non-computability in graphs - oscar levinmath.oscarlevin.com/talks/uw2013talk.pdf · computability...
TRANSCRIPT
![Page 1: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/1.jpg)
Non-Computability in Graphs
Oscar Levin
University of Northern Colorado
UW ColloquiumOctober 24, 2013
![Page 2: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/2.jpg)
Computability Theory
Interested in the nature of computable functionsAlternatively: recursive functions, lambda calculus, Turingmachines, algorithms, etc.The meat: how can we talk about non-computablefunctions?Connection to logic: the more non-computable a functionis, the more quantifiers we need to define it.
![Page 3: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/3.jpg)
Computability Theory
Interested in the nature of computable functionsAlternatively: recursive functions, lambda calculus, Turingmachines, algorithms, etc.The meat: how can we talk about non-computablefunctions?Connection to logic: the more non-computable a functionis, the more quantifiers we need to define it.
![Page 4: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/4.jpg)
Computability Theory
Interested in the nature of computable functionsAlternatively: recursive functions, lambda calculus, Turingmachines, algorithms, etc.The meat: how can we talk about non-computablefunctions?Connection to logic: the more non-computable a functionis, the more quantifiers we need to define it.
![Page 5: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/5.jpg)
Computability Theory
Interested in the nature of computable functionsAlternatively: recursive functions, lambda calculus, Turingmachines, algorithms, etc.The meat: how can we talk about non-computablefunctions?Connection to logic: the more non-computable a functionis, the more quantifiers we need to define it.
![Page 6: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/6.jpg)
Computability and Graph Theory
Gain insight into computability theory using graphs as a canvas.Example: computable dimension.
Gain insight into graph theory using computability as a tool.Example: the Four Color Theorem.
![Page 7: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/7.jpg)
Computability and Graph Theory
Gain insight into computability theory using graphs as a canvas.Example: computable dimension.
Gain insight into graph theory using computability as a tool.Example: the Four Color Theorem.
![Page 8: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/8.jpg)
Computable Dimension
Given two computable copies A, B of the same structure, theremight or might not be a computable isomorphism between Aand B.
The number of copies of a structure up to computableisomorphism is the computable dimension of the structure.
![Page 9: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/9.jpg)
Computable Dimension
Given two computable copies A, B of the same structure, theremight or might not be a computable isomorphism between Aand B.
The number of copies of a structure up to computableisomorphism is the computable dimension of the structure.
![Page 10: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/10.jpg)
1 or ω
A graph with computable dimension 1:
A graph with computable dimension ω:
Question: Are there structures which have finite computabledimension greater than 1?
![Page 11: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/11.jpg)
1 or ω
A graph with computable dimension 1:
A graph with computable dimension ω:
Question: Are there structures which have finite computabledimension greater than 1?
![Page 12: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/12.jpg)
1 or ω
A graph with computable dimension 1:
A graph with computable dimension ω:
Question: Are there structures which have finite computabledimension greater than 1?
![Page 13: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/13.jpg)
Computable chromatic number
Any planar graph has a 4-coloring
There are computable planar graphs with no computablek-coloring for any k.
![Page 14: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/14.jpg)
Computable chromatic number
Any planar graph has a 4-coloring
There are computable planar graphs with no computablek-coloring for any k.
![Page 15: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/15.jpg)
Computable chromatic number
Any planar graph has a 4-coloring
There are computable planar graphs with no computablek-coloring for any k.
![Page 16: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/16.jpg)
Dominating Sets in Graphs
Given a graph, we look for sets of vertices close to everything.
A set is dominating if every vertex of G is in, or adjacent to avertex in, the set.
![Page 17: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/17.jpg)
Dominating Sets in Graphs
Given a graph, we look for sets of vertices close to everything.
A set is dominating if every vertex of G is in, or adjacent to avertex in, the set.
![Page 18: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/18.jpg)
Dominating Sets in Graphs
Given a graph, we look for sets of vertices close to everything.
A set is dominating if every vertex of G is in, or adjacent to avertex in, the set.
![Page 19: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/19.jpg)
Domatic Partitions
Definition
A domatic k-partition of a graph G is a partition of (all) thevertices of G into k (disjoint) dominating sets.
The domatic number d(G) is the size of a largest domaticpartition.
![Page 20: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/20.jpg)
Domatic Partitions
Definition
A domatic k-partition of a graph G is a partition of (all) thevertices of G into k (disjoint) dominating sets.
The domatic number d(G) is the size of a largest domaticpartition.
![Page 21: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/21.jpg)
Domatic Partitions
Definition
A domatic k-partition of a graph G is a partition of (all) thevertices of G into k (disjoint) dominating sets.
The domatic number d(G) is the size of a largest domaticpartition.
![Page 22: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/22.jpg)
Domatic Partitions
Definition
A domatic k-partition of a graph G is a partition of (all) thevertices of G into k (disjoint) dominating sets.
The domatic number d(G) is the size of a largest domaticpartition.
