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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© Amit Mitra & Amar Gupta
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
May be pattern of 0 or more[be contained in 0 or more]
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
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© Amit Mitra & Amar Gupta
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]
May be pattern of 0 or more[be contained in 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]
Map to 1[mapped from 0 or 1]
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)
Map to 1 [mapped from 0 or 1]
(subtype)
QuantitativeRule Expression
Quantitative Value
Quantitative Value(inherited)
Subtype of
Map to 1[mapped from 0 or 1]
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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© Amit Mitra & Amar Gupta
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]
May be represented by 0 or more[may represent 0 or more]
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© Amit Mitra & Amar Gupta
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 Number Symbol
May be represented by 0 or more[may represent 0 or more]
May be represented by 0 or more[may represent 0 or more]
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 Number Symbol
May be represented by 0 or more[may represent 0 or more]
May be represented by 0 or more[may represent 0 or more]
QuantitativeValue
QuantitativeValue NumberNumber SymbolSymbol
May be represented by 0 or more[may represent 0 or more]
May be represented by 0 or more[may represent 0 or more]
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:
GOLDEN RULES OF MEASUREMENT
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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© Amit Mitra & Amar Gupta
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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© Amit Mitra & Amar Gupta
Same object
Sam
e ob
ject
Number
Map to 1 [mapped from 0 or 1]
Mapped by 0 or more[map 1]
x +
(term)
NumberNumber
(term)
RuleExpression
RULEMEANING
Expressed by 1 or more[express 1]
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
GOLDEN RULES OF MEASUREMENT
See Supplementary materials Box
40
GOLDEN RULES OF MEASURE
MENT
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© Amit Mitra & Amar Gupta
Conflicting Subtypes?
• Rule for conversion of difference scaled values is a subtype of the rule for converting ratio scaled values
RULEMEANING
Expressed by 1 or more[express 1]
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
Expressed by 1 or more[express 1]
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