Abstract Machines (laracon)

From cellular automata to turing machines.

Transcript

1. Abstrakte Maschinen
2. @igorwhiletrue
5. Computers are grown-up tamagotchis
6. Programming is hard
7. Why?
8. • 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?
9. • Abstract machine is a model of computation • Or a really simple interpreter • Cellular automata are abstract machines
10. Conway’s Game of Life
12. • if alive • 2 or 3 neighbours to survive • if dead • exactly 3 neighbours to spawn • else • cell is dead
14. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
15. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
16. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
17. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
31. • Still lifes • Oscillators • Spaceships • Guns, puffers, breeders
32. • Cellular automaton • Metaphor for life • Complexity, emergence & stuff
33. • Other cellular automata • Codd’s automaton (8 states) • Langton’s loops (8 states) • Wireworld (4 states)
34. Deterministic finite automaton
36. • Endlicher automat • Regular expressions • Directed state transition graph
38. (refs|fixes|closes) #\d*
39. (refs|fixes|closes) #\d*
40. (refs|fixes|closes) #\d*
41. fixes #1234
42. ixes #1234
43. xes #1234
44. es #1234
45. s #1234
46. #1234
47. #1234
54. • M = (Q, Σ, δ, q0, F) • Rule δ = (qi, a → qi1) • O(1) space, O(n) time • Can accept regular languages
55. • Regular expressions • Lexical analysis • Network protocols • Game states • Business rules • Workflows
56. $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);
57. Nondeterministic finite automaton
58. baz
59. baz
60. az
61. z
65. • Does not add computational power • Can be compiled to a DFA • Previous DFA example already showed this • Parallel timelines on fixed time scale
68. Pushdown automaton
70. • Kellerautomat • Introduces a stack • Can accept nested structures • e.g. balanced parens, JSON, PHP
71. S ::= ε | (S) | SS
74. e ( ( ( ( ) ) ) ( ) )
75. x e ( ( ( ) ) ) ( ) )
76. x x e ( ( ) ) ) ( ) )
77. x x x e ( ) ) ) ( ) )
78. x x x x e ) ) ) ( ) )
79. x x x e ) ) ( ) )
80. x x e ) ( ) )
81. x e ( ) )
82. x x e ) )
83. x e )
84. e
85. e
86. • 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 • Can emulate a DFA
87. • Validation • Parsers • Stack machines
88. $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, $accept_states);
89. Turing Machine
96. 0 0 1 1
97. 0 0 1 1
98. 0 0 1 0
99. 0 0 0 0
100. 0 1 0 0
101. 0 1 0 0
102. 0 1 0 0
103. 0 1 0 0
104. 0 1 0 0
105. 0 1 0 0
106. 0 1 0 0
107. 0 1 0 1
108. 0 1 0 1
109. 0 1 0 1
110. 0 1 0 1
111. • M = (Q, Σ, Γ, δ, q0, b, F) • Rule δ = (qi, aj → qi1, aj1, dk) • Can accept recursively enumerable languages • Can emulate a PDA • Or loop forever
112. This machine can run any algorithm
113. This machine can run any algorithm etsy.com/shop/sharpwriter
114. while (!in_array($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. Big picture
169. Grammar Language Automaton Type-0 Recursively enumerable Turing machine Type-1 Context-sensitive Linear-bounded non- deterministic Turing machine Type-2 Context-free Non-deterministic pushdown automaton Type-3 Regular Finite state automaton Power
170. Accidentally Turing Complete • Magic: The Gathering • Minecraft • Border Gateway Protocol (BGP) • Microsoft Excel • Electronic circuits
171. Church-Turing Thesis
172. The brain is a machine
173. Beyond Turing Machines
174. Hypercomputation • Turing’s Oracle Machine • Zeno Machine • Turing Machine orbiting around a black hole • Quantum Computer • … and beyond
175. • Is our universe really turing complete? • Are the possible paths finite and pre- determined? • Or do even stronger forces exist? ! • Example: Truly random numbers.
176. The universe is a machine
178. Questions? • joind.in/11341 • github.com/igorw • conway-php • turing-php • lambda-php • @igorwhiletrue
179. 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])); }