Abstract Machines (bephpug)

Transcript

1. Abstrakte Maschinen
2. @igorwhiletrue
4. Programming is hard
5. Why?
6. • Link between our universe and computational universe • Cellular automata are self-replicating abstract machines • Humans are self-replicating biological machines (down to the cellular level) • Or is the entire universe a single machine?
7. • Abstract machine is a model of computation • Cellular automata are abstract machines
8. Conway’s Game of Life
10. • if alive • 2 or 3 neighbours to survive • if dead • exactly 3 neighbours to spawn • else • cell is dead
12. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
13. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
14. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
15. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
24. • Still lifes • Oscillators • Spaceships • Guns, puffers, breeders
25. • Cellular automaton • Metaphor for life • Complexity, emergence & stuff
26. • Other cellular automata • Codd’s automaton (8 states) • Langton’s loops (8 states) • Wireworld (4 states)
27. Deterministic finite automaton
28. • Endlicher automat • Regular expressions • Directed state transition graph
30. (refs|fixes|closes) #\d*
31. (refs|fixes|closes) #\d*
32. (refs|fixes|closes) #\d*
33. fixes #1234
34. ixes #1234
35. xes #1234
36. es #1234
37. s #1234
38. #1234
39. #1234
46. • M = (Q, Σ, δ, q0, F) • Rule δ = (qi, a → qi1) • Can accept regular languages
47. • Regular expressions • Network protocols • Game states • Business rules • Workflows • Queues
48. $rules = [ 0 => ['c' => 1, 'f' => 7, 'r' => 9], 1 => ['l' => 2], 2 => ['o' => 3], ... ]; ! $tokens = ['f', 'i', 'x', 'e', 's', ' ', '#', '1', '2', '3', '4', 'EOF']; ! foreach ($tokens as $token) { if (!isset($rules[$state][$token])) { throw new NoTransitionException(); } ! $state = $rules[$state][$token]; } ! $accepted = in_array($state, $accept_states);
49. Nondeterministic finite automaton
50. baz
51. baz
52. az
53. z
57. • Does not add computational power • Can be compiled to a DFA • Previous DFA example already showed this • Basic quantum physics
59. Pushdown automaton
60. • Kellerautomat • Introduces a stack • Can determine balanced parens
63. e ( ( ( ( ) ) ) ( ) )
64. x e ( ( ( ) ) ) ( ) )
65. x x e ( ( ) ) ) ( ) )
66. x x x e ( ) ) ) ( ) )
67. x x x x e ) ) ) ( ) )
68. x x x e ) ) ( ) )
69. x x e ) ( ) )
70. x e ( ) )
71. x x e ) )
72. x e )
73. e
74. e
75. • M = (Q, Σ, Γ, δ, q0, Zo, F) • Rule δ = (qi, a, sj → qi1, sj1) • Can accept context-free languages
76. • Validation • Parsers • Stack machines
77. Turing Machine
83. 0 0 1 1
84. 0 0 1 1
85. 0 0 1 0
86. 0 0 0 0
87. 0 1 0 0
88. 0 1 0 0
89. 0 1 0 0
90. 0 1 0 0
91. 0 1 0 0
92. 0 1 0 0
93. 0 1 0 0
94. 0 1 0 1
95. 0 1 0 1
96. 0 1 0 1
97. 0 1 0 1
98. • M = (Q, Σ, Γ, δ, q0, b, F) • Rule δ = (qi, aj → qi1, aj1, dk) • Can accept recursively enumerable languages • Or loop forever
99. while (!in_array($state, $accept_states)) { $read_val = isset($tape[$position]) ? $tape[$position] : '_'; ! if (!isset($rules[$state][$read_val])) { throw new NoTransitionException(); } ! list($write_val, $move_dir, $new_state) = $rules[$state][$read_val]; ! $tape[$position] = $write_val; ! if ('l' === $move_dir) { $position--; if ($position < 0) { $position++; array_unshift($tape, '_'); } } else if ('r' === $move_dir) { $position++; if ($position >= count($tape)) { array_push($tape, '_'); } } ! $state = $new_state; }
100. This machine can run any algorithm
101. This machine can run any algorithm etsy.com/shop/sharpwriter
102. Universality
103. add increment one-third
104. add increment one-third add increment one-third
105. add increment one-third
106. add increment one-third U
107. U M
109. • Stored-program computer (John von Neumann) • Programs as data • FPGA • PHPPHP
110. Turing completeness
111. • System capable of emulating a turing machine • Unbounded storage • Conditional branching • Recursion
112. • Universal Turing Machine • λ-calculus (Alonzo Church) • Game of Life • Brainfuck • PHP
113. • If PHP can only do as much as a turing machine, why bother? • Beware of the Turing tar-pit in which everything is possible but nothing of interest is easy. • Epigrams on Programming by Alan Perlis
114. • Is our universe really turing complete? • Or are the possible paths finite and pre- determined? • Do even stronger forces exist?
115. Self-reference
116. <?php $data = <<<'DATA' $program = <<<PROGRAM <?php \$data = <<<'DATA'\n$data\nDATA; $data ! PROGRAM; echo $program; DATA; $program = <<<PROGRAM <?php \$data = <<<'DATA'\n$data\nDATA; $data ! PROGRAM; echo $program;
117. Recursion
119. call_user_func( function ($x) { return $x($x); }, function ($x) { return $x($x); } );
120. while (true);
122. Russell’s paradox
123. • Let R be the set of all sets that do not contain themselves • Does R contain itself? • If yes, then R’s definition is incorrect • If no, R is not in the set, so it must contain itself
124. • Liar paradox: “This sentence is false.” • Type theory • Hierarchy of types avoids self-reference
125. Entscheidungsproblem
126. • David Hilbert asks for an algorithm that decides if a statement in first-order logic is universally valid • Halting problem can be reduced to Entscheidungsproblem • Machine that determines if another machine will halt
127. Halts?
128. Halts? Negate
129. Halts? Negate Copy
130. Halts? Negate Copy { X
131. ( ) X X
132. Halts? Negate Copy X
133. Halts? Negate Copy X X
134. Halts? Negate Copy true X X
135. Halts? Negate Copy X ∞ true X
136. Halts? Negate Copy X ∞ true X X X }
137. Halts? Negate Copy false X X
138. Halts? Negate Copy false halting now X X
139. Halts? Negate Copy false halting now X X X X }
140. • Proof by contradiction • Decision machine cannot exist • We are screwed
141. Ways to cope
144. Use finite state machines in parts of your programs
145. Build restricted subsets of computing such as type systems that cannot loop forever
146. Conclusion
147. Programming is hard
149. Questions? • github.com/igorw • conway-php • turing-php • lambda-php • @igorwhiletrue