From 82dd24d55b038b4c4f848e0ecc987e5d0df866f3 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Thu, 20 Mar 2014 19:20:30 +0100 Subject: [PATCH] Prolog --- documents/Programmierparadigmen/Lambda.tex | 71 +++++++++++++++++- .../Programmierparadigmen.pdf | Bin 803711 -> 811688 bytes .../Programmiersprachen.tex | 11 ++- documents/Programmierparadigmen/Prolog.tex | 18 +++++ .../Symbolverzeichnis.tex | 5 +- .../scripts/prolog/dl-faerbbarkeit | Bin 0 -> 1049941 bytes .../scripts/prolog/dl-faerbbarkeit.pl | 18 +++++ .../scripts/prolog/hello-world.pl | 2 +- .../scripts/prolog/praedikat-farbe.pl | 4 + .../scripts/prolog/simple-term.pl | 1 + 10 files changed, 124 insertions(+), 6 deletions(-) create mode 100755 documents/Programmierparadigmen/scripts/prolog/dl-faerbbarkeit create mode 100644 documents/Programmierparadigmen/scripts/prolog/dl-faerbbarkeit.pl create mode 100644 documents/Programmierparadigmen/scripts/prolog/praedikat-farbe.pl create mode 100644 documents/Programmierparadigmen/scripts/prolog/simple-term.pl diff --git a/documents/Programmierparadigmen/Lambda.tex b/documents/Programmierparadigmen/Lambda.tex index feabfeb..f0e823c 100644 --- a/documents/Programmierparadigmen/Lambda.tex +++ b/documents/Programmierparadigmen/Lambda.tex @@ -294,6 +294,75 @@ Insbesondere ist also $f \ g$ ein Fixpunkt von $g$. $\qed$ \end{beweis} +\section{Let-Polymorphismus}\xindex{let-Polymorphismus}\footnote{WS 2013 / 2014, Folie 205ff}% +Das Programm $P = \text{let } f = \lambda x.\ 2 \text{ in } f\ (f\ \text{\texttt{true}})$ +ist eine polymorphe Hilfsfunktion, da sie beliebige Werte auf 2 Abbildet. +Auch solche Ausdrücke sollen typisierbar sein. +Die Kodierung +\[\text{let} x = t_1 \text{ in } t_2\] +ist bedeutungsgleich mit +\[(\lambda x.\ t_2) t_1\] -Für den Turing-Operator gilt \ No newline at end of file +Das Problem ist, dass +\[P = \lambda f. \ f (f\ \text{\texttt{true}})\ (\lambda x.\ 2)\] +so nicht typisierbar ist, da in +\[\ABS \frac{f: \tau_f \vdash f\ (f\ \text{\texttt{true}}): \dots}{\vdash \lambda f.\ f\ (f\ \text{\texttt{true}}): \dots}\] +müsste +\[\tau_f = \text{bool} \rightarrow \text{int}\] +und zugleich +\[\tau_f = \text{int} \rightarrow \text{int}\] +in den Typkontext eingetragen werden. Dies ist jedoch nicht möglich. Stattdessen +wird +\[\text{let} x = t_1 \text{ in } t_2\] +als neues Konstrukt im $\lambda$-Kalkül erlaubt. + +\begin{definition}[Typschema]\xindex{Typschema}% + Ein Typ der Gestalt $\forall \alpha_1.\ \forall \alpha_2.\ \dots\ \forall \alpha_n. \tau$ + heißt \textbf{Typschema}. Es bindet freie Variablen $\alpha_1, \dots, \alpha_n$ + in $\tau$. +\end{definition} + +\begin{beispiel}[Typschema] + Das Typschema $\forall \alpha.