1. 1 © Amit Mitra & Amar Gupta FORMATS, SYMBOLS & UNITS OF MEASURE Continuation of our discussion of Pattern and its semantics

2. 2 © Amit Mitra & Amar Gupta Object May be contained in 0 to many [contain 1 to many] May be used by 0 to many [involve values in 1] RULE MEANING Expressed by 1 or more [expression of 1] Rule Expression Object Set Map to 1 [result of 0 or more] REPRESENT/ENCRYPT (REPRESENTATION RULE) May be pattern of 0 or more [be contained in 0 or more] FORMATTING DOMAIN (Domain of Symbols) Member of Object May be pattern of 0 or more [be contained in 0 or more] FORMATTING DOMAIN (Domain of Symbols) Member of Symbol TRANSLATE TO (TRANSLATION RULE) CONTEXT INFORMATION CAPACITY MUST EQUAL OR EXCEED INFORMATION PAYLOAD OF DEGREES OF FREEDOM Symbol FORMAT/ENCRYPT DEGREES OF FREEDOM ENCRYPT GOLDEN RULE OF ENCRYPTION

3. 3 © Amit Mitra & Amar Gupta Object May be contained in 0 to many [contain 1 to many] May be used by 0 to many [involve values in 1] RULE MEANING Expressed by 1 or more [expression of 1] Rule Expression Object Set Map to 1 [result of 0 or more] May be pattern of 0 or more [be contained in 0 or more] FORMATTING DOMAIN (Domain of Symbols) Member of May be pattern of 0 or more [be contained in 0 or more] FORMATTING DOMAIN (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

4. 4 © Amit Mitra & Amar Gupta 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 GOLDEN RULE OF ENCRYPTION Information payload of the object being represented should not exceed information capacity of the object representing it implies…

5. 5 © Amit Mitra & Amar Gupta Metamodel of Representation METAMODEL OF REPRESENTATION

6. 6 © Amit Mitra & Amar Gupta CONVERSIONFORMAT Measure MEASURE

7. 7 © 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 Quantitative Value 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]

8. 8 © Amit Mitra & Amar Gupta Measure of meaning Truncation Limited extent in state space of symbol Rounding Constraints 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 Quantitative Value 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)

9. 9 © Amit Mitra & Amar Gupta 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. EXAMPLES

10. 10 © Amit Mitra & Amar Gupta MAPPING RULES 1.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: 0 o 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 100 o Celsius or 212 o 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

11. 11 © Amit Mitra & Amar Gupta 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” ARITHMETIC Nil/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”

12. 12 © Amit Mitra & Amar Gupta METAMODEL OF UNIT OF MEASURE Eg: ft., ‘, $, USD, etc. Eg: 1,2, IV, FULL FORMAT

13. 13 © Amit Mitra & Amar Gupta Same object 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

14. 14 © Amit Mitra & Amar Gupta Conflicting Subtypes? Rule for conversion of difference scaled values is a subtype of the rule for converting ratio 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

15. 15 © Amit Mitra & Amar Gupta Object Attribute Value Attribute Rule Expression Nominal Rule Expression Ordinal Rule Expression Must take only 1 [of 0 or more] Nominal Value Ordinal Value (inherited) Map to 1 [mapped from 0 or 1] (subtype) term in 0 or more [conjoined via operator with 0 or more] Subtype of Map to 1 [mapped from 0 or 1] Subtype of Mapped by 0 or many [map 1] (subtype) Object Set influence 0 to many [influenced by 0 or 1] (subtype) * RULE MEANING * 1 Object Value DOMAIN Is role of 1 Is member of Attribute Must take only 1 [of 0 or more] Value DOMAIN Is role of 1 Is member of Must take only 1 [of 0 or more] Object Attribute Value Object May participate in 0 or more [contain 0 or more] Map to 1 [mapped from 0 or 1] REPRESENTED BY FORMATTING DOMAIN 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 Information Capacity Information Capacity Object Must take only 1 [of 0 or more] Sets are equal Expressed by 1 or more [expression of 1] 0..* Mapped by 0 or more [map 1 or more] influence 0 to many [influenced by 0 or 1] (subtype) Map to 1 [mapped from 0 or 1] (subtype) Quantitative Rule Expression Quantitative Value (inherited) Subtype of Map to 1 [mapped from 0 or 1] Subtype of

16. 16 © Amit Mitra & Amar Gupta Conversion Map - Table

17. 17 © Amit Mitra & Amar Gupta Conversion Map - Table

18. 18 © Amit Mitra & Amar Gupta 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] RULE MEANING Expressed by 1 or more [express 1] May be pattern in 1 or more [be contained in 0 or more]

19. 19 © Amit Mitra & Amar Gupta Measure of meaning Quantitative Value Number Symbol May be represented by 0 or more [may represent 0 or more] Truncation Limited extent in state space of symbol Rounding Constraints 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 May be represented by 0 or more [may represent 0 or more]

20. 20 © Amit Mitra & Amar Gupta Number May be mapped by 0 to many [map 1 to many] Value Attribute Must take only 1 [of 0 or more] DOMAIN OF 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 OF NUMBERS Member of Expressed by 1 or more [expression of 1] Rule Expression RULE MEANING EXPRESSED BY (MEASURE) EXPRESSED BY Expression of Rule UML SYNTAX 0..* * Meaning of Rule 1 Number Symbol (inherited from Symbol) may be pattern of 1 or more [be contained in 0 or more] FORMATTING DOMAIN (Domain of Symbols) Member of (inherited from Symbol) Unit of Measure Symbol (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) 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)

21. 21 © Amit Mitra & Amar Gupta Same object Same Object

22. 22 © Amit Mitra & Amar Gupta Number Map to 1 [mapped from 0 or 1] Mapped by 0 or more [map 1] x + (term) Number (term) Rule Expression RULE MEANING Expressed by 1 or more [express 1] (subtype of)

23. 23 © Amit Mitra & Amar Gupta Number Map to 1 [mapped from 0 or 1] Mapped by 0 or more [map 1] x + (term) Number (term) Rule Expression RULE MEANING Expressed by 1 or more [express 1]

24. 24 © Amit Mitra & Amar Gupta NumberValue 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

25. 25 © Amit Mitra & Amar Gupta RULE MEANING Expressed by 1 or more [express 1] x + (term A) Number (term B) Rule Expression x + (term A) Number (term B) Rule Expression Number must be 0 (Number Constraint) Number Subtype of (CONSTRAINT ADDED TO SUBTYPE) RULE EXPRESSION FOR CONVERTING DIFFERENCE SCALED VALUES RULE EXPRESSION FOR CONVERTING RATIO SCALED VALUES

26. 26 © Amit Mitra & Amar Gupta RULE MEANING Expressed by 1 or more [express 1] x (term) Number Rule Expression x + (term) Number Rule Expression Number Subtype of (TERM ADDED TO SUBTYPE) RULE EXPRESSION FOR CONVERTING RATIO SCALED VALUES RULE EXPRESSION FOR CONVERTING DIFFERENCE SCALED VALUES

27. 27 © Amit Mitra & Amar Gupta RULE MEANING Expressed by 1 or more [express 1] x (term A) Number Rule Expression x + (term A) Number Rule Expression Number Subtype of (TERM B ADDED TO SUBTYPE) RULE EXPRESSION FOR CONVERTING RATIO SCALED VALUES RULE EXPRESSION FOR CONVERTING DIFFERENCE SCALED VALUES