Improved intro
This commit is contained in:
@ -1,18 +1,82 @@
|
||||
\section{Definitions}
|
||||
\section{Introduction}
|
||||
\textit{Lambda calculus} is a model of computation, much like the Turing machine.
|
||||
As we're about to see, it works in a fundamentally different way, which has a few
|
||||
practical applications we'll discuss at the end of class.
|
||||
|
||||
\generic{$\lm$ Notation:}
|
||||
$\lm$ notation is used to define functions, and looks like $(\lm ~ \text{input} ~ . ~ \text{output})$. \par
|
||||
Consider the following statement:
|
||||
$$
|
||||
I = \lm a . a
|
||||
$$
|
||||
This tells us that $I$ is a function that takes its input, $a$, to itself. We'll call this the \textit{identity function}.
|
||||
\vspace{2mm}
|
||||
|
||||
To apply functions, put them next to their inputs. We'll omit the usual parentheses to save space.
|
||||
|
||||
A lambda function starts with a lambda ($\lm$), followed by the names of any inputs used in the expression,
|
||||
followed by the function's output.
|
||||
|
||||
For example, $\lm x . x + 3$ is the function $f(x) = x + 3$ written in lambda notation.
|
||||
|
||||
\vspace{4mm}
|
||||
Let's disect $\lm x . x + 3$ piece by piece:
|
||||
|
||||
\begin{itemize}[itemsep=3mm]
|
||||
\item \say{$\lm$} tells us that this is the beginning of an expression. \par
|
||||
$\lm$ here doesn't have a special value or definition; \par
|
||||
it's just a symbol that tells us \say{this is the start of a function.}
|
||||
|
||||
|
||||
\item \say{$\lm x$} says that the variable $x$ is \say{bound} to the function (i.e, it is used for input).\par
|
||||
Whenever we see $x$ in the function's output, we'll replace it with the input of the same name.
|
||||
|
||||
\vspace{1mm}
|
||||
|
||||
This is a lot like normal function notation: In $f(x) = x + 3$, $(x)$ is \say{bound}
|
||||
to $f$, and we replace every $x$ we see with our input when evaluating.
|
||||
|
||||
|
||||
|
||||
\item The dot tells us that what follows is the output of this expression. \par
|
||||
This is much like $=$ in our usual function notation: \par
|
||||
The symbols after $=$ in $f(x) = x + 3$ tell us how to compute the output of this function.
|
||||
\end{itemize}
|
||||
|
||||
\problem{}
|
||||
Rewrite the following functions using this notation:
|
||||
\begin{itemize}
|
||||
\item $f(x) = 7x + 4$
|
||||
\item $f(x) = x^2 + 2x + 1$
|
||||
\end{itemize}
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
To evaluate $\lm x . x + 3$, we need to input a value:
|
||||
$$
|
||||
(I~\star) =
|
||||
(\lm \tzm{b}a. a)~\tzmr{a}\star =
|
||||
\star
|
||||
(\lm x . x + 3)~5
|
||||
$$
|
||||
This is very similar to the usual way we call functions: we usually write $f(5)$. \par
|
||||
Above, we define our function $f$ \say{in-line} using lambda notation, \par
|
||||
and we ommit the parentheses around 5 for the sake of simpler notation.
|
||||
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
|
||||
We evaluate this by removing the \say{$\lm$} prefix and substituting $3$ for $x$ wheverever it appears:
|
||||
$$
|
||||
(\lm x . \tzm{b}x + 3)~\tzmr{a}5 =
|
||||
5 + 3 = 8
|
||||
\begin{tikzpicture}[
|
||||
overlay,
|
||||
remember picture,
|
||||
@ -24,10 +88,89 @@ $$
|
||||
(a.center) to (b.center);
|
||||
\end{tikzpicture}
|
||||
$$
|
||||
Functions are left-associative: If $A$ and $B$ are functions, $(A~B~\star)$ is equivalent to $((A~B)~\star)$.
|
||||
As usual, we'll use parentheses to group terms if we want to override this order: $(A~(B~\star)) \neq (A~B~\star)$ \par
|
||||
In this handout, all types of parentheses ( $(), [~],$ etc ) are equivalent.
|
||||
|
||||
\problem{}
|
||||
Evaluate the following:
|
||||
\begin{itemize}[itemsep = 2mm]
|
||||
\item $(\lm x. 2x + 1)~4$
|
||||
\item $(\lm x. x^2 + 2x + 1)~3$
|
||||
\item $(\lm x. (\lm y. 9y) x + 3)~2$ \par
|
||||
\hint{
|
||||
This function has a function inside, but the evaluation process doesn't change. \\
|
||||
Replace all $x$ with 2 and evaluate again.
