From 936d9ffac684e54f78fee88bee9e029258cb9f1d Mon Sep 17 00:00:00 2001 From: Sankalp Gambhir <sankalp.gambhir@epfl.ch> Date: Wed, 26 Mar 2025 10:10:37 +0000 Subject: [PATCH] Add exercise set 4 --- info/exercises/ex-01-sol.pdf | Bin 137770 -> 137660 bytes info/exercises/ex-01.pdf | Bin 90718 -> 90561 bytes info/exercises/ex-02-sol.pdf | Bin 111251 -> 111277 bytes info/exercises/ex-02.pdf | Bin 81469 -> 81504 bytes info/exercises/ex-03-sol.pdf | Bin 112422 -> 112465 bytes info/exercises/ex-03.pdf | Bin 80462 -> 80511 bytes info/exercises/ex-04-sol.pdf | Bin 0 -> 149317 bytes info/exercises/ex-04.pdf | Bin 0 -> 110306 bytes info/exercises/src/ex-04/ex/grammar.tex | 263 +++++++++++++++++ info/exercises/src/ex-04/ex/semantics.tex | 332 ++++++++++++++++++++++ info/exercises/src/ex-04/ex/types.tex | 226 +++++++++++++++ info/exercises/src/ex-04/main.tex | 34 +++ info/exercises/src/macro.tex | 10 +- 13 files changed, 859 insertions(+), 6 deletions(-) create mode 100644 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First, expressing them as inductive relations, with rules named as on the + right: + % + \addtolength{\jot}{1ex} + \begin{gather*} + \AxiomC{\phantom{\(w_1 \in S\)}} + \RightLabel{\(S_{num}\)} + \UnaryInfC{\(\num \in S\)} + \DisplayProof + \quad + \AxiomC{\(w_1 \in S\)} + \AxiomC{\(w_2 \in S\)} + \RightLabel{\(S_+\)} + \BinaryInfC{\(w_1 + w_2 \in S\)} + \DisplayProof \\ + % + \AxiomC{\(w \in S\)} + \RightLabel{\(A_{1}^{start}\)} + \UnaryInfC{\(w \in A_1\)} + \DisplayProof \\ + % + \AxiomC{\(w \in R'\)} + \RightLabel{\(R_{num}\)} + \UnaryInfC{\(\num~ w\in R\)} + \DisplayProof \\ + % + \AxiomC{\(w \in R\)} + \AxiomC{\(w' \in R'\)} + \RightLabel{\(R'_{+}\)} + \BinaryInfC{\(+w ~ w' \in R'\)} + \DisplayProof + \quad + \AxiomC{\phantom{\(w' \in R'\)}} + \RightLabel{\(R'_{\epsilon}\)} + \UnaryInfC{\(\epsilon \in R'\)} + \DisplayProof \\ + % + \AxiomC{\(w \in R\)} + \RightLabel{\(A_{2}^{start}\)} + \UnaryInfC{\(w \in A_2\)} + \DisplayProof + \end{gather*} + + We must show that for any word \(w\), \(w \in A_1\) if and only if \(w \in + A_2\). For this, it must be the case that there is a derivation tree for + \(w \in A_1\) (equivalently, \(w \in S\)) if and only if there is a + derivation tree for \(w \in A_2\) (equivalently, \(w \in R\)) according to + the inference rules above. + \begin{enumerate} + \item \(w \in S \implies w \in R\): we induct on the depth of the + derivation tree. + \begin{itemize} + \item Base case: derivation tree of depth 1. The tree must be + \begin{gather*} + \AxiomC{} + \RightLabel{\(S_{num}\)} + \UnaryInfC{\(\num \in S\)} + \DisplayProof + \end{gather*} + We can show that there is a corresponding derivation tree for \(w \in R\): + \begin{gather*} + \AxiomC{} + \RightLabel{\(R'_{\epsilon}\)} + \UnaryInfC{\(\epsilon \in R'\)} + \RightLabel{\(R_{num}\)} + \UnaryInfC{\(\num \in R\)} + \DisplayProof + \end{gather*} + \item Inductive case: derivation tree of depth \(n+1\), given that for + every derivation of depth \( \le n\) of \(w' \in S\) for any \(w'\), there + is a corresponding derivation of \(w' \in R\). The last rule applied + in the derivation must be \(S_+\): + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(w_1 \in S\)} + \AxiomC{\ldots} + \UnaryInfC{\(w_2 \in S\)} + \RightLabel{\(S_+\)} + \BinaryInfC{\(w_1 + w_2 \in S\)} + \DisplayProof + \end{gather*} + By the inductive hypothesis, since \(w_1 \in S\) and \(w_2 \in S\) + have a derivation tree of smaller depth, there are derivation trees + for \(w_1 \in R\) and \(w_2 \in R\). In particular, the derivation for + \(w_1 \in R\) must end with the rule \(R_{num}\) (only case), so there + must be a derivation tree for \(\num ~w_1' \in R\) with \(w_1' \in + R'\) and \(num~w_1' = w_1\). We have the following pieces: + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(w_1' \in R'\)} + \RightLabel{\(R_{num}\)} + \UnaryInfC{\(\num ~w_1' \in R\)} + \DisplayProof + \quad + \AxiomC{\ldots} + \UnaryInfC{\(w_2 \in R\)} + \DisplayProof + \end{gather*} + To show that \(w_1 + w_2 \in R\), i.e. \(\num ~w_1' + w_2 \in R\), we + must first show that \(w_1' + w_2 \in R'\), as required by the rule + \(R_{num}\). Note that words in \(R'\) are of the form \((+ \num)^*\). + We will prove this separately for all pairs of words at the end + (\(R'_{Lemma}\)). Knowing this, however, we can construct the + derivation tree for \(w_1 + w_2 \in R\): + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(w_1' \in R'\)} + \AxiomC{\ldots} + \UnaryInfC{\(w_2 \in R\)} + \RightLabel{\(R'_{Lemma}\)} + \BinaryInfC{\(w_1' + w_2 \in R'\)} + \RightLabel{\(R_{num}\)} + \UnaryInfC{\(\num ~w_1' + w_2 \in R\)} + \DisplayProof + \end{gather*} + \(\num ~w_1' + w_2 = w_1 + w_2 = w\), as required. + + Finally, we will show the required lemma. We will prove a stronger + property \(R'_{concat}\) first, that for any pair of words \(w_1, w_2 + \in R'\), \(w_1 ~w_2 \in R'\) as well. We induct on the derivation of + \(w_1 \in R'\). + + Base case: derivation ends with \(R'_\epsilon\). Then \(w_1 = + \epsilon\), and \(w_1 ~w_2 = w_2 \in R'\) by assumption. + + Inductive case: derivation ends with \(R'_+\). Then \(w_1 = + v v'\) + for some \(v \in R\) and \(v' \in R'\): + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(v \in R\)} + \AxiomC{\ldots} + \UnaryInfC{\(v' \in R'\)} + \RightLabel{\(R'_+\)} + \BinaryInfC{\(+ v ~v' \in R'\)} + \DisplayProof + \end{gather*} + + Since \(v' \in R'\) has a smaller derivation tree than \(w_1\), by the + inductive hypothesis, we can prove that \(v'~w_2 \in R'\). We get: + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(v \in R\)} + \AxiomC{\ldots} + \UnaryInfC{\(v' \in R'\)} + \AxiomC{\ldots} + \UnaryInfC{\(w_2 \in R'\)} + \RightLabel{\(R'_{concat}\)} + \BinaryInfC{\(v' ~w_2 \in R'\)} + \RightLabel{\(R'_+\)} + \BinaryInfC{\(+ v ~v' ~w_2 \in R'\)} + \DisplayProof + \end{gather*} + + So, \(R'_{concat}\) is proven. We can show \(R'_{lemma}\), i.e. \(w_1' + + w_2 \in R'\) if \(w_1' \in R'\) and \(w_2 \in R\) as: + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(w_1' \in R'\)} + \AxiomC{\ldots} + \UnaryInfC{\(w_2 \in R\)} + \AxiomC{} + \RightLabel{\(R'_\epsilon\)} + \UnaryInfC{\(\epsilon \in R'\)} + \RightLabel{\(R'_+\)} + \BinaryInfC{\(+ w_2 \in R'\)} + \RightLabel{\(R'_{concat}\)} + \BinaryInfC{\(w_1' + w_2 \in R'\)} + \DisplayProof + \end{gather*} + + Thus, the proof is complete. + \end{itemize} + \item \(w \in R \implies w \in S\): we induct on the depth of the + derivation tree for \(w \in R\). This direction is simpler than the other, + but the general method is similar. + + \begin{itemize} + \item Base case: derivation tree of depth 2 (minimum). The tree must be + \begin{gather*} + \AxiomC{} + \RightLabel{\(R'_{\epsilon}\)} + \UnaryInfC{\(\epsilon \in R'\)} + \RightLabel{\(R_{num}\)} + \UnaryInfC{\(\num \in R\)} + \DisplayProof + \end{gather*} + + We have the corresponding derivation tree for \(w \in S\): + \begin{gather*} + \AxiomC{} + \RightLabel{\(S_{num}\)} + \UnaryInfC{\(\num \in S\)} + \DisplayProof + \end{gather*} + + \item Inductive case: derivation tree of depth \(n+1\), given that for + every derivation of depth \(\le n\) of \(w' \in R\) for any \(w'\), + there is a corresponding derivation of \(w' \in S\). The last rules + applied must be \(R_{num}\) and \(R'_{+}\) (otherwise the derivation + would be of the base case): + \begin{gather*} + \AxiomC{\ldots} + \UnaryInfC{\(w_1 \in R\)} + \AxiomC{\ldots} + \UnaryInfC{\(w_2 \in R'\)} + \RightLabel{\(R'_{+}\)} + \BinaryInfC{\(+ w_1 ~w_2 \in R'\)} + \RightLabel{\(R_{num}\)} + \UnaryInfC{\(\num + w_1~ w_2 \in R\)} + \DisplayProof + \end{gather*} + % + where \(w = \num + w_1 ~w_2\). However, we are somewhat stuck here, as + we have no way to relate \(R'\) and \(S\). We will separately show that + if \(+w' \in R'\), then there is a derivation of \(w' in S\) (lemma + \(R'_{S}\)). This will allow us to complete the proof: + \begin{gather*} + \AxiomC{} + \RightLabel{\(S_{num}\)} + \UnaryInfC{\(\num \in S\)} + \AxiomC{\ldots} + \UnaryInfC{\(+w_1 ~w_2 \in R'\)} + \RightLabel{\(R'_{S}\)} + \UnaryInfC{\(w_1 ~w_2 \in S\)} + \RightLabel{\(S_{+}\)} + \BinaryInfC{\(\num + w_1 ~w_2 \in S\)} + \DisplayProof + \end{gather*} + + The proof of the lemma \(R'_S\) is by induction again, and not shown + here. This completes the original proof. + \end{itemize} + + \end{enumerate} + \item Argument similar to Exercise Set 2 Problem 4 (same pair of + grammars). \(B_1 \subset B_2\) as relations can be seen by producing a + derivation tree for each possible case in \(B_1\). For the other + direction, \(B_2 \subseteq B_1\), it is first convenient to prove + that \(B_1\) is closed under concatenation, i.e., if \(w_1, w_2 \in B_1\) + then there is a derivation tree for \(w_1 ~ w_2 \in B_1\). + \end{enumerate} + \end{solution} + +\end{exercise} + diff --git a/info/exercises/src/ex-04/ex/semantics.tex b/info/exercises/src/ex-04/ex/semantics.tex new file mode 100644 index 0000000..889275d --- /dev/null +++ b/info/exercises/src/ex-04/ex/semantics.tex @@ -0,0 +1,332 @@ + + +\begin{exercise}{} + + Consider the following expression language over naturals, and a \emph{halving} + operator: + % + \begin{align*} + expr ::= \half(expr) \mid expr + expr \mid \num + \end{align*} + where \(\num\) is any natural number constant \(\ge 0\). + + We will design the operational semantics of this language. The semantics + should define rules that apply to as many expressions as possible, while being + subjected to the following safety conditions: + \begin{itemize} + \item the semantics should \emph{not} permit halving unless the argument is even + \item they should evaluate operands from left-to-right + \end{itemize} + + Of the given rules below, choose a \emph{minimal} set that satisfies the + conditions above. A set is \emph{not} minimal if removing any rule does not + change the set of expressions that can be evaluated by the semantics, i.e. the + domain of \(\leadsto, \{x \mid \exists y.\; x \leadsto y\}\), remains + unchanged. The removed rule is said to be \emph{redundant}. + + + % \begin{multicols}{2} + \allowdisplaybreaks + \addtolength{\jot}{2ex} + \begin{gather*} + \AxiomC{\(e \leadsto e'\)} + \UnaryInfC{\(\half(e) \leadsto e'\)} + \DisplayProof \tag{A} \\ + % + \AxiomC{\(n\) is a value} + \AxiomC{\(n = 2k\)} + \BinaryInfC{\(\half(n) \leadsto k\)} + \DisplayProof \tag{B} \\ + % + \AxiomC{\(n\) is a value} + \UnaryInfC{\(\half(n) \leadsto \lfloor \frac{n}{2} \rfloor\)} + \DisplayProof \tag{C} \\ + % + \AxiomC{\(\half(e) \leadsto \half(e')\)} + \UnaryInfC{\(\half(e) \leadsto e'\)} + \DisplayProof \tag{D} \\ + % + \AxiomC{\(e \leadsto e'\)} + \UnaryInfC{\(\half(e) \leadsto \half(e')\)} + \DisplayProof \tag{E} \\ + % + \AxiomC{\(e' \leadsto \half(e)\)} + \UnaryInfC{\(half(e) \leadsto e'\)} + \DisplayProof \tag{F} \\ + % + \AxiomC{\(n_1\) is a value} + \AxiomC{\(n_2\) is a value} + \AxiomC{\(n_1 + n_2 = k\)} + \AxiomC{\(n_1\) is odd} + \QuaternaryInfC{\(n_1 + n_2 \leadsto k\)} + \DisplayProof \tag{G} \\ + % + \AxiomC{\(e \leadsto e'\)} + \AxiomC{\(n\) is a value} + \BinaryInfC{\(n + e \leadsto n + e'\)} + \DisplayProof \tag{H} \\ + % + \AxiomC{\(e_2 \leadsto e_2'\)} + \UnaryInfC{\(e_1 + e_2 \leadsto e_1 + e_2'\)} + \DisplayProof \tag{I} \\ + % + \AxiomC{\(n_1\) is a value} + \AxiomC{\(n_2\) is a value} + \AxiomC{\(n_1 + n_2 = k\)} + \AxiomC{\(n_1, n_2\) are even} + \QuaternaryInfC{\(n_1 + n_2 \leadsto k\)} + \DisplayProof \tag{J} \\ + % + \AxiomC{\(n_1\) is a value} + \AxiomC{\(n_2\) is a value} + \AxiomC{\(n_1 + n_2 = k\)} + \TrinaryInfC{\(n_1 + n_2 \leadsto k\)} + \DisplayProof \tag{K} \\ + % + \AxiomC{\(e_1 \leadsto e_1'\)} + \UnaryInfC{\(e_1 + e_2 \leadsto e_1' + e_2\)} + \DisplayProof \tag{L} + \end{gather*} + % \end{multicols} + + \begin{solution} + A possible such minimal set of rules is \(\{B, E, H, K, L\}\). % TODO: + + On what happens when the other rules are added to this set: + \begin{itemize} + \item A: incorrect; allows deducing \(\half(\half(10)) \leadsto 5\) with + rule B. + \item C: incorrect; allows deducing \(\half(3) \leadsto 1\). + \item D: incorrect; allows deducing \(\half(\half(10)) \leadsto 5\) with + rules B and E. + \item F: redundant; reverses a reduction. + \item G: redundant; special case of rule K. + \item I: incorrect; does not reduce the expression left-to-right. + \item J: redundant; special case of rule K. + \end{itemize} + \end{solution} + +\end{exercise} + +\begin{exercise}{} + + Consider a simple programming language with integer arithmetic, boolean + expressions, and user-defined functions: + + \begin{align*} + expr ::= &~ true \mid false \mid \num \\ + &~ expr == expr \mid expr + expr \\ + &~ expr ~\&\& ~expr \mid if ~(expr) ~expr ~else ~expr \\ + &~ f(expr, \ldots, expr) \mid x + \end{align*} + % + where \(f\) represents a (user-defined) function, \(x\) represents a + variable, and \(\num\) represents an integer. + + \begin{enumerate} + \item Inductively define a substitution operation for the terms in this + language, which replaces every free occurrence of a variable \(x\) with a + given expression \(e\). + + The rule for substitution in an addition is provided as an example. Here, + \(t[x := e]\) represents the term \(t\), with every free occurrence of \(x\) + simultaneously replaced by \(e\). + + \begin{equation*} + \AxiomC{\(t_1[x := e] \to t_1'\)} + \AxiomC{\(t_2[x := e] \to t_2'\)} + \BinaryInfC{\(t_1 + t_2[x := e] \to t_1' + t_2'\)} + \DisplayProof + \end{equation*} + + \item Write the rules for the operational semantics for this language, + assuming \emph{call-by-name} semantics for function calls. In call-by-name + semantics, function arguments are not evaluated before the call. Instead, + the parameters are merely substituted into the function body. + + \item Under the following environment (with function names, parameters, and + bodies): + \begin{gather*} + (sum, [x], if ~ (x == 0) ~ then ~ 0 ~ else ~ x + sum(x + (-1))) \\ + (rec, [~], rec()) \\ + (default, [b, x], if ~ b ~ then ~ x ~ else ~ 0) + \end{gather*} + + evaluate each of the following expressions, showing the derivations: + \begin{enumerate} + \item \(sum(2)\) + \item \(if ~(1 == 2) ~ then ~ 3 ~ else ~ 4\) + \item \(sum(sum(0))\) + \item \(rec()\) + \item \(default(false, rec())\) + \end{enumerate} + + How would the evaluations in each case change if we used \emph{call-by-value} + semantics instead? + \end{enumerate} + + \begin{solution} + \allowdisplaybreaks + + \begin{enumerate} + \item Substitution rules: + \begin{gather*} + \AxiomC{} + \UnaryInfC{\(true[x := e] \to true\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(false[x := e] \to false\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(\num[x := e] \to \num\)} + \DisplayProof \\ + % + \AxiomC{\(t_1[x := e] \to t_1'\)} + \AxiomC{\(t_2[x := e] \to t_2'\)} + \BinaryInfC{\(t_1 == t_2[x := e] \to t_1' == t_2'\)} + \DisplayProof \\ + % + \AxiomC{\(t_1[x := e] \to t_1'\)} + \AxiomC{\(t_2[x := e] \to t_2'\)} + \BinaryInfC{\(t_1 + t_2[x := e] \to t_1' + t_2'\)} + \DisplayProof \\ + % + \AxiomC{\(t_1[x := e] \to t_1'\)} + \AxiomC{\(t_2[x := e] \to t_2'\)} + \BinaryInfC{\(t_1 ~\&\& ~t_2[x := e] \to t_1' ~\&\& ~t_2'\)} + \DisplayProof \\ + % + \AxiomC{\(t_1[x := e] \to t_1'\)} + \AxiomC{\(t_2[x := e] \to t_2'\)} + \AxiomC{\(t_3[x := e] \to t_3'\)} + \TrinaryInfC{\(if ~(t_1) ~t_2 ~else ~t_3[x := e] \to if ~(t_1') ~t_2' ~else ~t_3'\)} + \DisplayProof \\ + % + \AxiomC{\(t_1[x := e] \to t_1'\)} + \AxiomC{\(\ldots\)} + \AxiomC{\(t_n[x := e] \to t_3'\)} + \TrinaryInfC{\(f(t_1, \ldots, t_n)[x := e] \to f(t_1', \ldots, t_n')\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(x[x := e] \to e\)} + \DisplayProof \\ + % + \AxiomC{\(x \neq y\)} + \UnaryInfC{\(y[x := e] \to y\)} + \DisplayProof + \end{gather*} + + \item Operational semantics: + \begin{itemize} + \item Equality: + \begin{gather*} + \AxiomC{\(n_1, n_2\) are integer values} + \AxiomC{\(n_1 = n_2\)} + \BinaryInfC{\(n_1 == n_2 \leadsto true\)} + \DisplayProof \\ + % + \AxiomC{\(n_1, n_2\) are integer values} + \AxiomC{\(n_1 \neq n_2\)} + \BinaryInfC{\(n_1 == n_2 \leadsto false\)} + \DisplayProof + \end{gather*} + \item Addition: + \begin{gather*} + \AxiomC{\(t_1 \leadsto t_1'\)} + \UnaryInfC{\(t_1 + t_2 \leadsto t_1' + t_2\)} + \DisplayProof \\ + % + \AxiomC{\(n\) is an integer value} + \AxiomC{\(t_2 \leadsto t_2'\)} + \BinaryInfC{\(n + t_2 \leadsto n + t_2'\)} + \DisplayProof \\ + % + \AxiomC{\(n_1, n_2\) are integer values} + \AxiomC{\(n_1 + n_2 = k\)} + \BinaryInfC{\(n_1 + n_2 \leadsto k\)} + \DisplayProof + \end{gather*} + \item Conjunction: + \begin{gather*} + \AxiomC{\(t_1 \leadsto t_1'\)} + \UnaryInfC{\(t_1 ~\&\& ~t_2 \leadsto t_1' ~\&\& ~t_2\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(true ~\&\&~ t \leadsto t\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(false ~\&\&~ t \leadsto false\)} + \DisplayProof + \end{gather*} + \item Conditionals: + \begin{gather*} + \AxiomC{\(t_1 \leadsto t_1'\)} + \UnaryInfC{\(if ~(t_1) ~t_2 ~else ~t_3 \leadsto if ~(t_1') ~t_2 ~else ~t_3\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(if ~(true) ~t_2 ~else ~t_3 \leadsto t_2\)} + \DisplayProof \\ + % + \AxiomC{} + \UnaryInfC{\(if ~(false) ~t_2 ~else ~t_3 \leadsto t_3\)} + \DisplayProof + \end{gather*} + \item Function call: + \begin{gather*} + \alwaysNoLine + \AxiomC{\(b_0\) is the body of \(f\)} + \UnaryInfC{\((x_1, \ldots, x_n)\) are parameters of \(f\)} + \UnaryInfC{\(b_0[x_1 := t_1] \to b_1 \quad \ldots \quad b_{n-1}[x_n := t_n] \to b_n\)} + \alwaysSingleLine + \UnaryInfC{\(f(t_1, \ldots, t_n) \leadsto b_n\)} + \DisplayProof + \end{gather*} + \end{itemize} + + \item Evaluations: + \begin{enumerate} + \item \(sum(2) \leadsto 3\): + \begin{align*} + &sum(2)\\ + &\leadsto if ~(2 == 0) ~ then ~ 0 ~ else ~ 2 + sum(2 + (-1))\\ + &\leadsto if ~(false) ~ then ~ 0 ~ else ~ 2 + sum(2 + (-1))\\ + &\leadsto 2 + sum(2 + (-1))\\ + &\leadsto 2 + if ~((2 + (-1)) == 0) ~ then ~ 0 ~ else ~ 1 + sum(2 + (-1))\\ + &\leadsto 2 + if ~(1 == 0) ~ then ~ 0 ~ else ~ 1 + sum(2 + (-1))\\ + &\leadsto 2 + if ~(false) ~ then ~ 0 ~ else ~ 1 + sum(2 + (-1) + (-1))\\ + &\leadsto 2 + (1 + sum(2 + (-1) + (-1)))\\ + & \ldots \\ + &\leadsto 2 + (1 + 0) \\ + &\leadsto 2 + 1 \\ + &\leadsto 3 + \end{align*} + \item \(sum(sum(0)) \leadsto 0\): + \begin{align*} + &sum(sum(0))\\ + &\leadsto if ~(sum(0) == 0) ~ then ~ 0 ~ else ~ 0 + sum(sum(0) + (-1))\\ + &\leadsto \ldots \textit{(expand \(sum(0)\) in the conditional)} \\ + &\leadsto if ~(0 == 0) ~ then ~ 0 ~ else ~ 0 + sum(sum(0) + (-1))\\ + &\leadsto if ~(true) ~ then ~ 0 ~ else ~ 0 + sum(sum(0) + (-1))\\ + &\leadsto 0 + \end{align*} + \item \(if ~(1 == 2) ~ then ~ 3 ~ else ~ 4 \leadsto 4\). + \item \(rec() \leadsto rec()\) (infinite loop). + \item \(default(false, rec()) \leadsto 0\). + \end{enumerate} + + Under call-by-value-semantics, the structure of the evaluations would be + different. In \(sum(sum(0))\), we would evaluate the inner \(sum(0)\) to + \(0\) before evaluating the outer \(sum(\cdot)\). In \(default(false, + rec())\), we would need to evaluate \(rec()\), which would lead to an + infinite loop. + \end{enumerate} + + \end{solution} + +\end{exercise} diff --git a/info/exercises/src/ex-04/ex/types.tex b/info/exercises/src/ex-04/ex/types.tex new file mode 100644 index 0000000..7e0f27a --- /dev/null +++ b/info/exercises/src/ex-04/ex/types.tex @@ -0,0 +1,226 @@ + +% perform actual type derivation of some terms + +\begin{exercise}{} + + Consider the following type system for a language with integers, conditionals, pairs, and + functions: + + \allowdisplaybreaks + \addtolength{\jot}{0.5em} + \begin{gather*} + \AxiomC{\(n\) is an integer literal} + \UnaryInfC{\(\Gamma \vdash n : \lstinline|Int|\)} + \DisplayProof \quad + % variable + \AxiomC{\(x : \tau \in \Gamma\)} + \UnaryInfC{\(\Gamma \vdash x : \tau\)} + \DisplayProof \\ + % + % addition + \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Int|\)} + \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Int|\)} + \BinaryInfC{\(\Gamma \vdash e_1 + e_2 : \lstinline|Int|\)} + \DisplayProof \\ + % + % multiplication + \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Int|\)} + \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Int|\)} + \BinaryInfC{\(\Gamma \vdash e_1 \times e_2 : \lstinline|Int|\)} + \DisplayProof \\ + % + % booleans + \AxiomC{\(b\) is a boolean literal} + \UnaryInfC{\(\Gamma \vdash b : \lstinline|Bool|\)} + \DisplayProof \quad + \AxiomC{\(\Gamma \vdash e : \lstinline|Bool|\)} + \UnaryInfC{\(\Gamma \vdash \lstinline|not e| : \lstinline|Bool|\)} + \DisplayProof \\ + % boolean ops + \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Bool|\)} + \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Bool|\)} + \BinaryInfC{\(\Gamma \vdash e_1 \land e_2 : \lstinline|Bool|\)} + \DisplayProof + \quad + \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Bool|\)} + \AxiomC{\(\Gamma \vdash e_2 : \lstinline|Bool|\)} + \BinaryInfC{\(\Gamma \vdash e_1 \lor e_2 : \lstinline|Bool|\)} + \DisplayProof \\ + % + % conditionals + \AxiomC{\(\Gamma \vdash e_1 : \lstinline|Bool|\)} + \AxiomC{\(\Gamma \vdash e_2 : \tau\)} + \AxiomC{\(\Gamma \vdash e_3 : \tau\)} + \TrinaryInfC{\(\Gamma \vdash if ~e_1~ then ~e_2~ else ~e_3 : \tau\)} + \DisplayProof \\ + % pairs + \AxiomC{\(\Gamma \vdash e_1 : \tau_1\)} + \AxiomC{\(\Gamma \vdash e_2 : \tau_2\)} + \BinaryInfC{\(\Gamma \vdash (e_1, e_2) : (\tau_1, \tau_2)\)} + \DisplayProof \\ + % + % projections + \AxiomC{\(\Gamma \vdash e : (\tau_1, \tau_2)\)} + \UnaryInfC{\(\Gamma \vdash fst(e) : \tau_1\)} + \DisplayProof + \quad + \AxiomC{\(\Gamma \vdash e : (\tau_1, \tau_2)\)} + \UnaryInfC{\(\Gamma \vdash snd(e) : \tau_2\)} + \DisplayProof \\ + % + % function + \AxiomC{\(\Gamma \oplus \{x : \tau_1\} \vdash e : \tau_2\)} + \UnaryInfC{\(\Gamma \vdash x \Rightarrow e : \tau_1 \to \tau_2\)} + \DisplayProof \quad + % + % application + \AxiomC{\(\Gamma \vdash e_1 : \tau_1 \to \tau_2\)} + \AxiomC{\(\Gamma \vdash e_2 : \tau_1\)} + \BinaryInfC{\(\Gamma \vdash e_1 e_2 : \tau_2\)} + \DisplayProof + % + \end{gather*} + + \pagebreak + + \begin{enumerate} + \item Given the following type derivation with type variables \(\tau_1, + \ldots, \tau_5\), choose the correct options: + \begin{equation*} + \AxiomC{\((x, \tau_4) \in \Gamma\)} + \UnaryInfC{\(\Gamma \vdash x: \tau_4\)} + \UnaryInfC{\(\Gamma \vdash fst(x): \tau_3\)} + \AxiomC{\((x, \tau_4) \in \Gamma\)} + \UnaryInfC{\(\Gamma \vdash x: \tau_4\)} + \UnaryInfC{\(\Gamma \vdash snd(x): \tau_5\)} + \BinaryInfC{\(\Gamma \vdash fst(x)(snd(x)) : \tau_2\)} + \UnaryInfC{\(\Gamma' \vdash x \Rightarrow fst(x)(snd(x)): \tau_1\)} + \DisplayProof + \end{equation*} + + \begin{enumerate} + \item There are no valid assignments to the type variables such that the + above derivation is valid. + \item In all valid derivations, \(\tau_2 = \tau_5\). + \item There are \emph{no} valid derivations where \(\tau_2 = \lstinline|Int|\). + \item In all valid derivations, \(\tau_4 = (\tau_3, \tau_5)\) + \item In all valid derivations, \(\tau_2 = \tau_4 \to \tau_1\) + \item There is a valid derivation where \(\tau_1 = \tau_2\). + \end{enumerate} + \item For each of the following pairs of terms and types, provide a valid + type derivation