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234 lines
9.4 KiB

5 months ago
  1. \hspace{-0.09cm}\scalebox{0.85}{
  2. \begin{dplltabular}{6}
  3. \dpllStep{1|2|3|4|5|6}
  4. \dpllDecL{0|1|2|2|2|2}
  5. \dpllAssi{ - |$\lnot a$|$\lnot a, \lnot b$|$\lnot a, \lnot b, \lnot c$|\makecell{$\lnot a, \lnot b, \lnot c, $ \\ $\lnot e$}|\makecell{$\lnot a, \lnot b, \lnot c, $ \\ $\lnot e, \lnot d$}}
  6. \dpllClause{1}{$\lnot b, \lnot e$}{$\lnot b, \lnot e$|$\lnot b, \lnot e$|\done|\done|\done|\done}
  7. \dpllClause{2}{$a, d, e$}{$a, d, e$|$d, e$|$d, e$|$d, e$|$d$|\conflict}
  8. \dpllClause{3}{$b, \lnot c$}{$b, \lnot c$|$b, \lnot c$|$\lnot c$|\done|\done|\done}
  9. \dpllClause{4}{$c, \lnot d, e$}{$c, \lnot d, e$|$c, \lnot d, e$|$c, \lnot d, e$|$\lnot d, e$|$\lnot d$|\done}
  10. \dpllClause{5}{$\lnot c, e$}{$\lnot c, e$|$\lnot c, e$|$\lnot c, e$|\done|\done|\done}
  11. \dpllClause{6}{$\lnot a, c$}{$\lnot a, c$|\done|\done|\done|\done|\done}
  12. \dpllClause{7}{$a, b, \lnot e$}{$a, b, \lnot e$|$b, \lnot e$|$\lnot e$|$\lnot e$|\done|\done}
  13. \dpllBCP{ - | - |$\lnot c$|$\lnot e$|$\lnot d$| - }
  14. \dpllPL{ - | - | - | - | - | - }
  15. \dpllDeci{$\lnot a$|$\lnot b$| - | - | - | - }
  16. \end{dplltabular}
  17. }
  18. Conflict in step 6\\
  19. \scalebox{0.75}{
  20. \begin{tikzpicture}[>=latex,line join=bevel,]
  21. \pgfsetlinewidth{1bp}
  22. %%
  23. \pgfsetcolor{black}
  24. % Edge: 1 -> 5
  25. \draw [->] (21.98bp,107.68bp) .. controls (35.878bp,110.71bp) and (61.357bp,116.28bp) .. (89.136bp,122.35bp);
  26. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  27. \pgfsetstrokecolor{strokecol}
  28. \draw (54.0bp,124.0bp) node {$$3$$};
  29. % Edge: 1 -> 7
  30. \draw [->] (21.223bp,100.91bp) .. controls (26.631bp,98.299bp) and (33.631bp,95.049bp) .. (40.0bp,92.5bp) .. controls (53.069bp,87.269bp) and (68.1bp,82.187bp) .. (89.278bp,75.441bp);
  31. \draw (54.0bp,100.0bp) node {$$7$$};
  32. % Edge: 2 -> 4
  33. \draw [->] (22.153bp,30.975bp) .. controls (40.514bp,31.883bp) and (80.448bp,34.085bp) .. (114.0bp,37.5bp) .. controls (132.29bp,39.362bp) and (153.03bp,42.23bp) .. (177.97bp,45.931bp);
  34. \draw (100.0bp,45.0bp) node {$$2$$};
  35. % Edge: 2 -> 7
  36. \draw [->] (21.311bp,34.995bp) .. controls (35.439bp,41.815bp) and (62.567bp,54.912bp) .. (89.756bp,68.038bp);
  37. \draw (54.0bp,64.0bp) node {$$7$$};
  38. % Edge: 4 -> 3
  39. \draw [->] (199.67bp,50.723bp) .. controls (207.24bp,53.246bp) and (218.02bp,56.839bp) .. (237.09bp,63.197bp);
  40. % Edge: 5 -> 6
  41. \draw [->] (110.72bp,121.02bp) .. controls (122.5bp,116.74bp) and (142.92bp,109.08bp) .. (160.0bp,101.5bp) .. controls (163.18bp,100.09bp) and (166.52bp,98.509bp) .. (178.77bp,92.384bp);
  42. \draw (146.0bp,120.0bp) node {$$4$$};