|
||||
}
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\vfill
|
||||
|
||||
As we saw above, we denote function application by simply putting functions next to their inputs. \par
|
||||
If we want to apply $f$ to $5$, we write \say{$f~5$}, without any parentheses around the function's argument.
|
||||
|
||||
\vspace{4mm}
|
||||
|
||||
You may have noticed that we've been using arithmetic in the last few problems.
|
||||
This isn't fully correct: addition is not defined in lambda calculus. In fact, nothing is defined:
|
||||
not even numbers!
|
||||
|
||||
In lambda calculus, we have only one kind of object: the function.
|
||||
The only action we have is function application, which works by just like the examples above.
|
||||
|
||||
\vspace{2mm}
|
||||
|
||||
Don't worry if this sounds confusing, we'll see a few examples soon.
|
||||
|
||||
\pagebreak
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\definition{}
|
||||
The first \say{pure} functions we'll define are $I$ and $M$:
|
||||
\begin{itemize}
|
||||
\item $I = \lm x . x$
|
||||
\item $M = \lm x . x x$
|
||||
\end{itemize}
|
||||
|
||||
Note that $I$ and $M$ don't have a meaning on their own. They are not formal functions. \par
|
||||
Rather, it's notation that says \say{write $\lm x.x$ whenever you see $I$.}
|
||||
|
||||
|
||||
|
||||
|
||||
\problem{}
|
||||
Reduce the following expressions.
|
||||
\begin{itemize}[itemsep=2mm]
|
||||
\item $I~I$
|
||||
\item $(I~I)~I$
|
||||
\item $\Bigl(~ \lm a .(a~(a~a)) ~\Bigr) ~ I$
|
||||
\item $\Bigl(~(\lm a . (\lm b . a)) ~ M~\Bigr) ~ I$
|
||||
\end{itemize}
|
||||
|
||||
|
||||
\vfill
|
||||
|
||||
In lambda calculus, functions are left-associative: \par
|
||||
$(f~g~h)$ is equivalent to $((f~g)~h)$, not $(f~(g~h))$
|
||||
|
||||
As usual, we use parentheses to group terms if we want to override this order: $(f~(g~h)) \neq ((f~g)~h)$ \par
|
||||
In this handout, all types of parentheses ( $(), [~],$ etc ) are equivalent.
|
||||
|
||||
\problem{}
|
||||
Rewrite the following expressions with as few parentheses as possible, without changing their meaning or structure.
|
||||
@ -53,63 +196,6 @@ Remember that lambda calculus is left-associative.
|
||||
|
||||
|
||||
|
||||
\generic{$\beta$-Reduction:}
|
||||
|
||||
$\beta$-reduction is a fancy name for \say{simplifying an expression.} We've already done it once above,
|
||||
while evaluating $(I \star)$. Let's make another function:
|
||||
$$
|
||||
M = \lm f . f f
|
||||
$$
|
||||
The function $M$ simply repeats its input. What is $(M~I)$?
|
||||
$$
|
||||
(M~I) =
|
||||
((\lm \tzm{b}f.f f)~\tzmr{a}I) =
|
||||
(I~I) =
|
||||
((\lm \tzm{d}a.a)~\tzmr{c}I) =
|
||||
I
|
||||
\begin{tikzpicture}[
|
||||
overlay,
|
||||
remember picture,
|
||||
out=225,
|
||||
in=315,
|
||||
distance=0.5cm
|
||||
]
|
||||
\draw[->,gray,shorten >=5pt,shorten <=3pt]
|
||||
(a.center) to (b.center);
|
||||
\draw[->,gray,shorten >=5pt,shorten <=3pt]
|
||||
(c.center) to (d.center);
|
||||
\end{tikzpicture}
|
||||
$$
|
||||
We cannot go any further, so we stop. Our expression is now in \textit{$\beta$-normal} or \say{reduced} form.
|
||||
|
||||
|
||||
|
||||
\problem{}
|
||||
Reduce the following expressions:
|
||||
\begin{itemize}
|
||||
\item $I~I$
|
||||
\item $I~I~I$
|
||||
\item $(\lm a .(a~a~a)) ~ I$
|
||||
\item $(\lm a . (\lm b . a)) ~ M ~ I$
|
||||
\end{itemize}
|
||||
|
||||
\vfill
|
||||
\pagebreak
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
\generic{Currying:}
|
||||
|
||||
|
||||
Reference in New Issue
Block a user