Abstract Machines (nomadphp)

Transcript

1. Abstrakte Maschinen
2. https://twitter.com/old_sound/status/432365241352331264
4. #nomadphp irc.freenode.net
6. @igorwhiletrue
7. “PHP things”
9. Computers are grown-up tamagotchis
10. Programming is hard
11. Why?
12. • 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?
13. • Abstract machine is a model of computation • Or a really simple interpreter • Cellular automata are abstract machines
14. Conway’s Game of Life
16. • if alive • 2 or 3 neighbours to survive • if dead • exactly 3 neighbours to spawn • else • cell is dead
18. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
19. 1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1
20. 1 2 1 3 5 2 2 2 2 2 2 3 2 1
21. 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
34. • Endlicher automat • Regular expressions • Directed state transition graph
36. (refs|fixes|closes) #\d*
37. (refs|fixes|closes) #\d*
38. (refs|fixes|closes) #\d*
39. fixes #1234
40. ixes #1234
41. xes #1234
42. es #1234
43. s #1234
44. #1234
45. #1234
52. • M = (Q, Σ, δ, q0, F) • Rule δ = (qi, a → qi1) • Can accept regular languages
53. • Regular expressions • Network protocols • Game states • Business rules • Workflows • Queues
54. $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);
55. Nondeterministic finite automaton
56. baz
57. baz
58. az
59. z
63. • Does not add computational power • Can be compiled to a DFA • Previous DFA example already showed this • Parallel timelines on fixed time scale
65. Pushdown automaton
66. • Kellerautomat • Introduces a stack • Can determine balanced parens
69. e ( ( ( ( ) ) ) ( ) )
70. x e ( ( ( ) ) ) ( ) )
71. x x e ( ( ) ) ) ( ) )
72. x x x e ( ) ) ) ( ) )
73. x x x x e ) ) ) ( ) )
74. x x x e ) ) ( ) )
75. x x e ) ( ) )
76. x e ( ) )
77. x x e ) )
78. x e )
79. e
80. e
81. • M = (Q, Σ, Γ, δ, q0, Zo, F) • Rule δ = (qi, a, sj → qi1, sj1) • Can accept context-free languages
82. • Validation • Parsers • Stack machines
83. $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, $accept_states);
84. Turing Machine
90. 0 0 1 1
91. 0 0 1 1
92. 0 0 1 0
93. 0 0 0 0
94. 0 1 0 0
95. 0 1 0 0
96. 0 1 0 0
97. 0 1 0 0
98. 0 1 0 0
99. 0 1 0 0
100. 0 1 0 0
101. 0 1 0 1
102. 0 1 0 1
103. 0 1 0 1
104. 0 1 0 1
105. • M = (Q, Σ, Γ, δ, q0, b, F) • Rule δ = (qi, aj → qi1, aj1, dk) • Can accept recursively enumerable languages • Or loop forever
106. This machine can run any algorithm
107. This machine can run any algorithm etsy.com/shop/sharpwriter
108. while (!in_array($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; }
112. Universality
113. • For every problem there is a special purpose brain that solves it as fast as possible.
 
 — Konrad Zuse, 1937
114. add increment one-third
115. add increment one-third add increment one-third
116. add increment one-third
117. add increment one-third U
118. U M
120. • Stored-program computer (John von Neumann) • Programs as data • PHPPHP
121. Turing completeness
122. • System capable of emulating a turing machine • Unbounded storage • Conditional branching • Recursion
123. • Universal Turing Machine • λ-calculus (Alonzo Church) • Game of Life • Brainfuck • PHP
124. • 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
125. Self-reference
126. <?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 $program;
127. Recursion
128. call_user_func( function ($x) { return $x($x); }, function ($x) { return $x($x); } );
129. while (true);
131. Russell’s paradox
132. • Let R be the set of all sets that do not contain themselves • Does R contain itself?
133. • 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?
134. • 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
135. Entscheidungsproblem
136. • 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
137. Halts?
138. Halts? Negate
139. Halts? Negate Copy
140. Halts? Negate Copy { X
141. ( ) X X
142. Halts? Negate Copy X
143. Halts? Negate Copy X X
144. Halts? Negate Copy true X X
145. Halts? Negate Copy X ∞ true X
146. Halts? Negate Copy X ∞ true X X X }
147. Halts? Negate Copy false X X
148. Halts? Negate Copy false halting now X X
149. Halts? Negate Copy false halting now X X X X }
150. • Proof by contradiction • Decision machine cannot exist • Rice’s theorem generalises this • We are screwed
151. Ways to cope
154. Use finite state machines in parts of your programs
155. Introduce high-level concepts such as bounded loops
156. Build restricted subsets of computing such as type systems
157. Conclusion I
158. Programming is hard
159. Big picture
160. 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
161. Accidentally Turing Complete • Magic: The Gathering • Minecraft • Border Gateway Protocol (BGP) • Microsoft Excel • Electronic circuits
162. Hypercomputation • Turing’s Oracle Machine • Zeno Machine • Turing Machine orbiting around a black hole • Quantum Computer • … and beyond
163. • Is our universe really turing complete? • Are the possible paths finite and pre- determined? • Or do even stronger forces exist? ! • Example: Truly random numbers.
164. Conclusion II
165. Machines are human-like
166. Humans are machine-like
167. The universe is a machine
169. Questions? • joind.in/11180 • github.com/igorw • conway-php • turing-php • lambda-php • @igorwhiletrue
170. λ-calculus
171. The first functional programming language
172. Abstraction λ.x … function ($x) { ... } Application f x $f($x) Variables x return $x;
173. Identity λx.x function ($x) { return $x; }
174. Currying λx.λy.x function ($x) { return function ($y) use ($x) { return $x; } }
175. Booleans λx.λy.x λx.λy.y function ($x) { return function ($y) use ($x) { return $x; } } function ($x) { return function ($y) use ($x) { return $y; } }
176. Church numerals λf.λx.x λf.λx.f x λf.λx.f (f x) function ($f) { return function ($x) use ($f) { return $x; } } function ($f) { return function ($x) use ($f) { return $f($x); } } function ($f) { return function ($x) use ($f) { return $f($f($x)); } }
177. • increment • λn.λf.λx.f (n f x) • addition • λm.λn.m SUCC n • if • λx.λy.x • λx.λy.y • bool already acts as an if statement
179. Y combinator λf.(λx.f (λv.((x x) v))) (λx.f (λv.((x x) v))) function ($f) { return call_user_func( function ($x) use ($f) { return $f(function ($v) use ($x) { return $x($x)($v) }); }, function ($x) use ($f) { return $f(function ($v) use ($x) { return $x($x)($v) }); } ); }
180. function evaluate($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])); }
181. That’s it, folks!