENTATION RULE)

May be pattern of 0 or more[be contained in 0 or more]

FORMATTINGDOMAIN

(Domain of Symbols)

Mem

ber of

Object


FORMATTINGDOMAIN

(Domain of Symbols)

Member of

Symbol

TRANSLATE TO(TRANSLATION RULE)

CO

NT

EX

T

INFORMATION CAPACITY MUST EQUAL

OR EXCEED INFORMATION PAYLOAD OF

DEGREESOF

FREEDOM

Symbol

FORMAT/ENCRYPT

DEGREESOF

FREEDOM

ENCRYPT

GOLDEN RULE OF

ENCRYPTION

3


Object

May b

e contain

ed in

0 to man

y[con

tain 1 to m

any]

May be used by 0 to many[involve values in 1]

RULE MEANING Expressed by 1 or more[expression of 1]

Rule ExpressionObject SetMap to 1

[result of 0 or more]


FORMATTINGDOMAIN

(Domain of Symbols)

Mem

ber ofMay be pattern of 0 or more

[be contained in 0 or more]

FORMATTINGDOMAIN

(Domain of Symbols)

Member of

Symbol

FORMAT(FORMATTING RULE)

•Extent=Scope•Delimiter (when present) delimits Extent•Degrees of freedom determines discrimination•Proximity determines resolution

See Supplementary Materials Box

38: “Formatting Constraints” to

end

•Extent=Size•Delimiter (when present) delimits Extent•Degrees of freedom determines precision•Proximity determines cohesion

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RULES OF SIMPLE REPRESENTATION

• Each attribute of the object being represented will map to exactly one attribute of the object that represents it.

• A single value may not be represented by several values.

• Multiple discrete values may not be mapped to a single discrete value

• Attributes that have a continuum of values may not me mapped to attributes with discrete values.

• Ratio scaled attributes must be mapped to only ratio scaled attributes.

• Difference scaled attributes may be mapped to difference, or ratio scaled attributes

• Ordinally scaled attributes may be mapped to ratio, difference, or ordinally scaled attributes

• Nominally scaled attributes may be mapped to ratio, difference, ordinally, or nominally scaled attributes

Normalized information

stays normalized

Prevent Information

Loss

REPRESENTED ATTRIBUTE REPRESENTING ATTRIBUTE Nominal Ordinal

Difference Scaled Ratio Scaled

Nominal Ordinal Difference Scaled Ratio Scaled

GOLDEN RULE OF

ENCRYPTION

Information payload of the object being represented should not exceed information capacity of the object representing it implies…

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Metamodel of Representation

ObjectAttribute

Value

ObjectAttribute

Value

Attribute

Rule Expression

NominalRule Expression

OrdinalRule Expression

Must take only 1

[of 0 or more]

Nominal Value

Nominal Value

OrdinalValue

OrdinalValue

(inherited)

(inherited)

Map to 1 [mapped from 0 or 1]

(subtype)

term in 0 or more[conjoined via operator with 0 or more]

Subtype of

Subtype of

Map to 1[mapped from 0 or 1]

Su

btyp

e of

Su

btyp

e of

Mapped by 0 or many[map 1]

(subtype)

Object Set

infl

uen

ce 0

to

man

y[i

nfl

uen

ced

by

0 or

1]

(su

btyp

e)

*

RULEMEANING

RULEMEANING

*

1

ObjectValueValue

DOMAIN

Is r

ole

of 1

Is member

ofAttribute

Mu

st take only 1[of 0 or m

ore]

ValueValue

DOMAIN

Is role of 1

Is member

of

Must take only 1

[of 0 or more]

ObjectAttribute

Value

ObjectAttribute

Value

Object

May particip

ate in 0 or more

[contain 0 or m

ore]


REPRESENTED BYREPRESENTED BY

FORMATTINGDOMAIN

Is member of

is property of

Is member of

is property of

THIS SNAP-ON KNOWLEDGE COMPONENT WILL MAKE “REPRESENTED BY” INTO “CONVERT FORMAT”

