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Embedding computations in tilings (Part 2) Andrei Romashchenko the 2nd June 2016
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Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·}
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Embedding computations in tilings (Part 2) Andrei Romashchenko the - - PowerPoint PPT Presentation
Embedding computations in tilings (Part 2) Andrei Romashchenko the - - PowerPoint PPT Presentation
Embedding computations in tilings (Part 2) Andrei Romashchenko the 2nd June 2016 1 / 13 Color: an element of a finite set C = { , , , , , , } 2 / 13 Color: an element of a finite set C = { , , , , , , }
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Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides.
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Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides. i.e, an element of C 4, e.g.,
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Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides. i.e, an element of C 4, e.g., Set of Wang tiles: a set τ ⊂ C 4
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Color: an element of a finite set C = {·, ·, ·, ·, ·, ·, ·} Wang Tile: a unit square with colored sides. i.e, an element of C 4, e.g., Set of Wang tiles: a set τ ⊂ C 4 Tiling: a mapping f : Z2 → τ that respects the matching rules
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- Theorem. There exists a tile set τ such that
(i) τ-tilings exist, and
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- Theorem. There exists a tile set τ such that
(i) τ-tilings exist, and (ii) all τ-tilings are aperiodic.
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Existence of an aperiodic tile set:
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Existence of an aperiodic tile set:
◮ define self-similar tile sets
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
We know simple explicit example of self-similar tile sets
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
We know simple explicit example of self-similar tile sets even in Math. Intelligencer, Durand–Levin–Shen [2004]
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
We know simple explicit example of self-similar tile sets even in Math. Intelligencer, Durand–Levin–Shen [2004] (for kids!)
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
We know simple explicit example of self-similar tile sets even in Math. Intelligencer, Durand–Levin–Shen [2004] (for kids!) last lecture: the fixed-point construction from Durand-R.-Shen
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
We know simple explicit example of self-similar tile sets even in Math. Intelligencer, Durand–Levin–Shen [2004] (for kids!) last lecture: the fixed-point construction from Durand-R.-Shen Funny, but...
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Existence of an aperiodic tile set:
◮ define self-similar tile sets ◮ observe that every self-similar tile set is aperiodic ◮ construct some self-similar tile set
We know simple explicit example of self-similar tile sets even in Math. Intelligencer, Durand–Levin–Shen [2004] (for kids!) last lecture: the fixed-point construction from Durand-R.-Shen Funny, but... WHY???
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A tile set that simulates itself:
Universal TM m-colors ◦ program ◦ bin(N)
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A tile set that simulates itself:
Universal TM m-colors ◦ program ◦ bin(N)
Parameters:
◮ N = zoom factor ◮ k = #[bits in a macro-color] ◮ m = [size of the computational zone]
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A tile set that simulates itself:
Universal TM m-colors ◦ program ◦ bin(N)
Parameters:
◮ N = zoom factor ◮ k = #[bits in a macro-color] := 2 log N + O(1) ◮ m = [size of the computational zone]:= poly(log N)
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A tile set that simulates itself:
Universal TM m-colors ◦ program ◦ bin(N)
Parameters:
◮ N = zoom factor ◮ k(N) = #[bits in a macro-color] := 2 log N + O(1) ◮ m(N) = [size of the computational zone]:= poly(log N)
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A tile set τN that simulates itself:
Universal TM m-colors ◦ program ◦ bin(N)
Parameters:
◮ N = zoom factor (works for all large enough N) ◮ k = #[bits in a macro-color] := 2 log N + O(1) ◮ m = [size of the computational zone]:= poly(log N)
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A tile set τN that simulates itself with variable zoom :
Universal TM m-colors ◦ program ◦ bin(N)
◮ level 1 (macro-tiles): zoom=N, ◮ level 2 (macro-maro-tiles): zoom=N + 1, ◮ level 3 (macro-maro-macro-tiles): zoom=N + 2, ◮ . . .
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[Turing machine π] → tile set τ(π)
Service computrations m-colors ◦ program ◦ bin(N) Useful computation
Useful computation = simulating machine π on available space and time
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[Turing machine π] → tile set τ(π)
Service computrations m-colors ◦ program ◦ bin(N) Useful computation
Useful computation = simulating machine π on available space and time τ-tiling exists ⇐ ⇒ π never stops
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[Turing machine π] → tile set τ(π) τ-tiling exists ⇐ ⇒ π never stops Theorem [Berger 66]. The tiling problem is undecidable
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[Turing machine π] → tile set τ(π) τ-tiling exists ⇐ ⇒ π never stops Theorem [Berger 66]. The tiling problem is undecidable (given a tile set we cannot decide algorithmically whether it can tile the plane).
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a sequence embedded in a tiling:
Service computrations m-colors ◦ program ◦ bin(N) Useful computation with ω
ω = ω0ω1 . . . ωn . . .
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a sequence embedded in a tiling:
Service computrations m-colors ◦ program ◦ bin(N) Useful computation with ω
ω = ω0ω1 . . . ωn . . . N-macro-colors include the prefix ω[0:log N]
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- Definition. ω = ω0ω1 . . . ωn . . . is a separator if
◮ ωn = 0 for every n s.t. the n-th Turing machine(n) = 0, ◮ ωn = 1 for every n s.t. the n-th Turing machine(n) = 1.
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- Definition. ω = ω0ω1 . . . ωn . . . is a separator if
◮ ωn = 0 for every n s.t. the n-th Turing machine(n) = 0, ◮ ωn = 1 for every n s.t. the n-th Turing machine(n) = 1.
- Lemma. Every separator is non computable.
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- Definition. ω = ω0ω1 . . . ωn . . . is a separator if
◮ ωn = 0 for every n s.t. the n-th Turing machine(n) = 0, ◮ ωn = 1 for every n s.t. the n-th Turing machine(n) = 1.
- Lemma. Every separator is non computable.
This is a very standard fact: a pair of r.e. non separable sets.
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- Definition. ω = ω0ω1 . . . ωn . . . is a separator if
◮ ωn = 0 for every n s.t. the n-th Turing machine(n) = 0, ◮ ωn = 1 for every n s.t. the n-th Turing machine(n) = 1.
- Lemma. Every separator is non computable.
This is a very standard fact: a pair of r.e. non separable sets. Theorem [Hanf, Myers 74]. There exists a tile set τ such that
◮ τ-tilings of the plane exist, ◮ every τ-tiling is non computable.
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- Definition. ω = ω0ω1 . . . ωn . . . is a separator if
◮ ωn = 0 for every n s.t. the n-th Turing machine(n) = 0, ◮ ωn = 1 for every n s.t. the n-th Turing machine(n) = 1.
- Lemma. Every separator is non computable.
This is a very standard fact: a pair of r.e. non separable sets. Theorem [Hanf, Myers 74]. There exists a tile set τ such that
◮ τ-tilings of the plane exist, ◮ every τ-tiling is non computable.
Proof:
◮ embed an ω in our tiling ◮ useful computation: simulate in parallel n-th TM(n) and check
that the embedded ω is a separator
◮ every (infinite) tiling must include an incomputable ω
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