177 lines
8.1 KiB
TeX
177 lines
8.1 KiB
TeX
\documentclass[11pt,a4paper]{book}
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\usepackage{isabelle,isabellesym}
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\usepackage{graphicx}
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\graphicspath {{figures/}}
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% further packages required for unusual symbols (see also
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% isabellesym.sty), use only when needed
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\usepackage{latexsym}
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\usepackage{amssymb}
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%for \<leadsto>, \<box>, \<diamond>, \<sqsupset>, \<mho>, \<Join>,
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%\<lhd>, \<lesssim>, \<greatersim>, \<lessapprox>, \<greaterapprox>,
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%\<triangleq>, \<yen>, \<lozenge>
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%\usepackage[greek,english]{babel}
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%option greek for \<euro>
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%option english (default language) for \<guillemotleft>, \<guillemotright>
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%\usepackage[latin1]{inputenc}
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%for \<onesuperior>, \<onequarter>, \<twosuperior>, \<onehalf>,
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%\<threesuperior>, \<threequarters>, \<degree>
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%\usepackage[only,bigsqcap]{stmaryrd}
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%for \<Sqinter>
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%\usepackage{eufrak}
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%for \<AA> ... \<ZZ>, \<aa> ... \<zz> (also included in amssymb)
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%\usepackage{textcomp}
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%for \<cent>, \<currency>
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% this should be the last package used
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\usepackage{pdfsetup}
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% urls in roman style, theory text in math-similar italics
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\urlstyle{rm}
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\isabellestyle{it}
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\newcommand{\HOL}[1]{\verb{HOL}}
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\newcommand{\eg}[1]{e.g.}
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\renewcommand{\isasymdegree}{XXX}
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\begin{document}
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\title{Type and Proof Support for SI Units}
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\author{ Simon Foster \and Burkhart Wolff}
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\maketitle
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\textbf{ Abstract }
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The International System of Units
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(SI, abbreviated from the French Syst\`eme international (d’unit\'es)) is the modern form of
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the metric system and is the most widely used system of measurement. It comprises a coherent
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system of units of measurement built on seven base units, which are the second, metre, kilogram,
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ampere, kelvin, mole, candela, and a set of twenty prefixes to the unit names and unit symbols
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that may be used when specifying multiples and fractions of the units. The system also specifies
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names for 22 derived units, such as lumen and watt, for other common physical quantities.
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This theory represents a formal model of SI Units together with a deep integration in Isabelle's
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type system: unitswere represented in a way that they have a \emph{unit type} comprising its
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\emph{magnitude type} and its physical \emph{dimension}. Congruences on dimensions were supported.
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Our construction is validated by a test-set of known equivalences between SI Units.
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Moreover, the presented theory can be used for type-safe conversions between the SI system and
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others, like the British Imperial System (BIS).
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\tableofcontents
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% sane default for proof documents
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\parindent 0pt\parskip 0.5ex
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\chapter{SI Units in Isabelle \\ An Introduction}
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The International System of Units (SI, abbreviated from the French
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Système international (d'unités)) is the modern form of the metric
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system and is the most widely used system of measurement. It comprises
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a coherent system of units of measurement built on seven base units,
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which are the second, metre, kilogram, ampere, kelvin, mole, candela,
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and a set of twenty prefixes to the unit names and unit symbols that
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may be used when specifying multiples and fractions of the units.
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The system also specifies names for 22 derived units, such as lumen and
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watt, for other common physical quantities.
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(cited from \url{https://en.wikipedia.org/wiki/International_System_of_Units}).
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In more detail, the SI provides the following fundamental concepts:
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%
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\begin{enumerate}%
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\item \emph{quantities}, i.e. \emph{time}, \emph{length}, \emph{mass}, \emph{electric current},
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\emph{temperature}, \emph{amount of substance},\emph{luminous intensity},
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and other derived quantities such as \emph{volume};
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\item \emph{dimensions}, i.e. a set of the symbols \isa{T}, \isa{L}, \isa{M}, \isa{I}, \isa{{\isasymTheta}}, \isa{N}, \isa{J} corresponding
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to the above mentioned base quantities, indexed by an integer exponent
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(dimensions were also called \emph{base unit names} or just \emph{base units});
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\item \emph{magnitudes}, i.e. a factor or \emph{prefix}
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(typically integers, reals, vectors on real or complex numbers);
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\item \emph{units}, which are basically pairs of magnitudes and dimensions denoting quantities.
