## QUDT Ontologies Overview

The QUDT Specification is more than a list of quantities, units, dimensions, data types, enumerations, and structures. In order to provide for interoperability and data exchange between information systems, the specification needs to be available in a machine processable form, with no ambiguities. For these reasons, the QUDT approach to specifying quantities, units, dimensions, data types, enumerations, and other data structures is to use precise semantically grounded specifications in an ontology model with translation into machine-processable representations.

Ontologies provide the object-oriented strengths of encapsulation, inheritance, and polymorphism, strengths which are unavailable in other structured modeling approaches. The characteristics modeled in QUDT require a model-based approach because they are functionally dependent. Modeling one without modeling its dependency on the other requires that the understanding of those dependencies be carried by the observer, which injects ambiguity into the modeling approach. These models (dimensions, coordinate systems, etc.), like everything else, are hierarchical, so using a language to model them which doesn't support inheritance imposes constraints on the models and their use which, again, results in ambiguity.

QUDT supports system interoperability in four ways:

1. The unit ontologies provide a formal way of specifying units explicitly, thereby avoiding tacit conventions that are prone to misinterpretation.
2. QUDT distinguishes between variants of a given unit. For example, the English word "day" interpreted as a unit of measure may refer to a mean solar day, a sidereal day, or the length of time equivalent to exactly 86,400 seconds. Each of these interpretations of "day" appears as a distinct unit in the ontology.
3. QUDT distinguishes between units of different types that are commonly referred to with the same name. For example, "second" may refer to a measure of time or a measure of angle. Again, each usage appears as a distinct term in the ontology.
4. The ontology provides explicit conversion information, serving as a single point of reference for such conversions.

QUDT is based largely on the international standard for metric units (SI), as described in "BIPM International System of Units", the "ISO standards on Units and Quantities", and "The NIST Guide for the use of the International System of Units". In addition, QUDT includes units from other systems, such as CGS units for mechanics, CGS EMU (electromagnetic) units, CGS-ESU (electrostatic) units, and Gaussian units for electrodynamics, and the Planck system of natural units. Most US Customary and British Imperial units for length, weight, and heat are also included.

Wherever applicable, the SI standard is used for conversions between non-SI units. To convert from unit U1 to U2, one first converts U1 to SI (the equivalent value in the appropriate SI unit), then converts SI to U2. In the QUDT Ontologies, each unit has a corresponding conversion multiplier, which multiplied to quantities to convert from the current unit to the corresponding SI unit. So, if N1 and N2 are the conversion multipliers for U1 and U2 respectively, then the proper factor to convert from U1 to U2 is N1/N2. Unit conversion data was largely derived from the values given by the National Institute of Standards and Technology (NIST) for fundamental constants, as documented in "The NIST Reference on Constants, Units, and Uncertainty".

QUDT semantics are based on dimensional analysis expressed in the OWL Web Ontology Language (OWL). The dimensional approach relates each unit to a system of base units using numeric factors and a vector of exponents defined over a set of fundamental dimensions. In this way, the role of each base unit in the derived unit is precisely defined. A further relationship establishes the semantics of units and quantity kinds. By this means, QUDT supports reasoning over quantities as well as units. QUDT models may be translated into other representations for machine processing, or other programming language structures according to need. The following sections briefly define the primary objects of interest in the QUDT ontology and their relevance to the formal specification of quantities, units, dimensions and data types.

### 1.1: Ontology Architecture

QUDT ontologies are organized as collections of different types of graphs, as listed in the QUDT catalog. Vocabulary graphs hold different domains of quantities and units, which import the appropriate QUDT schemas. The core schema of QUDT imports the VAEM, DTYPE and SKOS ontologies.

### 1.2: Quantity Kind

A Quantity Kind is any observable property that can be measured and quantified numerically. Familiar examples include physical properties such as length, mass, time, force, energy, power, electric charge, etc. Less familiar examples include currency, interest rate, price to earning ratio, and information capacity.

qudt:belongsToSystemOfQuantities
qudt:hasQuantityKind
qudt:applicableUnit
qudt:belongsToSystemOfQuantities
qudt:hasDimensionVector
qudt:hasQuantityKind
qudt:hasQuantityKind

### 1.3: Quantity

A quantity is the measurement of an observable property of a particular object, event, or physical system. A quantity is always associated with the context of measurement (i.e. the thing measured, the measured value, the accuracy of measurement, etc.) whereas the underlying quantity kind is independent of any particular measurement. Thus, length is a quantity kind while the height of a rocket is a specific quantity of length; its magnitude that may be expressed in meters, feet, inches, etc. Examples of physical quantities include physical constants, such as the speed of light in a vacuum, Planck's constant, the electric permittivity of free space, and the fine structure constant.

In other words, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, momentum, energy, and weight, and units are used to describe their measure. Many of which are related to each other by various physical laws, and as a result the units of some of the quantities can be expressed as products (or ratios) of powers of other units (e.g., momentum is mass times velocity and velocity is measured in distance divided by time). These relationships are discussed in dimensional analysis. Those that cannot be so expressed can be regarded as "fundamental" in this sense.

A quantity is distinguished from a "quantity kind" in that the former carries a value and the latter is a type specifier.

qudt:belongsToSystemOfQuantities
qudt:belongsToSystemOfQuantities
qudt:hasQuantityKind
qudt:quantityValue
qudt:hasQuantityKind
qudt:hasQuantityKind

### 1.4: Quantity value

A Quantity Value expresses the magnitude and kind of a quantity and is given by the product of a numerical value n and a unit of measure U. The number multiplying the unit is referred to as the numerical value of the quantity expressed in that unit. Refer to NIST SP 811 section 7 for more on quantity values.

qudt:applicableUnit
qudt:hasQuantityKind
qudt:quantityValue
qudt:hasQuantityKind
qudt:hasQuantity
qudt:hasQuantityKind
qudt:unit

### 1.5: Quantity Dimension Vector

A Quantity Dimension Vector is a relationship between a quantity system, a quantity kind of that system, and one or more dimension vectors. The dimensions of a quantity are expressed as a product of the basic physical dimensions mass ($$M$$), length ($$L$$), time ($$T$$) current ($$I$$), amount of substance ($$N$$), luminous intensity ($$J$$) and absolute temperature ($$\theta$$) as $$dim \, Q = L^{\alpha} \, M^{\beta} \, T^{\gamma} \, I ^{\delta} \, \theta ^{\epsilon} \, N^{\eta} \, J ^{\nu}$$.

The rational powers of the dimensional exponents, $$\alpha, \, \beta, \, \gamma, \, \delta, \, \epsilon, \, \eta, \, \nu$$, are positive, negative, or zero.

For example, the dimension of the physical quantity $$\it{speed}$$ is $$\boxed{length/time}$$, $$L/T$$ or $$LT^{-1}$$, and the dimension of the physical quantity force is $$\boxed{mass \times acceleration}$$ or $$\boxed{mass \times (length/time)/time}$$, $$ML/T^2$$ or $$MLT^{-2}$$ respectively.

### 1.6: Unit

A unit of measure, or unit, is a particular quantity value that has been chosen as a scale for measuring other quantities the same kind (more generally of equivalent dimension). For example, the meter is a quantity of length that has been rigorously defined and standardized by the BIPM (International Board of Weights and Measures). Any measurement of the length can be expressed as a number multiplied by the unit meter. More formally, the value of a physical quantity Q with respect to a unit (U) is expressed as the scalar multiple of a real number (n) and U, as $$Q = nU$$.