Abstract Machines (dpc14)

Transcript

1. Abstrakte Maschinen
2. @igorwhiletrue
4. Computers are grown-up tamagotchis
5. Programming is hard
6. Why?
7. • 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?
8. • Abstract machine is a model of computation • Or a really simple interpreter • Cellular automata are abstract machines
9. Conway’s Game of Life
11. • if alive • 2 or 3 neighbours to survive • if dead • exactly 3 neighbours to spawn • else • cell is dead
13. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
14. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
15. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
16. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
30. • Still lifes • Oscillators • Spaceships • Guns, puffers, breeders
31. • Cellular automaton • Metaphor for life • Complexity, emergence & stuff
32. • Other cellular automata • Codd’s automaton (8 states) • Langton’s loops (8 states) • Wireworld (4 states)
33. Deterministic finite automaton
35. • Endlicher automat • Regular expressions • Directed state transition graph
36. /foo/
37. /foo|bar/
38. /\d*/
39. /\d+/
40. /tro(lo)+/
41. “trololo”
42. “rololo”
43. “ololo”
44. “lolo”
45. “olo”
46. “lo”
47. “o”
48. “”
49. Anas Ramadan, from The Noun Project
50. “trolu”
51. “rolu”
52. “olu”
53. “lu”
54. “u”
55. Anas Ramadan, from The Noun Project
56. • M = (Q, Σ, δ, q0, F) • Rule δ = (qi, a → qi1) • O(1) space, O(n) time • Can accept regular languages
57. • Regular expressions • Lexical analysis • Network protocols • Game states • Business rules • Workflows
58. $rules = [ 0 => ['t' => 1], 1 => ['r' => 2], 2 => ['o' => 3], ... ]; ! $tokens = ['t', 'r', 'o', 'l', ‘o', ‘l', 'o', 'EOF']; ! foreach ($tokens as $token) { if (!isset($rules[$state][$token])) { throw new NoTransitionException(); } ! $state = $rules[$state][$token]; } ! $accepted = in_array($state, $accept_states);
59. Pushdown automaton
61. • Kellerautomat • Introduces a stack • Can accept nested structures
63. { "look!": [ "it's a JSON", {"yes!": "nesting!"} ] }
65. (define Y (lambda (le) ((lambda (f) (f f)) (lambda (f) (le (lambda (x) ((f f) x)))))))
66. sexpr = atom | (sexpr*) atom = letter alphanum* | number+ alphanum = letter | number letter = a | b | ... | z number = 1 | 2 | ... | 9
68. S = ε | (S) | SS
72. e “( ( ( ( ) ) ) ( ) )”
73. x e “( ( ( ) ) ) ( ) )”
74. x x e “( ( ) ) ) ( ) )”
75. x x x e “( ) ) ) ( ) )”
76. x x x x e “) ) ) ( ) )”
77. x x x e “) ) ( ) )”
78. x x e “) ( ) )”
79. x e “( ) )”
80. x x e “) )”
81. x e “)”
82. e “”
83. e Anas Ramadan, from The Noun Project
84. • M = (Q, Σ, Γ, δ, q0, Zo, F) • Rule δ = (qi, a, sj → qi1, sj1) • O(n) space, O(n) time • Can accept context-free languages
85. • Parsers • Data formats • Network protocols • Interpreters • Query languages • Stack machines
86. $rules = [ 0 => ['(' => ['e' => [0, ['e', 'x']]]], ... ]; ! $stack = new SplStack(); $stack->push($init_stack); ! foreach ($tokens as $token) { $top = $stack->pop(); ! if (!isset($rules[$state][$token][$top])) { throw new NoTransitionException(); } ! list($state, $push_tokens) = $rules[$state][$token][$top]; ! foreach ($push_tokens as $push_token) { $stack->push($push_token); } } ! $accepted = in_array($state, $accept_states);
87. $rules = [ 0 => ['(' => ['e' => [0, ['e', 'x']]]], ... ]; ! $stack = new SplStack(); $stack->push($init_stack); ! foreach ($tokens as $token) { $top = $stack->pop(); ! if (!isset($rules[$state][$token][$top])) { throw new NoTransitionException(); } ! list($state, $push_tokens) = $rules[$state][$token][$top]; ! foreach ($push_tokens as $push_token) { $stack->push($push_token); } } ! $accepted = in_array($state, $accept_states);
88. Turing Machine
95. 0 0 1 1
96. 0 0 1 1
97. 0 0 1 0
98. 0 0 0 0
99. 0 1 0 0
100. 0 1 0 0
101. 0 1 0 0
102. 0 1 0 0
103. 0 1 0 0
104. 0 1 0 0
105. 0 1 0 0
106. 0 1 0 1
107. 0 1 0 1
108. 0 1 0 1
109. 0 1 0 1
110. 0 1 0 1 Anas Ramadan, from The Noun Project
111. • M = (Q, Σ, Γ, δ, q0, b, F) • Rule δ = (qi, aj → qi1, aj1, dk) • Can accept recursively enumerable languages • Can loop forever
112. This machine can run any algorithm
113. This machine can run any algorithm etsy.com/shop/sharpwriter
114. while (!in_array($state, $accept_states)) { $read_val = isset($tape[$position]) ? $tape[$position] : '_'; ! if (!isset($rules[$state][$read_val])) { throw new NoTransitionException(); } ! list($write_val, $move_dir, $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, '_'); } } }
115. 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, $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, '_'); } } } oh no! unbounded loop!
116. $rules = [ 1 => ['0' => ['1', 'r', 2], '_' => ['1', 'r', 2], '1' => ['0', 'l', 1]], 2 => ['0' => ['0', 'r', 2], '1' => ['1', 'r', 2], '_' => ['_', 'l', 3]], ];
120. Universality
121. • For every problem there is a special purpose brain that solves it as fast as possible.
 