CANNOT EXCEED

InformationCapacity

InformationCapacity

ObjectMust take only 1[of 0 or more]

Sets are equalSets are equalSets are equalSets are equal

Sets are equalSets are equal

Exp

ressed by 1 or m

ore[exp

ression of 1]

0..*

Mapped by 0 or more[map 1 or more]

infl

uen

ce 0

to

man

y[i

nfl

uen

ced

by

0 or

1]

(subtype)

(subtype)


(subtype)

QuantitativeRule Expression

Quantitative Value

Quantitative Value(inherited)

Subtype of


Su

btyp

e of

METAMODEL OF REPRESENTATION

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May be contained in 0 to many[contain 1 to many]

Symbol

Map to 1 [mapped by 0 or 1]

Expression of Rule

term in 0 or more[conjoined via operator with 0 or more]

Object Set

May be used in 0 to many[involve values in 1]

RULEMEANING

RULEMEANING

Expressed by 1 or more[express 1]

May be pattern in 1 or more[be contained in 0 or more]

CONVERSIONFORMAT

Measure

MEASURE

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FORMATTING QUANTITATIVE DOMAINS• Cardinality of Quantitative domain is infinite

– Quantitative domains are dense

• Cardinality of Formatting domains is finite– We cannot map infinite numbers of values to finite numbers of discrete symbols without

losing information

• Numbers are ratio scaled– Numbers are meaningless by themselves

– Joining numbers to domains lends them meaning

• Eg: length is 12 feet

• We could map quantitative domains to numbers without losing information– Must be formatted in physical space by symbols

REPRESENT

QuantitativeValue Number Symbol

May be represented by 0 or more[may represent 0 or more]


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Measure of meaning

TruncationLimited extent in state space of symbol

RoundingConstraints on permitted proximity between numbers

(Limitation on information carrying capacity)

Formatting rule

Scope of Format(Limited by inclusion of

value in an extent of state space)

REPRESENT




BEHAVIORS NORMALIZED BY DIFFERENT COMPONENTS WHEN EXPRESSING OF QUANTITATIVE VALUES

Unit Of Measure (UOM)

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• Rule: “Extremely large or extremely low values of temperature must be displayed in red and also voiced audibly to alert an operator of a furnace, regardless of the unit used to measure the temperature, such as Fahrenheit, Celsius, Kelvin, or any other number that displays the temperature”.

– The behavior of format depends on the value of the temperature, not the number displayed, nor its unit of measure. This formatting rule is normalized by a relationship between Value and Format (symbol).

• Rule: “Extremely large or extremely small numbers must be in exponential formats and those in between, in decimal format”.

– The behavior of format depends on the number, not value or unit of measure. The format depends on number alone. It is a relationship between number and Format that normalizes this rule.

• Rule: “All Roman Numerals are red”.

– Roman numerals are a visual format, a perceptible symbol that expresses a number. Format alone, not number, value or unit of measure, normalizes this rule.

• Rule: “All temperatures in degrees Celsius must be red”.

– The format depends on Unit of Measure alone, not formatting symbols, number, or value. A relationship between Unit of measure and Format normalizes this rule.

Measure of meaning

TruncationLimited extent in state space of symbol

RoundingConstraints on permitted proximity between numbers

(Limitation on information carrying capacity)

Formatting rule

Scope of Format(Limited by inclusion of

value in an extent of state space)

REPRESENT




QuantitativeValue

QuantitativeValue NumberNumber SymbolSymbol



Unit Of Measure (UOM)

EXAMPLES

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MAPPING RULES1. If values have different magnitudes, an internally consistent measure will not assign

the same number to them . – However, the same number in different measures may represent different values.

• This was the Mars Climate Orbiter’s problem.