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\end{enumerate}
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Note that \emph{quantities} are understood as idealized \emph{physical concepts}, and are sharply
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distinguished from \emph{units} representing \emph{a means to measure} them. The difference is
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roughly similar between words (or: symbols) and notions. This distinction reflects the fact that
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these concepts have been at times heavily questioned in the history of physics; the interested
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reader is referred to the debate around the "relativistic mass" at in the twenties of the last century.
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The purpose of the presented theory is to model SI units with polymorphic magnitudes inside the
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Isabelle/HOL\cite{nipkow.ea:isabelle:2002} system. The objective of our construction is to
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reflect the types of the magnitudes as well as their dimensions in order to allow type-safe
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calculations on SI units.
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As a result, it is possible to express "4500.0 kilogram times meter per second square" which can
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have the type
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\isa{{\isasymreal}\ {\isacharbrackleft}M\ \isactrlsup {\isachardot}\ L\ \isactrlsup
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{\isachardot}\ T\isactrlsup {\isacharminus}\isactrlsup {\isadigit{3}}
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\isactrlsup {\isachardot}\ T\isactrlsup {\isadigit{1}}{\isacharbrackright}}.
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For units of this type we can infer that this corresponds to the derived unit "4.5 kN" (kilo-Newton)
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of type \isa{{\isasymreal}\ {\isacharbrackleft}M\ \isactrlsup {\isachardot}\ L\ \isactrlsup
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{\isachardot}\ T\isactrlsup {\isacharminus}\isactrlsup {\isadigit{2}}{\isacharbrackright}}.
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This is an attempt to model the standard SI system and its derived entities (cf.
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\url{https://www.quora.com/What-are-examples-of-SI-units}),
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both from a type-checking as well as a proof-support perspective.
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These design objectives are for the case of Isabelle system somewhat contradictory.
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Since the Isabelle type system follows the Curry-style paradigm, which can be
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characterized by: "be as implicit as possible, inference as automatic as possible", it is not
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possible to do computations on type-terms (in contrast to, for example, Coq). We therefore
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need a more involved construction using Isabelle's type-classes to establish a semantic
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interpretation on certain classes of types. This paves the way to derive rules that
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establish and exploit type isomorphisms implicitely. For example, the implicit
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type isomorhisms for the dimensions $T^{-2} * T$ and $T^{-1}$ is dealt with appropriate
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rules on terms and a special form of equivalence.
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Our construction proceeds in three phases:
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\begin{enumerate}%
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\item We construct a type \isa{Dimension} which is basically a "semantic representation" or
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"semantic domain" of all SI dimensions. Since SI-types have an interpretation in this domain,
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it serves to give semantics to type-constructors by operations on this domain, too.
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We construct a multiplicative group on it.
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\item From \isa{Dimension} we build a language of type-constructors of \isa{dimS}-types,
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captured in a type class \isa{dim-types} giving it a pseudo-inductive structure. Types of
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this class are required to have an interpretation function into \isa{Dimension}'s, which
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allows for establishing equivalences on \isa{dim-types}.
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\item We construct a SI Unit-type as a pair of a polymorphic magnitude and a dimension type.
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This type will be inhabited by all basic SI units and predicates expressing their relationship.
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\end{enumerate}%
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On this basis, an algebra of SI units can be derived; while types can not be "touched" inside
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an Isabelle/HOL logic, it is possible to transform unit types by applying rules of this algebra.
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Tactical support over this algebra is provided enabling the construction of normal forms on
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both units and their type.
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\subsubsection{Previous Attempts.} The work of \cite{HayesBrendan95} represents to our knowledge a
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first attempt to formalize SI units in Z, thus a similar language of HOL. While our typing
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representation is more rigourous due to the use of type-classes, this works lacks any attempt
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to support formal and automated deduction on Si unit equivalences.
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MORE TO COME.
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\input{session}
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% optional bibliography
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\bibliographystyle{abbrv}
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\bibliography{adb-long,root}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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