\ \alpha \rightarrow \alpha$ steht für unendlich + viele Typen und insbesondere für folgende: + \begin{bspenum} + \item int $\rightarrow$ int, bool $\rightarrow$ bool, \dots + \item (int $\rightarrow$ int) $\rightarrow$ (int $\rightarrow$ int), \dots + \item \dots + \end{bspenum} +\end{beispiel} + +\begin{definition}[Typschemainstanziierung]\xindex{Typschemainstanziierung}% + Sei $\tau_2$ ein Nicht-Schema-Typ. Dann heißt der Typ + \[\tau[\alpha \mapsto \tau_2]\] + eine \textbf{Instanziierung} vom Typschema $\forall \alpha.\ \tau$ + und man schreibt: + \[(\forall \alpha.\ \tau) \succeq \tau [\alpha \mapsto \tau_2]\] +\end{definition} + +\begin{beispiel}[Typschemainstanziierung] + Folgendes sind Beispiele für Typschemainstanziierungen: + \begin{bspenum} + \item $\forall \alpha.\ \alpha \rightarrow \alpha \succeq \text{int} \rightarrow \text{int}$ + \item $\forall \alpha.\ \alpha \rightarrow \alpha \succeq (\text{int} \rightarrow \text{int}) \rightarrow (\text{int} \rightarrow \text{int})$ + \item $\text{int} \succeq \text{int}$ + \end{bspenum} + + Folgendes sind keine Typschemainstanziierungen: + \begin{bspenum} + \item $\alpha \rightarrow \alpha \nsucceq \text{int} \rightarrow \text{int}$ + \item $\alpha \nsucceq \text{bool}$ + \item $\forall \alpha.\ \alpha \rightarrow \alpha \nsucceq \text{bool}$ + \end{bspenum} +\end{beispiel} + +Zu Typschemata gibt es angepasste Regeln: + +\[\VAR \frac{\Gamma(x)= \tau' \;\;\; \tau' \succeq \tau}{\gamma \vdash x: \tau}\] + +und + +\[\ABS \frac{\Gamma, x: \tau_1 \vdash t: \tau_2 \;\;\; \tau_1 \text{ kein Typschema}}{\Gamma \vdash \lambda x. t: \tau_1 \rightarrow \tau_2}\] + +\todo[inline]{Folie 208ff} \ No newline at end of file diff --git a/documents/Programmierparadigmen/Programmierparadigmen.pdf b/documents/Programmierparadigmen/Programmierparadigmen.pdf index c484e397f541363f12cb578eaad7854328238c41..4da5baa2653ffc944ccb959b9e109d0278031de7 100644 GIT binary patch delta 214656 zcmZsCbBt$Evu)e9?P=S#ZQHhfZQHh|&1u`VHEr8^^WB^Ga$j|c3wMniMPXaXgUuwip+Jxx6=mpTb7gH*{t1c_{K^7J zA7v|P2&i1z7?S6>7)4KJ46GOst7s)@ZbmJNVpc6Vh%SseJ%I52MmbrUG={v9oy3gc zPO=CYus~qW04PHM@1XRU%{{dED-j5%4;a2-XuY5#u@w2YAuG5H@FOsEX@(RvxYk4p zNuH_-4U7CjNNfrUX)stkB5W!Z*6jc*b?8C`SUfW`s-b75P$K92JwaI73syZksBR3} zUIYvQo)K1yA^^_Z?uajgtq6B*8fdt$AVhk?p^He92B0XUHB=`GW^;s|Q8Nf-rbrLW zQlcmijW&f~rZjM*hzBl|lFmpRc@O%FAm01eSdlN1NrJ{{KP@=eFE|HG!