2. Conversely, given a measure of a domain, values with the same intrinsic magnitude will map to the same number .

– Eg: 0o Celsius will always mean the same temperature.

– The same temperature could map to different numbers if the measure is different. • Eg: the boiling point of water is 100o Celsius or 212o Fahrenheit. The Celsius measure maps the (magnitude of) temperature of

boiling water to the number 100, whereas the Fahrenheit measure maps the same temperature to the number 212.

3. The relative ordering of magnitudes must be consistent across measures– Sequencing of values must be preserved across measures.

• Eg: the freezing point of water is a lower temperature than that of boiling water. The number for the freezing point of water must always be lower than that of boiling water in every measure. Thus, in Celsius 0 is less than 100, and in Fahrenheit –32 is less than

212.

4. Each measure must have a unit of magnitude for gaps between magnitudes that maps to the number “1”.

– In (both ratio and) difference scaled domains magnitudes of gaps between measures are also meaningful, and we must be in a position to compare these gaps consistently (within a given measure).

– Therefore each measure must have a unit of magnitude for gaps between magnitudes that maps to the number “1”. Thus 1 o Celsius is a different magnitude from 1 o Fahrenheit, but both are units of measure of differences of temperature

5. When two values are equal, their difference must map to the number 0 (naturally!).

GOLDEN RULES OF MEASUREMENT

Friday, October 1, 1999“LOS ANGELES -- A mix-up over metric and Englishmeasurements..caused the destruction of the $125 millionMars Climate Orbiter..last week..The spacecraft flew tooclose to Mars and is believed to have broken apart orburned up in the atmosphere. NASA said the English-vs.-metric mix-up .. caused the navigation error.”

- THE ASSOCIATED PRESS(http://www.fas.org/mars/991001-mars01.htm)

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6. No value can be said to be of an infinitesimally small magnitude.– Values that naturally map to 0 signal the absence of a property (not the absence of meaning of the property!).

– For difference scaled values we do not know if a value naturally maps to the number 0. • Magnitudes of gaps between values may map to zero naturally when two or more values are coincident, but difference

scaled values have no natural zero.

• Therefore it is not mandatory that a single value must map to the number zero across all measures, nor is it mandatory that measures of difference scaled domains must have a zero– The number zero is arbitrarily imputed to an arbitrary value

• (Differences are valid, but addition, division and multiplication are meaningless; the results are “Unknown”)

– Eg: the length domain has a natural zero but not the domain of dates– We know differences between dates (and times) in days, hours, minutes, seconds etc, and can say which dates come before

which, but it is meaningless to talk about ratios between, or sums of dates. – 12 AM, Jan 1, 0 AD has been arbitrarily set to zero by convention.

6. A single value must map to the number zero across all measures because it represents the absence of magnitude of a property

• Not the absence of meaning of the property!

• Thus the number 0 means the same thing in all meaningful measures of ratio scaled domains.

• Eg: when two objects touch, their separation will be zero in every possible units of measure – feet, inches, meters etc; even units of measure not invented yet.

Ratio scaled values carry information on ratios between magnitudes and the kind of information conveyed by difference scaled values.

MAPPING RULE 6 FOR RATIO SCALED VALUES:


THE “UNKNOWN” ARITHMETICNil/Nil= “Unknown”

Infinity/Infinity = “Unknown”Infinity - Infinity = “Unknown”

“Unknown” (comparison) Nominal value = “Unknown”“Unknown” (ranking operation) Ordinal value = “Unknown”

“Unknown” (arithmetic operation) Quantitative value = “Unknown”

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NumberMay be mapped by 0 to many[map 1 to many]

ValueValue

Attribute

Must take only 1[of 0 or more]

DOMAINOF

MEANING

FRAGMENT FROM METAMODEL OF

ATTRIBUTE

is a subtype of

Equal sets

of only 1

Subtype of

Map to 1[mapped by 0 or more]

