@igorwhiletrue

Transcription

1 Abstrakte Maschinen

2 @igorwhiletrue

3

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

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10 if alive 2 or 3 neighbours to survive if dead exactly 3 neighbours to spawn else cell is dead

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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

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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

40 1234

41 234

42 34

43 4

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46 M = (Q, Σ, δ, q0, F) Rule δ = (q i, 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

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57 Does not add computational power Can be compiled to a DFA Previous DFA example already showed this Basic quantum physics

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59 Pushdown automaton

60 Kellerautomat Introduces a stack Can determine balanced parens

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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 = (q i, a, sj qi1, sj1) Can accept context-free languages

76 Validation Parsers Stack machines

77 Turing Machine

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98 M = (Q, Σ, Γ, δ, q0, b, F) Rule δ = (q i, 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 increment add one-third

104 increment add one-third increment add one-third

105 increment add one-third

106 U increment add one-third

107 U M

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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 predetermined? 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

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119 call_user_func( function ($x) { return $x($x); }, function ($x) { return $x($x); } );

120 while (true);

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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 Copy Halts? Negate

130 X { Copy Halts? Negate

131 X ( ) X

132 X Copy Halts? Negate

133 Copy X X Halts? Negate

134 Copy X X Halts? true Negate

135 Copy X X Halts? true Negate

136 Copy X X Halts? }X true X Negate

137 Copy X X Halts? false Negate

138 Copy X X Halts? false Negate halting now

139 Copy X X Halts? }X false X Negate halting now

140 Proof by contradiction Decision machine cannot exist We are screwed

141 Ways to cope

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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

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149 Questions? github.com/igorw conway-php turing-php