  43. % Edge: 6 -> 3
  44. \draw [->] (199.67bp,83.938bp) .. controls (207.41bp,81.085bp) and (218.51bp,76.995bp) .. (237.47bp,70.01bp);
  45. % Edge: 7 -> 4
  46. \draw [->] (110.01bp,67.766bp) .. controls (116.08bp,64.741bp) and (124.35bp,60.943bp) .. (132.0bp,58.5bp) .. controls (143.63bp,54.787bp) and (157.03bp,52.1bp) .. (177.91bp,48.768bp);
  47. \draw (146.0bp,66.0bp) node {$$2$$};
  48. % Edge: 7 -> 6
  49. \draw [->] (110.98bp,74.221bp) .. controls (124.8bp,76.604bp) and (150.09bp,80.963bp) .. (177.87bp,85.753bp);
  50. \draw (146.0bp,90.0bp) node {$$4$$};
  51. % Node: 1
  52. \begin{scope}
  53. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  54. \pgfsetstrokecolor{strokecol}
  55. \draw (11.0bp,105.5bp) ellipse (11.0bp and 11.0bp);
  56. \draw (11.0bp,105.5bp) node {$\lnot b$};
  57. \end{scope}
  58. % Node: 5
  59. \begin{scope}
  60. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  61. \pgfsetstrokecolor{strokecol}
  62. \draw (100.0bp,124.5bp) ellipse (11.0bp and 11.0bp);
  63. \draw (100.0bp,124.5bp) node {$\lnot c$};
  64. \end{scope}
  65. % Node: 7
  66. \begin{scope}
  67. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  68. \pgfsetstrokecolor{strokecol}
  69. \draw (100.0bp,72.5bp) ellipse (11.0bp and 11.0bp);
  70. \draw (100.0bp,72.5bp) node {$\lnot e$};
  71. \end{scope}
  72. % Node: 2
  73. \begin{scope}
  74. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  75. \pgfsetstrokecolor{strokecol}
  76. \draw (11.0bp,30.5bp) ellipse (11.0bp and 11.0bp);
  77. \draw (11.0bp,30.5bp) node {$\lnot a$};
  78. \end{scope}
  79. % Node: 4
  80. \begin{scope}
  81. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  82. \pgfsetstrokecolor{strokecol}
  83. \draw (189.0bp,47.5bp) ellipse (11.0bp and 11.0bp);
  84. \draw (189.0bp,47.5bp) node {$d$};
  85. \end{scope}
  86. % Node: 3
  87. \begin{scope}
  88. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  89. \pgfsetstrokecolor{strokecol}
  90. \draw (248.0bp,66.5bp) ellipse (11.0bp and 11.0bp);
  91. \draw (248.0bp,66.5bp) node {$\bot$};
  92. \end{scope}
  93. % Node: 6
  94. \begin{scope}
  95. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  96. \pgfsetstrokecolor{strokecol}
  97. \draw (189.0bp,87.5bp) ellipse (11.0bp and 11.0bp);
  98. \draw (189.0bp,87.5bp) node {$\lnot d$};
  99. \end{scope}
  100. %
  101. \end{tikzpicture}
  102. }
  103. \begin{prooftree}
  104. \AxiomC{$2. \; a \lor d \lor e$}
  105. \AxiomC{$4. \; c \lor \lnot d \lor e$}
  106. \BinaryInfC{$a \lor e \lor c$}
  107. \AxiomC{$7. \; a \lor b \lor \lnot e$}
  108. \BinaryInfC{$a \lor c \lor b$}
  109. \AxiomC{$3. \; b \lor \lnot c$}
  110. \BinaryInfC{$a \lor b$}
  111. \end{prooftree}