_Y%zAmEZl 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Es wird eine ausführbare Datei erzeugt. \section{Syntax} +In Prolog gibt es Prädikate, die Werte haben. Prädikate werden immer klein geschrieben. +So kann das Prädikat \texttt{farbe} mit den Werten \texttt{rot}, \texttt{gruen}, +\texttt{blau}, \texttt{gelb} - welche auch immer klein geschrieben werden - wie +folgt definiert werden: + +\inputminted[numbersep=5pt, tabsize=4]{prolog}{scripts/prolog/praedikat-farbe.pl} + +\begin{itemize} + \item Terme werden durch \texttt{,} mit einem logischem \textbf{und} verknüpft. + \item Ungleichheit wird durch \texttt{\=} ausgedrückt. +\end{itemize} + +So ist folgendes Prädikat \texttt{nachbar(X, Y)} genau dann wahr, wenn $X$ +und $Y$ Farben sind und $X \neq Y$ gilt: + +\inputminted[numbersep=5pt, tabsize=4]{prolog}{scripts/prolog/simple-term.pl} + + \section{Beispiele} \subsection{Humans} Erstelle folgende Datei: diff --git a/documents/Programmierparadigmen/Symbolverzeichnis.tex b/documents/Programmierparadigmen/Symbolverzeichnis.tex index a037df1..dff1d55 100644 --- a/documents/Programmierparadigmen/Symbolverzeichnis.tex +++ b/documents/Programmierparadigmen/Symbolverzeichnis.tex @@ -45,6 +45,7 @@ $\psi \vdash \varphi$ & Syntaktische Herleitbarkeit\newline Die Formel $\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax} \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}} -$\bot$ & Bottom\\ -$\Parr$ & TODO? +$\bot$ & Bottom\\ +$\Parr$ & TODO?\\ +$\succeq$& Typschemainstanziierung\\ \end{xtabular} \ No newline at end of file diff --git a/documents/Programmierparadigmen/scripts/prolog/dl-faerbbarkeit b/documents/Programmierparadigmen/scripts/prolog/dl-faerbbarkeit new file mode 100755 index 0000000000000000000000000000000000000000..ba7edbda388691a75dff84123d3cb4b8ff2f3dc9 GIT binary patch literal 1049941 zcmcG%cVJXS*FV1WP?Ug(u@V(ER6`Yk)hI3Wh=7P91PC1xlhBJ1QNR_Af{55CiiizR zl%f(q*oc&f4Fxn}n+vFjqFDKUK4)g`&W5bd`}^aU_x0I(&OLq3oH;Z1?s8X$ZXL^( zDHE`NWdqj*2pw;fA}Ia8fd8o`F#jhd&?--9Rs+FIHRFG^tg~MlG8+&dF@tRQ#rV(st_6M@;hL2#6wU;Xmd8@7KQP zyysfqKKfvt6Vt}c`>cPn*V7;9zFWwk4M0@7fsj>te`HkAeR-hF`zNzdi2sZja`wis z=f)WL{1|%wh%t^=#=uvIfqyK9-d-{2?~6fyJO=*q7;zXAL;k)P^h0CBVOI?NKQZuQ zVvN_tG3?(Q!_NC-*tsqSJ|Tvj$uaD_IEI`FG4L~E$k`o3Z^syN&WRC+tQhjIjX^&n z27N*d`sp#~?~P&SGcm^T%NTN|#E8ScG3kwekJ0+ zl7EL|=-nPe&Mz_WpTv-}JBFQoV&GqiA?FtuUfO!WCZ#m|J2B{w#jt;J3^|v^u;=+0 z_=7Rz4~>D}6hls(820=SBX0F$;6IKLhtwGIJH@cGLJT_#V(2XsLw+!Z{10Q`e~KYL zJBIxCV%Yg@j5w@|q4z)xIV~Y45&xC^dn|^Xf5yOn76bnP1ecc2qhjE%k0EDB41B*B z{)S@6UmAmcLk#*RG2%8ohCMT5$WMtezFnX<5&xC^dlZP$;b`%Jr^yxquOa#adA zBY$r~{-vPLFG0UP27VjjK>2e@(C;#FE4nfzA{MquE;;bSw$4a~_+&(0k%v{G>|bVE0C2rF_1naGbb=6b9i7##>g>&^qi670_nNw z;{rK%X6FWA*br@4PUeu&84xgR%=nxUft=jzAz70lV`Oe1J0pGQ(2>~yGKNfm1w995 zjn5qt$jQhZlQ|?EcIRZLj~f~omYJ0SskzzXhK$V$j2xGdJ1{*rQ!v>>Mr6Y~cshL4 z*sOs=MvM+*jm)CMARjtD3l}4y|JrK@=Hv`XA2%#8By-%jj3K#!A!9OgGN@a`jmynU z4-CuA7&9i2GjjO2bbL@7?1Vvx&Ac*%Mu@lXXIv$92yusl7{4F z=Vn3=KgI+`=A`H5P7aJ6o1PUQ)x>e>V*`khmXJM$(nk)@7@L(l8Q}|z%^1rMFysFN z*<-12;J}RR>~Wb0){ykvk%(XJ5YRwuc3|*GhB+rAV{~B12+~qT`tXdLkyA1P& z7|T!