or briefly argue why the typing is incorrect: + \begin{enumerate} + \item \(x \Rightarrow x + 5\): \lstinline|Int| \(\to\) \lstinline|Int| + \item \(x \Rightarrow y \Rightarrow x + y\): \lstinline|Int| \(\to\) + \lstinline|Int| \(\to\) \lstinline|Int| + \item \(x \Rightarrow y \Rightarrow y(2) \times x\): \lstinline|Int| \(\to\) + \lstinline|Int| \(\to\) \lstinline|Int| + \item \(x \Rightarrow (x, x)\): \lstinline|Int| \(\to\) \lstinline|(Int, Int)| + \item \(x \Rightarrow y \Rightarrow if ~fst(x) ~then ~snd(x) ~else~ y\): \lstinline|(Bool, Int)| \(\to\) \lstinline|(Int, Int)| \(\to\) \lstinline|Int| + \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Bool| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Bool| \(\to\) \lstinline|Bool|) + \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Int| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Int| \(\to\) \lstinline|Bool|) + \end{enumerate} + + \item Prove that there is \emph{no} valid type derivation for the term + \begin{equation*} + x \Rightarrow if~ fst(x) ~then~ snd(x)~ else~ x + \end{equation*} + \end{enumerate} + + \begin{solution} + + \begin{enumerate} + \item The correct statements are d and e. For the remaining: + \begin{itemize} + \item \textbf{a}: set \(\tau_2 = \lstinline|Int|\), \(\tau_5 = + \lstinline|Bool|\), \(\tau_3 = \tau_5 \to \tau_2\), \(\tau_4 = (\tau_3, + \tau_5)\), and \(\tau_1 = \tau_4 \to \tau_2\). + \item \textbf{b}: see (a). + \item \textbf{c}: see (a). + \item \textbf{f}: given \(\tau_1 = \tau_2\), we also know from the rule + for lambda abstraction that \(\tau_1 = \tau_4 \to \tau_2\), and hence + \(\tau_2 = \tau_4 \to \tau_2\) recursively, which is a contradiction. + \end{itemize} + + \item For the given terms and types: + \begin{enumerate} + \item \(x \Rightarrow x + 5\): \lstinline|Int| \(\to\) \lstinline|Int|: \cmark + \begin{equation*} + \AxiomC{\(x : \lstinline|Int| \in \Gamma\)} + \UnaryInfC{\(\Gamma \vdash x : \lstinline|Int|\)} + \AxiomC{} + \UnaryInfC{\(\Gamma \vdash 5 : \lstinline|Int|\)} + \BinaryInfC{\(\Gamma \vdash x + 5 : \lstinline|Int|\)} + \UnaryInfC{\(\Gamma' \vdash x \Rightarrow x + 5 : \lstinline|Int| \to \lstinline|Int|\)} + \DisplayProof + \end{equation*} + \item \(x \Rightarrow y \Rightarrow x + y\): \lstinline|Int| \(\to\) + \lstinline|Int| \(\to\) \lstinline|Int|: \cmark + \item \(x \Rightarrow y \Rightarrow y(2) \times x\): \lstinline|Int| + \(\to\) \lstinline|Int| \(\to\) \lstinline|Int|: \xmark. If \(y\) has + type \lstinline|Int|, then \(y(2)\) cannot not well-typed, as the + function application rule is not applicable. + \item \(x \Rightarrow (x, x)\): \lstinline|Int| \(\to\) \lstinline|(Int, Int)|: \cmark + \item \(x \Rightarrow y \Rightarrow if ~fst(x) ~then ~snd(x) ~else~ y\): + \lstinline|(Bool, Int)| \(\to\) \lstinline|(Int, Int)| \(\to\) + \lstinline|Int|: \xmark. The type of the two branches of a conditional + must match, but here they are \lstinline|Int| and (\lstinline|Int|, + \lstinline|Int|) respectively. + \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Bool| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Bool| \(\to\) \lstinline|Bool|): \cmark + \begin{equation*} + \AxiomC{\((y, \lstinline|Bool|) \in \Gamma\)} + \UnaryInfC{\(\Gamma \vdash y : \lstinline|Bool|\)} + \AxiomC{\((x, \lstinline|Bool| \to \lstinline|Bool|) \in \Gamma\)} + \UnaryInfC{\(\Gamma \vdash x : \lstinline|Bool| \to \lstinline|Bool|\)} + \AxiomC{\((y, \lstinline|Bool|) \in \Gamma \oplus \{(z, \lstinline|Bool|)\}\)} + \UnaryInfC{\(\Gamma \oplus \{(z, \lstinline|Bool|)\} \vdash y : \lstinline|Bool|\)} + \UnaryInfC{\(\Gamma \vdash z \Rightarrow y : \lstinline|Bool| \to \lstinline|Bool|\)} + \TrinaryInfC{\(\Gamma \vdash if ~y~ then~ (z \Rightarrow y) ~else~ x : \lstinline|Bool| \to \lstinline|Bool|\)} + \UnaryInfC{\(\Gamma' \vdash y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x : (\lstinline|Bool| \to \lstinline|Bool|) \to \lstinline|Bool| \to (\lstinline|Bool| \to \lstinline|Bool|)\)} + \UnaryInfC{\(\Gamma'' \vdash x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x : (\lstinline|Bool| \to \lstinline|Bool|) \to \lstinline|Bool| \to (\lstinline|Bool| \to \lstinline|Bool|)\)} + \DisplayProof + \end{equation*} + Note that the choice of type of \(z\) (and of the argument of \(x\)) is + arbitrary. Hence, the next typing is also valid. + \item \(x \Rightarrow y \Rightarrow if ~y~ then~ (z \Rightarrow y) ~else~ x \): (\lstinline|Int| \(\to\) \lstinline|Bool|) \(\to\) \lstinline|Bool| \(\to\) (\lstinline|Int| \(\to\) \lstinline|Bool|): \cmark + \end{enumerate} + \item Non-existence of a valid type derivation for the term: + \begin{equation*} + t = x \Rightarrow if~ fst(x) ~then~ snd(x)~ else~ x + \end{equation*} + Assume that there is a valid type derivation for the term. We will attempt + to derive a contradiction. We use the fact that if there exists a type + derivation, every step must use one of the rules above, and that the types + assigned to each variable must be consistent across the derivation. + + First, \(t\) has a type derivation \emph{if and only if} \(t_1 = if ~ + fst(x)~then~snd(x)~else~x\) has a type derivation, by using the function + abstraction rule. We will work with \(t_1\) directly. The function + abstraction rule here does not give us more information. + + Any type derivation for \(t_1\) must end in the conditional rule. For this + rule to be applicable, we must have that the following are derivable: + \begin{enumerate} + \item \(\Gamma \vdash fst(x) : \lstinline|Bool|\) + \item \(\Gamma \vdash snd(x) : \tau\) + \item \(\Gamma \vdash x : \tau\) + \end{enumerate} + where the type variable \(\tau\) is also the type of \(t_1\). + + By using the projection rule on (a) and (b), we learn that the type of + \(x\) must be \((\lstinline|Bool|, \tau_1)\) and \((\tau_2, \tau)\) for + two fresh variables \(\tau_1\) and \(\tau_2\) respectively. Matching the + two, as \(x\) may only have one type, we must have \(\tau_1 = \tau\), + \(\tau_2 = \lstinline|Bool|\), and thus the type of \(x\) is + \((\lstinline|Bool|, \tau)\). + + However, from (c), we learn that the type of \(x\) is \(\tau\). It must be + the case that \(\tau = (\lstinline|Bool|, \tau)\). This is not possible + for any type \(\tau\), and we have a contradiction. + + Hence, there is no valid type derivation for the term \(t\). + \end{enumerate} + + \end{solution} + +\end{exercise} diff --git a/info/exercises/src/ex-04/main.tex b/info/exercises/src/ex-04/main.tex new file mode 100644 index 0000000..af89da5 --- /dev/null +++ b/info/exercises/src/ex-04/main.tex @@ -0,0 +1,34 @@ +\documentclass[a4paper]{article} + +\input{../macro} + +\ifdefined\ANSWERS + \if\ANSWERS1 + \printanswers + \fi +\fi + +\DeclareMathOperator{\half}{half} +\newcommand{\num}{\ensuremath{\mathbf{num}}} + +\title{CS 320 \\ Computer Language Processing\\Exercise Set 4} +\author{} +\date{March 26, 2025} + +\begin{document} +\maketitle + +% inductive relations +%% grammars as inductive relations +\input{ex/grammar} + +% operational semantics +%% old ex 5 problems 1 and 2 +\input{ex/semantics} + +% type systems +%% type derivations +%% type system exte +\input{ex/types} + +\end{document} diff --git a/info/exercises/src/macro.tex b/info/exercises/src/macro.tex index 716f8d6..fca3008 100644 --- a/info/exercises/src/macro.tex +++ b/info/exercises/src/macro.tex @@ -106,10 +106,8 @@ \newcommand{\naturals}{\mathbb{N}} \newcommand{\booleans}{\mathbb{B}} -% church booleans -\newcommand{\tru}{\lstinline|true|\xspace} -\newcommand{\fls}{\lstinline|false|\xspace} - -% proof systems -\newcommand{\natded}{\mathcal{N}} % natural deduction +\usepackage{bussproofs} +\usepackage{pifont} +\newcommand{\cmark}{\ding{51}} +\newcommand{\xmark}{\ding{55}} -- GitLab