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zlOO%wxQeZ8@^~c;TyO8BZ4P~)a|yPNy2|cyoBdYKk6O!ia;^O` p&yE^uJG$`q*Y~5{tL2m4(I>? diff --git a/paper/paper.tex b/paper/paper.tex index de795f5..5b91de9 100644 --- a/paper/paper.tex +++ b/paper/paper.tex @@ -6,6 +6,8 @@ \usepackage[T1]{fontenc} \usepackage{listings} \usepackage{xcolor} +\usepackage{mathtools} +\usepackage{amsthm} \usepackage{collectbox} %\usepackage{tgpagella} % text only \usepackage{mathpazo} @@ -54,6 +56,27 @@ \fbox{\BOXCONTENT}% }% } +\def\C{\mathbb{C}} +\def\N{\mathbb{N}} +\def\Q{\mathbb{Q}} +\def\R{\mathbb{R}} +\def\Z{\mathbb{Z}} +\DeclarePairedDelimiter\abs{\lvert}{\rvert} + +\theoremstyle{plain}% default +\newtheorem{thm}{Theorem}[section] +\newtheorem{lem}[thm]{Lemma} +\newtheorem{prop}[thm]{Proposition} +\newtheorem*{cor}{Corollary} +\theoremstyle{definition} +\newtheorem{defn}{Definition}[section] +\newtheorem{conj}{Conjecture}[section] +\newtheorem{exmp}{Example}[section] +\newtheorem{exrc}[exmp]{Exercise} +\theoremstyle{remark} +\newtheorem*{comm}{Comment} +\newtheorem*{note}{Note} +\newtheorem{caso}{Case} \title{Minicaml, a purely functional, didactical programming language\\WORK IN PROGRESS DRAFT} \author{Alessandro Cheli\\Course taught by Prof. Gianluigi Ferrari\\and Prof. Francesca Levi} @@ -146,51 +169,82 @@ \section{Types} \section{Evaluation} \subsection{Operational Semantics} -\textbf{Dictionaries}\\ -Note that the evaluation step of the dictionary is considered implicit in most -cases. Primitives require that the arguments passed are already evaluated, -therefore dictionaries are considered already evaluated. An exception is made -for the \textit{Fold Left} catamorphism, where the key-value couples have -to be enumerated in order to show the set-theoretic representation of a left fold. +\begin{note} + The letter $e$ denotes an environment. \\ + The symbol $\_$ is used whenever a value exists but is content is irrelevant + to the semantical rule, or cannot be determined and therefore is discarded. +\end{note} + +\textbf{Dictionaries} +\begin{gather*} + \te{Creation} \\ + \hline \\ + <\te{e, d}> \Rightarrow \\ \{ (k,v) \in d \mid \forall i,j \in \N \wedge i,j \in \left[1, \abs{d}\right] \land i \neq j \\ \text{such that } k_i \neq k_j \} +\end{gather*} \begin{gather*} \te{Insertion} \\ \hline \\ - \dfrac{<\te{env, k} \in \te{dict}> \Rightarrow - \te{true}}{<\te{env, insert k v dict}> \Rightarrow \te{dict} \cup (\te{k,v})} \\ \\ - \dfrac{<\te{env, k} \in \te{dict}> \Rightarrow \te{false}}{<\te{env, insert k v dict}> \Rightarrow \te{dict}} + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, k} \in \te{ed}> \Rightarrow \te{false} \\ + \end{aligned} }{<\te{e, insert k v d}> \Rightarrow \te{ed} \cup (\te{k,v})} \\ \\ + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, k} \in \te{ed}> \Rightarrow \te{true} + \end{aligned}}{<\te{e, insert k v d}> \Rightarrow \te{ed} \textbackslash \{(\te{k,\_})\} \cup (k, v) } \end{gather*} \begin{gather*} \te{Deletion} \\ \hline \\ - \dfrac{<\te{env, k} \in \te{dict}> \Rightarrow \te{true}}{<\te{env, remove k dict}> \Rightarrow \te{dict} \textbackslash \{(\te{k,v})\}} \\ \\ - \dfrac{<\te{env, k} \in \te{dict}> \Rightarrow \te{false}}{<\te{env, remove k dict}> \Rightarrow \te{error}} + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, k} \in \te{ed}> \Rightarrow \te{true} + \end{aligned}}{<\te{e, remove k d}> \Rightarrow \te{ed} \textbackslash \{(\te{k,v})\}} \\ \\ + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, k} \in \te{ed}> \Rightarrow \te{false} + \end{aligned}}{<\te{e, remove k d}> \Rightarrow \te{error}} \end{gather*} \begin{gather*} \te{Contains key} \\ \hline \\ - <\te{env}, \te{haskey k dict}> \Rightarrow \te{k} \in \te{dict} + \dfrac{<\te{e, d}> \Rightarrow \te{ed}}{<\te{e}, \te{haskey k d}> \Rightarrow (\te{k, \_}) \in \te{ed}} \end{gather*} \begin{gather*} \te{Retreive a value} \\ \hline \\ - \dfrac{<\te{env, k} \in \te{dict}> \Rightarrow \te{true}}{<\te{env, getkey k dict}> \Rightarrow \te{v}} \\ \\ - \dfrac{<\te{env, k} \in \te{dict}> \Rightarrow \te{false}}{<\te{env, getkey k dict}> \Rightarrow \te{error}} + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, k} \in \te{ed}> \Rightarrow \te{true} + \end{aligned}}{<\te{e, getkey k d}> \Rightarrow \te{v}} \\ \\ + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, k} \in \te{ed}> \Rightarrow \te{false} + \end{aligned}}{<\te{e, getkey k d}> \Rightarrow \te{error}} \end{gather*} \begin{gather*} \te{Filter by keys} \\ \hline \\ - \dfrac{<\te{env, ks}> \Rightarrow \te{\{k1, ..., kn\}}}{<\te{env, filterkeys ks d}> \Rightarrow \{ (k, v) \in \te{d} \mid (k \in ks) \}} + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, ks}> \Rightarrow \te{\{k1, ..., kn\}} + \end{aligned}}{<\te{e, filterkeys ks d}> \Rightarrow \{ (k, v) \in \te{ed} \mid (k \in ks) \}} \end{gather*} \begin{gather*} \te{Map} \\ \hline \\ - \dfrac{<\te{env, f}> \Rightarrow \lambda(x)}{<\te{env, map f d}> \Rightarrow \{ (k, \lambda(v)) \mid (k, v) \in \te{d} \}} + \dfrac{ \begin{aligned} + <\te{e, d}> \Rightarrow \te{ed} \\ + <\te{e, f}> \Rightarrow \lambda(x) + \end{aligned}} + {<\te{e, map f d}> \Rightarrow \{ (k, \lambda(v)) \mid (k, v) \in \te{ed} \}} \end{gather*} \begin{gather*} \te{Fold Left} \\ \hline \\ \dfrac{\begin{aligned} - <\te{env, f}> \Rightarrow \lambda(x,y) \\ - <\te{env, d}> \Rightarrow \{(k_1, v_1), \hdots, (k_n, v_n)\} - \end{aligned}}{<\te{env, foldl f a d}> \Rightarrow \lambda( \hdots \lambda(\lambda(a, v_1), v_2), \hdots, v_n)} \\ \\ + <\te{e, f}> \Rightarrow \lambda(x,y) \\ + <\te{e, d}> \Rightarrow \{(k_1, v_1), \hdots, (k_n, v_n)\} + \end{aligned}}{<\te{e, foldl f a d}> \Rightarrow \lambda( \hdots \lambda(\lambda(a, v_1), v_2), \hdots, v_n)} \\ \\ \end{gather*} +\section{Javascript Transpiler} \section{Tests} Unit testing is extensively performed using the alcotest testing framework. Code