  112. \hspace{-0.09cm}\scalebox{0.85}{
  113. \begin{dplltabular}{5}
  114. \dpllStep{7|8|9|10|11}
  115. \dpllDecL{1|1|1|1|1}
  116. \dpllAssi{$\lnot a$|$\lnot a, b$|$\lnot a, b, \lnot e$|\makecell{$\lnot a, b, \lnot e, $ \\ $\lnot c$}|\makecell{$\lnot a, b, \lnot e, $ \\ $\lnot c, \lnot d$}}
  117. \dpllClause{1}{$\lnot b, \lnot e$}{$\lnot b, \lnot e$|$\lnot e$|\done|\done|\done}
  118. \dpllClause{2}{$a, d, e$}{$d, e$|$d, e$|$d$|$d$|\conflict}
  119. \dpllClause{3}{$b, \lnot c$}{$b, \lnot c$|\done|\done|\done|\done}
  120. \dpllClause{4}{$c, \lnot d, e$}{$c, \lnot d, e$|$c, \lnot d, e$|$c, \lnot d$|$\lnot d$|\done}
  121. \dpllClause{5}{$\lnot c, e$}{$\lnot c, e$|$\lnot c, e$|$\lnot c$|\done|\done}
  122. \dpllClause{6}{$\lnot a, c$}{\done|\done|\done|\done|\done}
  123. \dpllClause{7}{$a, b, \lnot e$}{$b, \lnot e$|\done|\done|\done|\done}
  124. \dpllClause{8}{$a, b$}{$b$|\done|\done|\done|\done}
  125. \dpllBCP{$b$|$\lnot e$|$\lnot c$|$\lnot d$| - }
  126. \dpllPL{ - | - | - | - | - }
  127. \dpllDeci{ - | - | - | - | - }
  128. \end{dplltabular}
  129. }
  130. Conflict in step 11\\
  131. \scalebox{0.75}{
  132. \begin{tikzpicture}[>=latex,line join=bevel,]
  133. \pgfsetlinewidth{1bp}
  134. %%
  135. \pgfsetcolor{black}
  136. % Edge: 1 -> 4
  137. \draw [->] (22.088bp,105.34bp) .. controls (27.406bp,105.99bp) and (34.044bp,106.68bp) .. (40.0bp,107.0bp) .. controls (128.76bp,111.74bp) and (151.4bp,114.1bp) .. (240.0bp,107.0bp) .. controls (243.53bp,106.72bp) and (247.27bp,106.26bp) .. (261.0bp,104.03bp);
  138. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  139. \pgfsetstrokecolor{strokecol}
  140. \draw (140.0bp,119.5bp) node {$$2$$};
  141. % Edge: 1 -> 7
  142. \draw [->] (21.216bp,99.701bp) .. controls (26.622bp,97.268bp) and (33.622bp,94.268bp) .. (40.0bp,92.0bp) .. controls (51.861bp,87.783bp) and (65.43bp,83.883bp) .. (85.966bp,78.457bp);
  143. \draw (54.0bp,99.5bp) node {$$8$$};
  144. % Edge: 3 -> 4
  145. \draw [->] (193.64bp,73.547bp) .. controls (207.62bp,78.689bp) and (233.81bp,88.321bp) .. (261.31bp,98.437bp);
  146. \draw (226.0bp,97.5bp) node {$$2$$};
  147. % Edge: 3 -> 5
  148. \draw [->] (194.22bp,69.303bp) .. controls (222.41bp,67.381bp) and (301.66bp,61.978bp) .. (349.75bp,58.699bp);
  149. \draw (272.0bp,72.5bp) node {$$4$$};
  150. % Edge: 3 -> 6
  151. \draw [->] (188.34bp,60.121bp) .. controls (192.93bp,50.793bp) and (201.03bp,37.066bp) .. (212.0bp,29.0bp) .. controls (223.36bp,20.648bp) and (238.65bp,16.125bp) .. (260.91bp,12.018bp);
  152. \draw (226.0bp,36.5bp) node {$$5$$};
  153. % Edge: 4 -> 2
  154. \draw [->] (283.04bp,100.42bp) .. controls (306.89bp,96.661bp) and (367.37bp,87.133bp) .. (408.89bp,80.592bp);
  155. % Edge: 5 -> 2