![Page 23: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/23.jpg)
Domatic Partitions
Definition
A domatic k-partition of a graph G is a partition of (all) thevertices of G into k (disjoint) dominating sets.
The domatic number d(G) is the size of a largest domaticpartition.
![Page 24: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/24.jpg)
Main Question
Question
Given a computable graph G with domatic number n, what isthe size of the largest computable domatic partition of G?In other words, what is dc(G), the computable domatic number?
![Page 25: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/25.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 26: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/26.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 27: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/27.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 28: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/28.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 29: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/29.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 30: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/30.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 31: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/31.jpg)
If d(G) = 2 then dc(G) = 2.
Suppose G has a domatic 2-partition (so no isolated vertices).There is an algorithm which produces a domatic 2-partition.
Vertices: {v0, v1, v2, . . .}
Put v0 ∈ A.
Put vn ∈ B iff there is an adjacent vertex vk ∈ A (with k < n)
A is a dominating set: if vn 6∈ A then . . .
B is a dominating set: if vn 6∈ B then . . .
![Page 32: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/32.jpg)
What if d(G) = 3?
Proposition
There is a computable graph with domatic number 3 butcomputable domatic number 2.
To prove this, we diagonalize against all computable functions.
![Page 33: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/33.jpg)
What if d(G) = 3?
Proposition
There is a computable graph with domatic number 3 butcomputable domatic number 2.
To prove this, we diagonalize against all computable functions.
![Page 34: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/34.jpg)
Some More Computability Theory
There is an effective list of all (partial) computable functions:
ϕ0, ϕ1, ϕ2, . . .
These can be simulated by a universal computable function
We can run these programs “simultaneously” to see if any looklike they compute a domatic 3-partition.
Meanwhile, we build a computable graph with a 3-partition
When some ϕe tries to compute a 3-partition, we thwart it.
![Page 35: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/35.jpg)
Some More Computability Theory
There is an effective list of all (partial) computable functions:
ϕ0, ϕ1, ϕ2, . . .
These can be simulated by a universal computable function
We can run these programs “simultaneously” to see if any looklike they compute a domatic 3-partition.
Meanwhile, we build a computable graph with a 3-partition
When some ϕe tries to compute a 3-partition, we thwart it.
![Page 36: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/36.jpg)
Some More Computability Theory
There is an effective list of all (partial) computable functions:
ϕ0, ϕ1, ϕ2, . . .
These can be simulated by a universal computable function
We can run these programs “simultaneously” to see if any looklike they compute a domatic 3-partition.
Meanwhile, we build a computable graph with a 3-partition
When some ϕe tries to compute a 3-partition, we thwart it.
![Page 37: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/37.jpg)
Some More Computability Theory
There is an effective list of all (partial) computable functions:
ϕ0, ϕ1, ϕ2, . . .
These can be simulated by a universal computable function
We can run these programs “simultaneously” to see if any looklike they compute a domatic 3-partition.
Meanwhile, we build a computable graph with a 3-partition
When some ϕe tries to compute a 3-partition, we thwart it.
![Page 38: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/38.jpg)
Some More Computability Theory
There is an effective list of all (partial) computable functions:
ϕ0, ϕ1, ϕ2, . . .
These can be simulated by a universal computable function
We can run these programs “simultaneously” to see if any looklike they compute a domatic 3-partition.
Meanwhile, we build a computable graph with a 3-partition
When some ϕe tries to compute a 3-partition, we thwart it.
![Page 39: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/39.jpg)
The Construction
G will start with copies of K4, one for each ϕe.
Build G in stages. At each stage, build a new K4 and checkwhether ϕe has halted on its copy of K4.
If ϕe looks like it computes a 3-partition on its K4, spring thetrap!
![Page 40: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/40.jpg)
The Construction
G will start with copies of K4, one for each ϕe.
Build G in stages. At each stage, build a new K4 and checkwhether ϕe has halted on its copy of K4.
If ϕe looks like it computes a 3-partition on its K4, spring thetrap!
![Page 41: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/41.jpg)
The Construction
G will start with copies of K4, one for each ϕe.
Build G in stages. At each stage, build a new K4 and checkwhether ϕe has halted on its copy of K4.
If ϕe looks like it computes a 3-partition on its K4, spring thetrap!
![Page 42: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/42.jpg)
The Trap
The sprung trap still has a 3-partition, but not the one ϕe claims.
![Page 43: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/43.jpg)
The Trap
The sprung trap still has a 3-partition, but not the one ϕe claims.
![Page 44: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/44.jpg)
The Trap
The sprung trap still has a 3-partition, but not the one ϕe claims.
![Page 45: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/45.jpg)
The Trap
The sprung trap still has a 3-partition, but not the one ϕe claims.
![Page 46: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/46.jpg)
The Trap
The sprung trap still has a 3-partition, but not the one ϕe claims.
![Page 47: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/47.jpg)
Worse Better than that. . .
Proposition
For any n, there is a computable graph with domatic number nbut computable domatic number 2.