 — Konrad Zuse, 1937
122. add increment one-third
123. add increment one-third add increment one-third
124. add increment one-third
125. add increment one-third U
126. U M
128. • Stored-program computer (John von Neumann) • Programs as data • PHPPHP
129. Turing completeness
130. • System capable of emulating a turing machine • Unbounded storage • Conditional branching • Recursion
131. • Universal Turing Machine • λ-calculus (Alonzo Church) • Game of Life • Brainfuck • PHP
132. • 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
133. Self-reference
134. <?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 $program;
135. Recursion
136. call_user_func( function ($x) { return $x($x); }, function ($x) { return $x($x); } );
137. while (true);
140. Russell’s paradox
141. • Let R be the set of all sets that do not contain themselves • Does R contain itself?
142. • Barber paradox • A town with just one barber • Everyone either shaves themselves or goes to the barber • Barber shaves all who do not shave themselves • Who shaves the barber?
143. • Liar paradox: “This sentence is false.” • Type theory • Hierarchy of types avoids self-reference • And then came Gödel in 1931 and smashed the foundation of mathematical reasoning
144. Entscheidungsproblem
145. • 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
146. Halts?
147. Halts? Negate
148. Halts? Negate Copy
149. Halts? Negate Copy { X
150. ( ) X X
151. Halts? Negate Copy X
152. Halts? Negate Copy X X
153. Halts? Negate Copy true X X
154. Halts? Negate Copy X ∞ true X
155. Halts? Negate Copy X ∞ true X X X }
156. Halts? Negate Copy false X X
157. Halts? Negate Copy false halting now X X
158. Halts? Negate Copy false halting now X X X X }
159. • Proof by contradiction • Decision machine cannot exist • Rice’s theorem generalises this • We are screwed
160. Ways to cope
163. Use finite state machines in parts of your programs
164. Introduce high-level concepts such as bounded loops
165. Build restricted subsets of computing such as type systems
166. Conclusion
167. Programming is hard
168. Church-Turing Thesis
169. The brain is a machine
170. The universe is a machine
172. Questions? • joind.in/10868 • github.com/igorw • conway-php • turing-php • lambda-php • @igorwhiletrue
173. function evaluate($exp, array $env = []) { if (is_string($exp)) { return $env[$exp]; } ! if ('λ' === first($exp)) { list($_, $arg, $body) = $exp; return ['λ', $arg, $body, $env]; } ! $f = evaluate(first($exp), $env); $arg = evaluate(second($exp), $env); return apply($f, $arg); } ! function apply($f, $x) { list($_, $arg, $body, $env) = $f; return evaluate($body, array_merge($env, [$arg => $x])); }