A Study of Quantities Sebastian Brandt brandt@cs.manchester.ac.uk - - PowerPoint PPT Presentation

A Study of Quantities

Sebastian Brandt brandt@cs.manchester.ac.uk

(Slides by Bijan Parsia, bparsia@cs.man.ac.uk)

Tuesday, 1 April 2014

Previous class

• OWL Datatypes

– Two worlds paradigm

• abstract logic world: classes, individuals, object properties
• concrete data world(s): numbers, strings, etc.

– Syntax for data

• datatypes, data restrictions (with facets)
• boolean connectives for datatypes, user-defined datatypes
• data properties link the logic world to the data world(s)

– Semantics for data

• interpretation domains of “data world” disjoint with “logic world”
• fixed semantics for datatypes,

matches standard semantics for integer, string, etc.

– Limitations: “admissible” datatypes – Existing reasoners can be extended to admissible datatypes

• Today

– What about units of measure? Quantities?

2

integer[>= 0 , <=150]

Tuesday, 1 April 2014

All About Quantities

Quantities are ubiquitous

• Length, time, charge, amount of money, acceleration, velocity,

interest rate...

• Measured of some sort
• Generally represented with a magnitude (a real number)

and a unit of measurement

• Implicitly, quantities have a dimension
• Dimensions are disjoint
• Quantities may be derived from other quantities
• (Except for "base" quantities)
• Quantities can be complex
• Consider a first order (plus maths) ontology
• http://www-ksl.stanford.edu/knowledge-sharing/papers/engmath.htm

3 Tuesday, 1 April 2014

One Definition

(defrelation UNIT-Of-MEASURE ;; units are scalar quantities (=> (unit-of-measure ?u) (scalar-quantity ?u)) ;; units are positive (=> (unit-of-measure ?u) (forall ?u2 (=> (and (unit-of-measure ?u2) (= (quantity.dimension ?u) (quantity.dimension ?u2))) (positive (magnitude ?u ?u2))))) ;; units can be combined using * (abelian-group unit-of-measure * identity-unit) ;; units can be combined using expt (=> (and (unit-of-measure ?u) (real-number ?r))

4 Tuesday, 1 April 2014

Formalising in OWL

• Much we can't hope to capture without extensions
• How might we try to capture with vanilla OWL?

– First thought: data properties – Yangtze Types: River – Facts: length-in-miles "3937.5"ˆˆxsd:decimal

• Advantages: Simple, flexible, extensible
• What happens with different units?

– Yangtze Facts: length-in-kilometers "6300"ˆˆxsd:decimal – Are these facts consistent? – How do we convert? – Linear (in)equations would help!

5 Tuesday, 1 April 2014

Capturing Conversion

• Equations capture the unit conversion

– SubClassOf(River – DataAllValuesFrom(length-in-miles length-in-kilometers – DataComparison(Arguments(mi km) eq( mi (* 1.609 km) ))))

• Dimensional analysis

– Are the properties disjoint? – Do we need a super-property "length"? – How about derived quantities?

• Big problem

– Horrific proliferation of strange properties

• -in-miles, -in-feet, -in-centimeters, ...
• length-, height-, depth-, circumference-

– Lots of extra stuff in the ontology – Error prone; computationally expensive

6 Tuesday, 1 April 2014

A Solution

• Datatypes!

– Exactly what Two World Philosophy called for

• Separates domain and quantity theories
• User friendly:
• Yangtze Types: River
• Facts: length Yangtze "6300 km"ˆˆowl:quantity,

length Yangtze "3937.5 miles"ˆˆowl:quantity

• Simple, direct, no clutter

– And an efficient engine works behind the scenes! – Doesn’t touch anything when it doesn’t have to – No n-ary!

7 Tuesday, 1 April 2014

More Examples

• Assuming height is functional...
• Consistent

– sheeva Facts: height "2 meters"ˆˆowl:quantity, height "200 cm"ˆˆowl:quantity

• Inconsistent

– sheeva Facts: height "2 m"ˆˆowl:quantity, – height "2 cm"ˆˆowl:quantity

• Entails that Sheeva is a TallThing

– TallThing equivalentTo: – height some owl:quantity[>= "6 feet"]

8 Tuesday, 1 April 2014

(Dis)Advantages Of A Datatype

• Advantages

– Very very user friendly – Can scale to sophisticated equations – Proper simplification and analysis – Solve the problem once – Implementation easy

• Disadvantages

– Doesn't help us to reason about quantities – Supports ontologies with quantities, not about quantities – Needs a fair bit of tool support

9 Tuesday, 1 April 2014