Commit 3b1b5046 by Miriam

### presentation draft finished

parent a90c51a8
 ... ... @@ -5,6 +5,8 @@ \usepackage[T1]{fontenc} \usepackage{graphicx,xspace} \usepackage{tikz} \usetikzlibrary{arrows,shapes,positioning} \usetikzlibrary{calc,decorations.markings} \usepackage{amsmath} \usepackage{url} \usepackage{bussproofs} ... ... @@ -22,7 +24,7 @@ \newcommand{\dneg}{{\neg\neg}} \usetheme{Madrid} %\usefonttheme{structuresmallcapsserif} \usefonttheme{structuresmallcapsserif} \title[DNegMod]{Double-Negation Translation of Intuitionistic Modal Logics in Coq} \author{Miriam Polzer \& Ulrich Rabenstein} ... ... @@ -48,14 +50,34 @@ \vspace{2cm} Now we need a proof system satisfying exactly: \only<1>{The intuitionistic modal logic $\KIbox \oplus Z$: \begin{itemize} \item All intuitionistic tautologies \item Closure under substitution and MP \item Closure under MP and substitution \item Closure under generalization: If $A$ valid, then $\Box A$ valid. \item $\Box (A \rightarrow B) \rightarrow (\Box A \rightarrow \Box B)$ \item Axiom $Z$ \end{itemize} \end{itemize}} \only<2>{Kripke-Semantics for $\KIbox \oplus Z$: \begin{itemize} \item Nonempty set of worlds \item Two relations: \\ \scalebox{1}{\begin{tikzpicture} \draw[->] (current bounding box.south west) ++(0, -2em) -- ++(2em, 0) node[right] {Intuitionistic relation $R_i$, preorder}; \draw[->, dashed] (current bounding box.south west) ++(0, -1em) -- ++(2em, 0) node[right] {Modal relation $R_m$}; \end{tikzpicture}} \item A \textit{frame condition}, e.g. $\forall w_1 w_2, (\exists w_3, w_1 R_i w_3 \land w_3 R_m w_2) \Rightarrow (\exists w_3, w_1 R_m w_3 \land w_3 R_i w_2)$ \end{itemize}} \end{frame} ... ... @@ -114,11 +136,6 @@ \BinaryInfC{$\KIbox \oplus Z \vdash G \Rightarrow \Box B$} \end{prooftree} TODO explanaition? \end{frame} \begin{frame}{Natural Deduction for $\KClbox \oplus Z$} Classical counterpart of an intuitionistic modal logic:\\ $\KClbox \oplus Z := \KIbox \oplus Z \land (\dneg a \rightarrow a)$ \end{frame} ... ... @@ -134,44 +151,100 @@ \end{frame} \begin{frame}{Glivenko's translation} $A^{glv} := \dneg A$ \begin{frame}{Glivienko's translation} $A^{glv} := \dneg A$ \begin{theorem} \it Formula $A$ is a classical tautology if and only if $A^{glv}$ is an intuitionistic tautology. \end{theorem} \only<1>{$\KIbox \oplus Z \vdash \dneg A \Leftrightarrow \KClbox \oplus Z \vdash A$ ?} \pause $\KIbox \oplus Z \vdash A^{glv} \Leftrightarrow \KClbox \oplus Z \vdash A$ ? \end{frame} \begin{frame}{Glivenko's translation} \sout{$\KIbox \oplus Z \vdash A^{glv} \Leftrightarrow \KClbox \oplus Z \vdash A$} \begin{example} $\Box (\dneg p \rightarrow p) \in \KClbox$ but $\dneg \Box (\dneg p \rightarrow p) \not \in \KIbox$\\ \begin{tikzpicture} \vspace{2em} \begin{tabular}{l|r} \scalebox{1.6}{\begin{tikzpicture} \node (a) {a}; \node (b) [below of = a] {b}; \node (c) [right of = b] {c}; \node (c) [right of = b] {c}; \path (a) edge [->] (b) (a) edge [->] (c) (a) edge [dashed, ->] (b) (a) edge [dashed, ->] (c) (b) edge [->] (c); \end{tikzpicture} TODO: a->c and a->b are modal. b->c is intuitionistic. c satisfies p, b does not. then b satisfies $\dneg$ p, but not p \end{tikzpicture}} & \scalebox{1}{\begin{tikzpicture} \node (1) {$V(p) = \{c\}$}; \draw[->] (current bounding box.south west) ++(0, -2em) -- ++(2em, 0) node[right] {Intuitionistic Relation}; \draw[->, dashed] (current bounding box.south west) ++(0, -1em) -- ++(2em, 0) node[right] {Modal Relation}; \end{tikzpicture}} \end{tabular} %a->c and a->b are modal. b->c is intuitionistic. %2c satisfies p, b does not. then b satisfies $\dneg$ p, but not p \end{example} \end{frame} \begin{frame}{The translation triangle} \begin{block}{Translation Properties} \begin{description} \item[$\dneg$-characterization] $\KIbox \vdash \dneg A \leftrightarrow A^t$ \item[adequateness] $\adequate{Z}{t}$ \scriptsize(..for a certain class of axioms) \item[characterization] $\KClbox \vdash A \leftrightarrow A^t$ \item[adequateness] $\adequate{Z}{t}$ {\scriptsize(..for a certain class of axioms)} \end{description} \end{block} Translations: \begin{itemize} \item Glivenko: $A^t := \dneg A$ \item Kolmogorov: $\dneg$ in front of \textsc{every} subformula \item Refined Gödel-Gentzen: simplfy Kolmogorov from the \textsc{outside} \item Kuroda: simplify Kolmogorov from the \textsc{inside} \end{itemize} TODO triangle picture including glv as well TODO adequateness equivalent \end{frame} \begin{frame}{The translation triangle} \center \scalebox{1}{\begin{tikzpicture}[node distance = 2cm] \node (ggr) {ggr}; \node (kol) [below right of= ggr] {kol}; \node (kur) [below left of= ggr] {kur}; \node (inv) [below of= ggr] {}; \draw (0,-1) circle (2cm) node[label={[label distance=2cm]\textit{The Triangle}}] (circ) {}; \node (glv) [left = 3cm of circ] {glv}; \path (ggr) edge [<->, double] (kol) (kol) edge [<->, double] (kur) (kur) edge [<->, double] (ggr) (glv) edge [->, double] ++(1.3cm,0) (circ.west); \end{tikzpicture}} %adequateness equivalent \begin{block}{} For any tanslations $t_1, t_2$ in \textit{The Triangle}: \begin{itemize} \item $\KIbox \oplus Z \vdash (A^{t_1} \leftrightarrow A^{t_2})$ \end{itemize} $\Rightarrow$ sufficient to show adequateness for one translation \end{block} \end{frame} %\begin{frame} ... ...
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