Abstract Machines (nomadphp)

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Abstrakte Maschinen

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https://twitter.com/old_sound/status/432365241352331264

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#nomadphp irc.freenode.net

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@igorwhiletrue

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“PHP things”

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Computers are grown-up tamagotchis

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Programming is hard

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Why?

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• 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?

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• Abstract machine is a model of computation • Or a really simple interpreter • Cellular automata are abstract machines

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Conway’s Game of Life

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

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1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1

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1 1 2 1 3 5 2 2 1 2 2 2 2 3 2 1

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1 2 1 3 5 2 2 2 2 2 2 3 2 1

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1 2 1 3 5 2 2 2 2 2 2 3 2 1

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• Still lifes • Oscillators • Spaceships • Guns, puffers, breeders

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• Cellular automaton • Metaphor for life • Complexity, emergence & stuff

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• Other cellular automata • Codd’s automaton (8 states) • Langton’s loops (8 states) • Wireworld (4 states)

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Deterministic ﬁnite automaton

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• Endlicher automat • Regular expressions • Directed state transition graph

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(refs|ﬁxes|closes) #\d*

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(refs|ﬁxes|closes) #\d*

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(refs|ﬁxes|closes) #\d*

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ﬁxes #1234

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ixes #1234

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xes #1234

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es #1234

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#1234

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1234

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234

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• M = (Q, Σ, δ, q0, F) • Rule δ = (qi, a → qi1) • Can accept regular languages

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• Regular expressions • Network protocols • Game states • Business rules • Workﬂows • Queues

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$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);

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Nondeterministic ﬁnite automaton

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baz

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baz

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• Does not add computational power • Can be compiled to a DFA • Previous DFA example already showed this • Parallel timelines on ﬁxed time scale

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Pushdown automaton

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• Kellerautomat • Introduces a stack • Can determine balanced parens

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e ( ( ( ( ) ) ) ( ) )

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x e ( ( ( ) ) ) ( ) )

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x x e ( ( ) ) ) ( ) )

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x x x e ( ) ) ) ( ) )

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x x x x e ) ) ) ( ) )

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x x x e ) ) ( ) )

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x x e ) ( ) )

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x e ( ) )

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x x e ) )

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x e )

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• M = (Q, Σ, Γ, δ, q0, Zo, F) • Rule δ = (qi, a, sj → qi1, sj1) • Can accept context-free languages

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• Validation • Parsers • Stack machines

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$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);

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Turing Machine

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0 0 1 1

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• M = (Q, Σ, Γ, δ, q0, b, F) • Rule δ = (qi, aj → qi1, aj1, dk) • Can accept recursively enumerable languages • Or loop forever

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This machine can run any algorithm

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This machine can run any algorithm etsy.com/shop/sharpwriter

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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; }

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Universality

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• For every problem there is a special purpose brain that solves it as fast as possible.    — Konrad Zuse, 1937

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add increment one-third

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add increment one-third add increment one-third

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add increment one-third

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add increment one-third U

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U M

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• Stored-program computer (John von Neumann) • Programs as data • PHPPHP

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Turing completeness

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• System capable of emulating a turing machine • Unbounded storage • Conditional branching • Recursion

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• Universal Turing Machine • λ-calculus (Alonzo Church) • Game of Life • Brainfuck • PHP

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• 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

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Self-reference

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Recursion

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

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while (true);

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Russell’s paradox

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• Let R be the set of all sets that do not contain themselves • Does R contain itself?

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• 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?

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• 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

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Entscheidungsproblem

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• 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

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Halts?

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Halts? Negate

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Halts? Negate Copy

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Halts? Negate Copy { X

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( ) X X

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Halts? Negate Copy X

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Halts? Negate Copy X X

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Halts? Negate Copy true X X

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Halts? Negate Copy X ∞ true X

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Halts? Negate Copy X ∞ true X X X }

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Halts? Negate Copy false X X

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Halts? Negate Copy false halting now X X

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

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• Proof by contradiction • Decision machine cannot exist • Rice’s theorem generalises this • We are screwed

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Ways to cope

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Use ﬁnite state machines in parts of your programs

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Introduce high-level concepts such as bounded loops

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Build restricted subsets of computing such as type systems

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Conclusion I

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Programming is hard

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Big picture

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Grammar Language Automaton Type-0 Recursively enumerable Turing machine Type-1 Context-sensitive Linear-bounded nondeterministic Turing machine Type-2 Context-free Non-deterministic pushdown automaton Type-3 Regular Finite state automaton Power

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Accidentally Turing Complete • Magic: The Gathering • Minecraft • Border Gateway Protocol (BGP) • Microsoft Excel • Electronic circuits

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Hypercomputation • Turing’s Oracle Machine • Zeno Machine • Turing Machine orbiting around a black hole • Quantum Computer • … and beyond

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• Is our universe really turing complete? • Are the possible paths ﬁnite and pre- determined? • Or do even stronger forces exist? ! • Example: Truly random numbers.

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Conclusion II

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Machines are human-like

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Humans are machine-like

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The universe is a machine

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Questions? • joind.in/11180 • github.com/igorw • conway-php • turing-php • lambda-php • @igorwhiletrue

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λ-calculus

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The ﬁrst functional programming language

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Abstraction λ.x … function ($x) { ... } Application f x $f($x) Variables x return $x;

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Identity λx.x function ($x) { return $x; }

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Currying λx.λy.x function ($x) { return function ($y) use ($x) { return $x; } }

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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; } }

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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)); } }

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• 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

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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) }); } ); }

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That’s it, folks!