  156. \draw [->] (371.67bp,61.562bp) .. controls (379.41bp,64.415bp) and (390.51bp,68.505bp) .. (409.47bp,75.49bp);
  157. % Edge: 6 -> 5
  158. \draw [->] (283.25bp,12.51bp) .. controls (295.55bp,14.674bp) and (316.47bp,19.492bp) .. (332.0bp,29.0bp) .. controls (337.58bp,32.419bp) and (342.82bp,37.21bp) .. (353.76bp,49.494bp);
  159. \draw (318.0bp,36.5bp) node {$$4$$};
  160. % Edge: 7 -> 3
  161. \draw [->] (108.3bp,75.264bp) .. controls (121.47bp,74.324bp) and (144.58bp,72.673bp) .. (171.88bp,70.723bp);
  162. \draw (140.0bp,81.5bp) node {$$1$$};
  163. % Node: 1
  164. \begin{scope}
  165. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  166. \pgfsetstrokecolor{strokecol}
  167. \draw (11.0bp,104.0bp) ellipse (11.0bp and 11.0bp);
  168. \draw (11.0bp,104.0bp) node {$\lnot a$};
  169. \end{scope}
  170. % Node: 4
  171. \begin{scope}
  172. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  173. \pgfsetstrokecolor{strokecol}
  174. \draw (272.0bp,102.0bp) ellipse (11.0bp and 11.0bp);
  175. \draw (272.0bp,102.0bp) node {$d$};
  176. \end{scope}
  177. % Node: 7
  178. \begin{scope}
  179. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  180. \pgfsetstrokecolor{strokecol}
  181. \draw (97.0bp,76.0bp) ellipse (11.0bp and 11.0bp);
  182. \draw (97.0bp,76.0bp) node {$b$};
  183. \end{scope}
  184. % Node: 2
  185. \begin{scope}
  186. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  187. \pgfsetstrokecolor{strokecol}
  188. \draw (420.0bp,79.0bp) ellipse (11.0bp and 11.0bp);
  189. \draw (420.0bp,79.0bp) node {$\bot$};
  190. \end{scope}
  191. % Node: 3
  192. \begin{scope}
  193. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  194. \pgfsetstrokecolor{strokecol}
  195. \draw (183.0bp,70.0bp) ellipse (11.0bp and 11.0bp);
  196. \draw (183.0bp,70.0bp) node {$\lnot e$};
  197. \end{scope}
  198. % Node: 5
  199. \begin{scope}
  200. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  201. \pgfsetstrokecolor{strokecol}
  202. \draw (361.0bp,58.0bp) ellipse (11.0bp and 11.0bp);
  203. \draw (361.0bp,58.0bp) node {$\lnot d$};
  204. \end{scope}
  205. % Node: 6
  206. \begin{scope}
  207. \definecolor{strokecol}{rgb}{0.0,0.0,0.0};
  208. \pgfsetstrokecolor{strokecol}
  209. \draw (272.0bp,11.0bp) ellipse (11.0bp and 11.0bp);
  210. \draw (272.0bp,11.0bp) node {$\lnot c$};
  211. \end{scope}
  212. %
  213. \end{tikzpicture}
  214. }
  215. \begin{prooftree}
  216. \AxiomC{$4. \; c \lor \lnot d \lor e$}
  217. \AxiomC{$2. \; a \lor d \lor e$}
  218. \BinaryInfC{$c \lor e \lor a$}
  219. \AxiomC{$5. \; \lnot c \lor e$}
  220. \BinaryInfC{$e \lor a$}
  221. \AxiomC{$1. \; \lnot b \lor \lnot e$}
  222. \BinaryInfC{$a \lor \lnot b$}
  223. \AxiomC{$8. \; a \lor b$}
  224. \BinaryInfC{$a$}
  225. \end{prooftree}