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Most of the commonly used programming languages use \textit{eager evaluation}, whereas functional programming languages tend to prefer \textit{lazy evaluation} +Tasks typically involve doing the compiler / interpreter's work, evaluating a given statement. +This can either be full haskell code, or in the mini-Haskell, often with lambda functions instead of pattern matching and recursion, +as that \textit{tends} to be more challenging due to the possibility of losing track of what you expanded or not. +An easy way around that is to use different colour highlighters. + + \paragraph{Lazy Evaluation} In Lazy Evaluation, expressions are substituted until all possible values are substituted and the expression is then evaluated. -At the exam, it is likely that one such task will appear. +\subparagraph{Lambda functions} +For expressions of form \texttt{t1 t2}, \texttt{t1} is evaluated by substituting every occurrence of the function arguments by \texttt{t2}, without evaluation \texttt{t2}, +e.g. \verb|(\x -> x y) (\x -> x)|, the evaluation will lead to \verb|(\x -> x) y| +Below a more complicated example, highlighted with colours for you to track what goes where: + +\begin{enumerate} + \item \texttt{{\color{red} ($\backslash$x ->} {\color{ForestGreen} x ($\backslash$y -> x y)}{\color{red})} {\color{orange} ($\backslash$x -> ($\backslash$y -> y) x)}} + \item Substitute \texttt{t2} into \texttt{t1} (no eval): + \texttt{{\color{orange} ($\backslash$x -> ($\backslash$y -> y) x)} {\color{ForestGreen}($\backslash$y -> {\color{orange} ($\backslash$x -> ($\backslash$y -> y) x)} y)}} + \item Recolour for next changes: + \texttt{{\color{orange} ($\backslash$x -> {\color{purple}($\backslash$y -> y)} x)} {\color{ForestGreen}($\backslash$y -> + {\color{Aquamarine} ($\backslash$x -> ($\backslash$y -> y) x)} y)}} + \item Substitute again: + \texttt{{\color{purple}($\backslash$y -> y)} {\color{ForestGreen}($\backslash$y -> + {\color{Aquamarine} ($\backslash$x -> ($\backslash$y -> y) x)} y)}} + \item And again: + \texttt{{\color{ForestGreen}($\backslash$y -> + {\color{Aquamarine} ($\backslash$x -> ($\backslash$y -> y) x)} y)}} +\end{enumerate} +Now, evaluation stops, because there is nothing left to apply. In some cases, this can go on until you get a final result, but \textit{not always}. +This evaluation example also makes it evident why using colours can be very handy. + +\subparagraph{Pattern Matching} +Pattern matching is fairly straight forward, however, conditional evaluation requires evaluation to the smallest extent possible to determine the option to pick from. + \paragraph{Eager Evaluation} +For expressions of form \texttt{t1 t2}, \texttt{t1} is evaluated by substituting every occurrence of the function arguments by an \textit{evaluated} \texttt{t2}, +i.e. \texttt{t2} is evaluated \textit{before} substituting it. +Coming back to the short example from above \verb|(\x -> x y) (\x -> x)| is evaluated to the same \verb|(\x -> x) y|. +The difference only becomes evident with a more complicated example. We'll use the same example again as above, again highlighted. +\begin{enumerate} + \item \texttt{{\color{red} ($\backslash$x ->} {\color{ForestGreen} x ($\backslash$y -> x y)}{\color{red})} {\color{orange} ($\backslash$x -> ($\backslash$y -> y) x)}} + \item Evaluate {\color{orange} orange parts} (now {\color{purple} purple}): + \texttt{{\color{red} ($\backslash$x ->} {\color{ForestGreen} x ($\backslash$y -> x y)}{\color{red})} {\color{purple} ($\backslash$x -> x)}} + \item Substitute {\color{purple} purple}: + \texttt{{\color{ForestGreen} {\color{purple} ($\backslash$x -> x)} ($\backslash$y -> {\color{purple} ($\backslash$x -> x)} y)}} + \item Evaluate {\color{ForestGreen} green} parts (now {\color{cyan} cyan}): + \texttt{{\color{purple} ($\backslash$x -> x)} {\color{cyan}($\backslash$y -> y)}} + \item Substitute {\color{cyan} cyan}: + \texttt{{\color{cyan}($\backslash$y -> y)}} +\end{enumerate} +As you can clearly see, both strategies don't result in the same evaluation. diff --git a/semester4/fmfp/parts/06_exercises/01_fp/01_haskell.tex b/semester4/fmfp/parts/06_exercises/01_fp/01_haskell.tex index e69de29..f5dea82 100644 --- a/semester4/fmfp/parts/06_exercises/01_fp/01_haskell.tex +++ b/semester4/fmfp/parts/06_exercises/01_fp/01_haskell.tex @@ -0,0 +1,35 @@ +\subsubsection{Haskell} +\begin{examdetails} + typically either short coding task or proof of program +\end{examdetails} + +\paragraph{Programming} +The best tip here is to read the Haskell book, and to solve the exercises during the semester. + +\subparagraph{Fold} +One of the most important functions to understand is \texttt{foldr} (and \texttt{foldl}). +If you have used \texttt{reduce} functions before, in e.g. JavaScript / TypeScript, +they are similar to a \texttt{map} combined with a \texttt{reduce}. + +For instance, in TypeScript, the \texttt{reduce} function has the type signature +\mint{typescript}|array.reduce( ( accumulator: A, current: A, idx: number, array: A[] ) => A, initialValue: A )| + +In the Haskell prelude, they are defined as follows: +\begin{code}{haskell} + foldr :: (a -> b -> b) -> b -> [a] -> b + foldr f z [] = z + foldr f z (x:xs) = f x (foldr f z xs) + + foldl :: (a -> b -> a) -> a -> [b] -> a + foldl f z [] = z + foldl f z (x:xs) = foldl f (f v x) xs +\end{code} + +\subparagraph{zipWith} +To combine a \texttt{map} and a \texttt{zip} function, use \texttt{zipWith}, type: +\mint{haskell}|zipWith :: (a -> b -> c) -> [a] -> [b] -> [c]| + +\TODO Add more remarks + +\paragraph{Proofs} +These proofs use structural induction, see Section~\ref{sec:induction-proofs} for that. diff --git a/semester4/fmfp/parts/06_exercises/01_fp/02_natural-deduction.tex b/semester4/fmfp/parts/06_exercises/01_fp/02_natural-deduction.tex index e69de29..83a9e23 100644 --- a/semester4/fmfp/parts/06_exercises/01_fp/02_natural-deduction.tex +++ b/semester4/fmfp/parts/06_exercises/01_fp/02_natural-deduction.tex @@ -0,0 +1,52 @@ +\subsubsection{Natural Deduction} +\paragraph{Parenthesis} +This task (if it were to even ever appear in the exams) is simply applying precedences, as well as remembering associativity. +The precedences are as follows: +\[ + \neg > \land > \lor > \rightarrow +\] +The associativities are right for $\land$ and $\lor$, whereas $\rightarrow$ is left associative. This means that +\[ + A \rightarrow B \rightarrow C \text{ is parenthesized as } A \rightarrow (B \rightarrow C) +\] + + +\paragraph{Derivation trees} +The most important tip here is to write as little as possible, by defining variables (such as $\Gamma$, $\Gamma'$, etc) for each step, +so you can simply write that instead of all the content. + + +\paragraph{Defining rules} +This is often the hardest type of task. First state the obvious cases, the basic introduction and elimination rules according to the given definition. +Remember that the rules are defined top-down, so, for introducing $A \leftrightarrow B$, defined as $(A \rightarrow B) \land (B \rightarrow A)$, we define the rules as follows: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash (A \rightarrow B) \land (B \rightarrow A)} + \infer1[$\leftrightarrow$-intro]{\Gamma \vdash A \leftrightarrow B} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash A \leftrightarrow B} + \infer1[$\leftrightarrow$-elim]{\Gamma \vdash (A \rightarrow B) \land (B \rightarrow A)} + \end{prooftree} +\] +The hard part is coming up with useful extra rules. Here, and with all rules defined by an AND, we can also define it slightly differently: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash A \rightarrow B} + \hypo{\Gamma \vdash B \rightarrow A} + \infer2[$\leftrightarrow$-I]{\Gamma \vdash A \leftrightarrow B} + \end{prooftree} +\] +Similarly, we can also provide elimination rules for the left or right: +\[ + \begin{prooftree} + \hypo{\Gamma \vdash A \leftrightarrow B} + \infer1[$\leftrightarrow$-EL]{\Gamma \vdash (A \rightarrow B)} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash A \leftrightarrow B} + \infer1[$\leftrightarrow$-ER]{\Gamma \vdash (B \rightarrow A)} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/06_exercises/01_fp/03_type-inference.tex b/semester4/fmfp/parts/06_exercises/01_fp/03_type-inference.tex index e69de29..d2614fa 100644 --- a/semester4/fmfp/parts/06_exercises/01_fp/03_type-inference.tex +++ b/semester4/fmfp/parts/06_exercises/01_fp/03_type-inference.tex @@ -0,0 +1,4 @@ +\subsubsection{Type Inference} +\begin{enumerate} + \item +\end{enumerate} diff --git a/semester4/fmfp/parts/06_exercises/03_checklist.tex b/semester4/fmfp/parts/06_exercises/03_checklist.tex new file mode 100644 index 0000000..6c9f699 --- /dev/null +++ b/semester4/fmfp/parts/06_exercises/03_checklist.tex @@ -0,0 +1,7 @@ +\subsection{Checklist} +\subsubsection{Learning} + +\subsubsection{For the exam} +\begin{itemize}[label=$\square$] + \item Multiple colours of highlighters (for eval tasks) +\end{itemize}