DOMAIN OFNUMBERS

Mem

ber of

Expressed by 1 or more[expression of 1]

Rule Expression

RULE MEANINGRULE MEANING

EXPRESSED BY(MEASURE)

EXPRESSED BY

Expression of Rule

UML SYNTAX

0..*

*

Meaning of Rule

1

NumberSymbol

(inherited from Symbol)may be pattern of 1 or more[be contained in 0 or more]

FORMATTINGDOMAIN

(Domain of Symbols)

Member of (inherited from Symbol)

Unit of MeasureSymbol

(inherited from Symbol)may be pattern of 1 or more[be contained in 0 or more]

]

Expressed by 1 or more[represent 0 or more]

Expressed by 1 or more[represent 0 or more]

(FORMATTING RULE)

(FORMATTING RULE)

FULL FORMAT

FULL FORMAT(ordered pair of symbols)

FULL FORMAT(ordered pair of symbols)

Number Symbol

Unit of Measure Symbol

Contain 1[contained in 0 or more]

Contain 1[contained in 0 or more]

Member of

Member of (inherited from Symbol)

(inherited from Symbol)

METAMODEL OF UNIT OF MEASURE

Eg: ft., ‘, $, USD, etc.

Eg: 1,2, IV,

FULL FORMAT

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Same object

Sam

e ob

ject

Number


Mapped by 0 or more[map 1]

x +

(term)

NumberNumber

(term)

RuleExpression

RULEMEANING


Same objec

t

Number

Value Value

Number

must equal[must equal]

Expressed by 0 or more[Express by 0 or more]

Expressed by 0 or more[Express by 0 or more]

Convert to 0 or more

UNIT OF MEASURE CONVERSION

•Units of measure can only be converted to other units of measure for the same domain

•Unit of Measure conversion must conform to the Golden Rules of Measurement

– Units of measure for ratio scaled domains can be converted to another unit of measure for the same domain by multiplying every number in the unit of measure by a fixed, non-zero conversion factor

– Units of measure for difference scaled domains can be converted to another unit of measure for the same domain by multiplying every number in the unit of measure by a fixed, non-zero conversion factor.

•Even if we add (or subtract) a fixed number from the result, it will stay a unit of measure (because different difference scaled units of measure need not map the same value to the number zero)

– More complex conversion rules may also conform to the Golden Rules of Measurement

•Eg: Decibels, Richter scale etc

•UOM conversion rules may change with time

– Eg: Currency conversion, indexing


See Supplementary materials Box

40

GOLDEN RULES OF MEASURE

MENT

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Conflicting Subtypes?

• Rule for conversion of difference scaled values is a subtype of the rule for converting ratio scaled values

RULEMEANING


x

(term A)

NumberRule

Expression

x +

(term A)

NumberRule

ExpressionNumber

Subtype of

(TERM B ADDEDTO SUBTYPE)

RULEEXPRESSION FORCONVERTINGRATIO SCALEDVALUES

RULE EXPRESSION FOR CONVERTINGDIFFERENCE SCALED VALUES

RULEMEANING


x +

(term A)

NumberNumber

(term B)

RuleExpression

x +

(term A)

Number

(term B)

RuleExpression

Numbermust be 0(Number

Constraint)Number

Subtype of(CONSTRAINT ADDED

TO SUBTYPE)

RULE EXPRESSIONFOR CONVERTINGDIFFERENCESCALED VALUES

RULE EXPRESSION FOR CONVERTINGRATIO SCALED VALUES

• Rule for conversion of ratio scaled values is a subtype of the rule for converting difference scaled values

• Both cannot be true; which is correct?• RULE OF THUMB: Meaning adds more information than a computational term • Arbitrary, possibly different difference scaled values have been mapped to the same number (0)

– The expression on the right has lost information because it has violated the first rule of simple representation: “Each attribute of the object being represented will map to exactly one attribute of the object that represents it”

The object paradigm is not enough!

• The model on the right is correct

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