(lQ9X5MObqgCPsanBYUj&Hg^ieGhEuKA-R*YG6oK&Thx{@ZURF}6Gvta$weF| zBlcqhW5frF8io;yO1??{9*Gmxe2F&+s0_?$5r88QGvCL&-Hvq$D; z1cotHH@(4JvK#vx>2bK*X5pv`4w0rlpvT=X0W%|l2Fi_>$pXShU80E76W zTworc44AqzHXV76U~zV0ltyMF1v_=?d}F(TO&d3D+`_+Z(!{^K#=mOP+`o)mHEZl# zqaK#UnOpf1|Ht#V=>MzW!;!1`0-hW|?{e%k*mkNu>6YOI9+&iubP;IxIV zos~r9JbL3r89Gzd(TJn2Gpr`~@Q`^W?dJit$$l z9xTQu@AL2BhZj~1;8c7U?h|+3$dP9S0&#&)4ZJ|^R|oM^UStLH~dQ5!2 zgHJfF_3}&>OR@i#=-{z*`+rFe9!tIdm+au-qW{;*!Be;WOL6e@+5V+Ec-mzD(j2_) zp?I&KgGYep=p_#R9EX0TgXfvM{afwe&$UQ=u6OXY9eja_XGFYx%E4dk&ND z%)uu)_{$x9vV*Vd;9EKPD;#`^gP-l-Qyu)34nEDn^9;)V^>gq^7KzV64!)j)AK~EZ zJNPUI-@w67a_|iue4c}E*!8dX6D;<1O2fy0EH*@go z9lUweCU*-Qd<%!Z(82SJ!u}OGcs`@DfBPN$H5Q4_Vh7*K!7B%Ut%E=A;9EQRQx3k3 zgAbgDND-N7e0 z_yG<+*}>o8;9EKPfet>!!4GoqsSZBf!KXR+!4AHkgCFAH2RZnm4t|7#&v5Wr4t|(} zpXA_2IQTpVKhnX^aqy!Ye7=Jp?cf(Vcs-)}<`M@#)}deN;Kw=m)eb(>!LN7lSq{Fy z!QbiN3mts6gD-OMISzimgU@yF#SVVFgI5lIf`dQq;3qlwQx1NzgAe=~$^R)1KF-0< zckuBJeyW2{aPaO66p0R=&jjsXl7pXNk@!q@@OL@*Rt`ST!KXO*yB&P0gP-Z((;WOg z4!)m*pXJ~OIrw`WeCj>N;!;BucJFK$NWHfpw}P5lJ`hN~=iRt%n&@-4<%q@wRC2(27+=_UE1pf0qz}$j(X#$@h%q@nOBJdHy z+*)|a0v{&KErpjT@aKdtCLAyDKEm8WcmaWT66V&yJN^&--%2=wif}u^s|8+0xIN(|0xu@qfpEUSj}T^y z>g5T1KVi0}UY5W!38xYsB=8i%oe8H2oK3h3;S_;K5oYV@B?~;5a5ut<0{17}op8Ls zy$JUp91yq*VYZgu@jqq!3A1(diUn>>m@T7MBydy0Y!$r%f$I~#neb|XFC)wr&|4z# z1%%o9dHDj@B+Qo2%M-W?VYYf+mcV5Rv&HiU3H)amFk3q>P2dxR+0uC_0v{pFR?bTn z_%LC%a9*OopA#NHI9}j=gxRur0fBcCW~=5M|3k)~@F2p)0&gUoPPj691oA4;Y zDFTloJeqK_z=H|1S@04C?oXI4l@~8?FT!l4ynw)62(yLqj-Qh8C(PE#D;BslVYW$Xg=t1%%lWdHDj@B+ORG%M-W?;faK^1TIUMEs-}! z;6HZ&W-H{S34DSuTOcn*;3I_D;&{mdA12J!#!D3VbHdXJ#|ylVFk2ZfAn;DYY+=0P zzsvX&&Ldnb@J7OS6D|^XE#aAj3j|(8_#VQm1ztvY7U3lVFD861;e3G~A$%XvVDTI(fvc zCuHfjoGJ(U)VW(zw;Z`XwMwwg`J2d}c!=D$dya$WV_!@n{ zsb(Oc19nvSZFj>!nP0ZCze>F)*l#lkaxd{ssMTk4Jl@I%{pq3s-c+zuuM_Zc^k?VL zg1T{Wfxzj(r^KJsaQK3&Att<|0zT9a{i?wB)KKtOr~tA&EK~jZ3o;yTN0QG6R0)9K z00=sVn%7mWy8=cl_}mSrpRJn+(GZzM!lk2xaHA!xp$V(cngmlE;-^TQtBD`A#NTug zsnut7b%@m$86t9`rFzFwUExp-!(Zd-b(k_@Oj-I+{+22*w^rTGp(pDm(a5t5?-X@Vi7OW zaFNf8ez=&!i$S=^sqA1FK zq?}Ybr4E#pIg^wHE)+gnP(kY{5_9o@D(ka(R$(%vTUZqpciK1Ld%B0dM^P?#Dk@8=3WRCCFqeW4a zf0Oc~XiC^8zDIj>WiwE&DxGpkl*}|z=9Nwv?a>rc_Kl{5R?)sXO6IeqY*;#FwDvuu zEE`P;pG9VLlunUyKl(N&N|rJ&O8aHaLHSBFCA5mnXpg#)a#pF7GOeP-PA2g!(Zo

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