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2/7/01 Professor Richard Fikes Example of Multi-Use Ontology: Units and Measures Computer Science Department Stanford University CS222 Winter 2001 Knowledge Systems Laboratory, Stanford University

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2 Units and Measures Mathematical models of engineered systems äPhysical-Quantities E.g., 3 meters, 80 kilometers/hour äPhysical-Dimensions E.g., length, velocity, weight, time äUnits-Of-Measure E.g., meters, kilometers/hour äMagnitudes E.g., scalars, vectors, tensors, and functions thereof

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Knowledge Systems Laboratory, Stanford University3 Magnitude of a Quantity in a Unit Physical-Dimension Instance-Of: Class *Standard-Unit Slot-Cardinality: 1 Value-Type: Unit-Of-Measure Unit-Of-Measure Instance-Of: Class *Unit-Dimension Slot-Cardinality: 1 Value-Type: Physical-Dimension Physical-Quantity Instance-Of: Class *Quantity-Dimension Slot-Cardinality: 1 Value-Type: Physical-Dimension Magnitude Instance-Of: Class

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Knowledge Systems Laboratory, Stanford University4Unit* (Deffunction Unit* Unit* is an associative commutative mapping of all pairs of units to units (and (=> (Instance-Of ?u1 Unit-Of-Measure) (Instance-Of ?u2 Unit-Of-Measure) (Unit-Of-Measure (Unit* ?u1 ?u2))) (= (Unit* ?u1 ?u2) (Unit* ?u2 ?u1)) (= (Unit* ?u1 (Unit* ?u2 ?u3)) (Unit* (Unit* ?u1 ?u2) ?u3))))

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Knowledge Systems Laboratory, Stanford University5Unit^ (Deffunction Unit^ Unit^ is a mapping of reals to units that has the algebraic properties of exponentiation (and (=> (Instance-Of ?u Unit-Of-Measure) (Instance-Of ?r Real-Number) (Unit-Of-Measure (Unit^ ?u ?r))) (= (Unit^ ?u 1) ?u) (= (Unit* (Unit^ ?u ?r1) (Unit^ ?u ?r2)) (Unit^ ?u (+ ?r1 ?r2))) (= (Unit^ (Unit* ?u1 ?u2) ?r) (Unit* (Unit^ ?u1 ?r) (Unit^ ?u2 ?r))))) Example (Defobject Meter/Second (Unit* Meter (Unit^ Second –1)))

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Knowledge Systems Laboratory, Stanford University6 Magnitude of Physical Quantities Quantity-Magnitude (Deffunction Quantity-Magnitude The magnitude of a physical quantity in a given unit of measure (=> (and (Quantity-Magnitude ?q ?u ?m) (/= ?m (and (Instance-Of ?q Physical-Quantity) (Instance-Of ?u Unit-Of-Measure) (Instance-Of ?m Magnitude) (= (Quantity-Dimension ?q) (Unit-Dimension ?u))) The-Quantity (Deffunction The-Quantity The physical quantity with a given magnitude in a given unit of measure ( (The-Quantity ?m ?u ?q) (Quantity-Magnitude ?q ?u ?m)) E.g., (Debobject Real-Fast (The-Quantity Meter/Second)) (=> (Instance-Of ?q1 Physical-Quantity) (Instance-Of ?q2 Physical-Quantity) (= ?q1 ?q2) ( (= ?q1 ?q2) (and (Quantity-Dimension ?q1 (Quantity-Dimension ?q2)) (=> (Instance-Of ?u Unit-Of-Measure) (Unit-Dimension ?u (Quantity-Dimension ?q1)) (Quantity-Magnitude ?q1 ?u (Quantity-Magnitude ?q2 ?u))))))

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Knowledge Systems Laboratory, Stanford University7 Example Definitions Length-Dimension Instance-of: Physical-Dimension Standard-Unit: Meter Length-Unit-Of-Measure Subclass-Of: Unit-Of-Measure *Unit-Dimension: Length-Dimension Meter Instance-Of: Length-Unit-Of-Measure Kilometer Instance-Of: Length-Unit-Of-Measure ============================================================ (=> (and (Instance-Of ?q1 Physical-Quantity) (Quantity-Dimension ?q1 Length-Dimension)) (= (Quantity-Magnitude ?q1 Kilometer) (/ (Quantity-Magnitude ?q1 Meter) 1000)))

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