mirror of
https://github.com/achlipala/frap.git
synced 2024-11-27 23:06:20 +00:00
Interpreter chapter: expressions and substitution
This commit is contained in:
parent
c8ff080a20
commit
2134aa2477
1 changed files with 92 additions and 1 deletions
93
frap.tex
93
frap.tex
|
@ -1,6 +1,6 @@
|
|||
\documentclass{amsbook}
|
||||
|
||||
\usepackage{hyperref,url,amsmath,proof}
|
||||
\usepackage{hyperref,url,amsmath,proof,stmaryrd,tikz-cd}
|
||||
|
||||
\newtheorem{theorem}{Theorem}[chapter]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
|
@ -463,9 +463,100 @@ The general patterns should soon become clear, as they are somehow already famil
|
|||
\end{quote}
|
||||
The quoted remark could just as well be in Spanish instead of English, in which case we have two languages nested in a nontrivial way.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\chapter{Semantics via Interpreters}
|
||||
|
||||
That's enough about what programs \emph{look like}.
|
||||
Let's shift our attention to what programs \emph{mean}.
|
||||
|
||||
\section{Semantics for Arithmetic Expressions via Finite Maps}
|
||||
|
||||
\newcommand{\mempty}[0]{\bullet}
|
||||
\newcommand{\msel}[2]{#1(#2)}
|
||||
\newcommand{\mupd}[3]{#1[#2 \mapsto #3]}
|
||||
|
||||
To explain the meaning of one of last chapter's arithmetic expressions, we need a way to indicate the value of each variable.
|
||||
A theory of \emph{finite maps}\index{finite map} is helpful here.
|
||||
We apply the following notations throughout the book: \\
|
||||
|
||||
\begin{tabular}{rl}
|
||||
$\mempty$ & empty map, with $\emptyset$ as its domain \\
|
||||
$\msel{m}{k}$ & mapping of key $k$ in map $m$ \\
|
||||
$\mupd{m}{k}{v}$ & extension of map $m$ to also map key $k$ to value $v$
|
||||
\end{tabular} \\
|
||||
|
||||
As the name advertises, finite maps are functions with finite domains, where the domain may be expanded by each extension operation.
|
||||
Two axioms explain the essential interactions of the basic operators.
|
||||
|
||||
$$\infer{\msel{\mupd{m}{k}{v}}{k} = v}{}
|
||||
\quad
|
||||
\infer{\msel{\mupd{m}{k_1}{v}}{k_2} = m(k_2)}{
|
||||
k_1 \neq k_2
|
||||
}$$
|
||||
|
||||
\newcommand{\denote}[1]{{\left \llbracket #1 \right \rrbracket}}
|
||||
|
||||
With these operators in hand, we can write a semantics for arithmetic expressions.
|
||||
This is a recursive function that \emph{maps variable valuations to numbers}.
|
||||
We write $\denote{e}$ for the meaning of $e$; this notation is often referred to as \emph{Oxford brackets}\index{Oxford brackets}.
|
||||
Recall that we allow notations like this as syntactic sugar for arbitrary functions, even when giving the equations that define those functions.
|
||||
We write $v$ for a valuation (finite map).
|
||||
\begin{eqnarray*}
|
||||
\denote{n}v &=& n \\
|
||||
\denote{x}v &=& v(x) \\
|
||||
\denote{e_1 + e_2}v &=& \denote{e_1}v + \denote{e_2}v \\
|
||||
\denote{e_1 - e_2}v &=& \denote{e_1}v - \denote{e_2}v \\
|
||||
\denote{e_1 \times e_2}v &=& \denote{e_1}v \times \denote{e_2}v
|
||||
\end{eqnarray*}
|
||||
|
||||
Note how parts of the definition feel a little bit like cheating, as we just ``push notations inside the brackets.''
|
||||
It's important to remember that plus \emph{inside} the brackets is syntax, while plus \emph{outside} the brackets is the normal addition of math!
|
||||
|
||||
\newcommand{\subst}[3]{[#3/#2]#1}
|
||||
|
||||
To test our semantics, we define a \emph{variable substitution} function\index{substitution}.
|
||||
A substitution $\subst{e}{x}{e'}$ stands for the result of running through the syntax of $e$, replacing every occurrence of variable $x$ with expression $e'$.
|
||||
\begin{eqnarray*}
|
||||
\subst{n}{x}{e} &=& n \\
|
||||
\subst{x}{x}{e} &=& e \\
|
||||
\subst{y}{x}{e} &=& y \textrm{, when $y \neq x$} \\
|
||||
\subst{(e_1 + e_2)}{x}{e} &=& \subst{e_1}{x}{e} + \subst{e_2}{x}{e} \\
|
||||
\subst{(e_1 - e_2)}{x}{e} &=& \subst{e_1}{x}{e} - \subst{e_2}{x}{e} \\
|
||||
\subst{(e_1 \times e_2)}{x}{e} &=& \subst{e_1}{x}{e} \times \subst{e_2}{x}{e}
|
||||
\end{eqnarray*}
|
||||
|
||||
We can prove a key compatibility property of these two recursive functions.
|
||||
|
||||
\begin{theorem}
|
||||
For all $e$, $e'$, $x$, and $v$, $\denote{\subst{e}{x}{e'}}{v} = \denote{e}{(\mupd{v}{x}{\denote{e'}{v}})}$.
|
||||
\end{theorem}
|
||||
|
||||
That is, in some sense, the operations of interpretation and substitution \emph{commute} with each other.
|
||||
That intuition gives rise to the common notion of a \emph{commuting diagram}\index{commuting diagram}, like the one below for this particular example.
|
||||
|
||||
\[
|
||||
\begin{tikzcd}
|
||||
(e, v) \arrow{r}{\subst{\ldots}{x}{e'}} \arrow{d}{\mupd{\ldots}{x}{\denote{e'}v}} & (\subst{e}{x}{e'}, v) \arrow{d}{\denote{\ldots}} \\
|
||||
(e, \mupd{v}{x}{\denote{e'}v}) \arrow{r}{\denote{\ldots}} & \denote{\subst{e}{x}{e'}}v
|
||||
\end{tikzcd}
|
||||
\]
|
||||
|
||||
We start at the top left, with a given expresson $e$ and valuation $v$.
|
||||
The diagram shows the equivalence of \emph{two different paths} to the bottom right.
|
||||
Each individual arrow is labeled with some description of the transformation it performs, to get from the term at its source to the term at its destination.
|
||||
The right-then-down path is based on substituting and then interpreting, while the down-then-right path is based on extending the valuation and then interpreting.
|
||||
Since both paths wind up at the same spot, the diagram indicates an equality between the corresponding terms.
|
||||
|
||||
It's a matter of taste whether the theorem statement or the diagram expresses the property more clearly!
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\appendix
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\chapter{The Coq Proof Assistant}
|
||||
|
||||
Coq\index{Coq} is a proof-assistant software package developed as open source, primarily by Inria\index{Inria}, the French national computer-science lab.
|
||||
|
|
Loading…
Reference in a new issue