Use K3(n−2)+1 as the trap to diagonalize against all possiblecomputable domatic 3-partitions.
![Page 48: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/48.jpg)
Worse Better than that. . .
Proposition
For any n, there is a computable graph with domatic number nbut computable domatic number 2.
Use K3(n−2)+1 as the trap to diagonalize against all possiblecomputable domatic 3-partitions.
![Page 49: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/49.jpg)
Stupid ϕe
Why does ϕe partition its trap so soon?
Just because G is computable, doesn’t mean we can computethe degree of a given vertex!
But what if we could?
![Page 50: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/50.jpg)
Stupid ϕe
Why does ϕe partition its trap so soon?
Just because G is computable, doesn’t mean we can computethe degree of a given vertex!
But what if we could?
![Page 51: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/51.jpg)
Stupid ϕe
Why does ϕe partition its trap so soon?
Just because G is computable, doesn’t mean we can computethe degree of a given vertex!
But what if we could?
![Page 52: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/52.jpg)
Highly computable graphs
Definition
A graph is highly computable if it is computable and degreefunction is computable.
Does this extra information help ϕe compute a domaticpartition?
Proposition
There is a highly computable graph with domatic number 3 butcomputable domatic number 2.
![Page 53: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/53.jpg)
Highly computable graphs
Definition
A graph is highly computable if it is computable and degreefunction is computable.
Does this extra information help ϕe compute a domaticpartition?
Proposition
There is a highly computable graph with domatic number 3 butcomputable domatic number 2.
![Page 54: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/54.jpg)
The idea: remotely sprung traps
Wait for ϕe to partition some fixed vertices. Then act.
Our action cannot change the degree of any vertex in the graph.
ϕe might never partition its vertices, but we don’t know that atany finite stage.
We must be able to force ϕe’s partition to be wrong, bymodifying the graph arbitrarily far away from ϕe’s vertices.
![Page 55: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/55.jpg)
The idea: remotely sprung traps
Wait for ϕe to partition some fixed vertices. Then act.
Our action cannot change the degree of any vertex in the graph.
ϕe might never partition its vertices, but we don’t know that atany finite stage.
We must be able to force ϕe’s partition to be wrong, bymodifying the graph arbitrarily far away from ϕe’s vertices.
![Page 56: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/56.jpg)
The idea: remotely sprung traps
Wait for ϕe to partition some fixed vertices. Then act.
Our action cannot change the degree of any vertex in the graph.
ϕe might never partition its vertices, but we don’t know that atany finite stage.
We must be able to force ϕe’s partition to be wrong, bymodifying the graph arbitrarily far away from ϕe’s vertices.
![Page 57: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/57.jpg)
The idea: remotely sprung traps
Wait for ϕe to partition some fixed vertices. Then act.
Our action cannot change the degree of any vertex in the graph.
ϕe might never partition its vertices, but we don’t know that atany finite stage.
We must be able to force ϕe’s partition to be wrong, bymodifying the graph arbitrarily far away from ϕe’s vertices.
![Page 58: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/58.jpg)
Highly intricate trap
A path:
Every third vertex must be colored the same.
![Page 59: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/59.jpg)
Springing the trap
![Page 60: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/60.jpg)
Springing the trap
![Page 61: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/61.jpg)
Springing the trap
![Page 62: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/62.jpg)
Springing the trap
![Page 63: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/63.jpg)
Springing the trap
![Page 64: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/64.jpg)
Springing the trap
![Page 65: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/65.jpg)
Springing the trap
![Page 66: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/66.jpg)
If d(G) = 4 then. . .
Proposition
There is a highly computable graph with domatic number 4 butcomputable domatic number 3.
![Page 67: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/67.jpg)
If d(G) = 4 then. . .
Proposition
There is a highly computable graph with domatic number 4 butcomputable domatic number 3.
![Page 68: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/68.jpg)
Can we do better?
Is it easier to find smaller domatic partitions in highlycomputable graphs?
Conjecture
Any highly computable graph with domatic number n hascomputable domatic number at least f (n).
Maybe f (n) = n− 1. Or f (n) = (n + 1)/2
![Page 69: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/69.jpg)
Can we do better?
Is it easier to find smaller domatic partitions in highlycomputable graphs?
Conjecture
Any highly computable graph with domatic number n hascomputable domatic number at least f (n).
Maybe f (n) = n− 1. Or f (n) = (n + 1)/2
![Page 70: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/70.jpg)
Can we do better?
Is it easier to find smaller domatic partitions in highlycomputable graphs?
Conjecture
Any highly computable graph with domatic number n hascomputable domatic number at least f (n).
Maybe f (n) = n− 1. Or f (n) = (n + 1)/2
![Page 71: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/71.jpg)
The End
Thanks for listening
![Page 72: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/72.jpg)
Partial results towards and away from the conjecture
Proposition
Let k ≥ 3. For any non-computable c.e. set A, there is anA-computable graph G = (V,E) such that G has a domatick-partition, but no computable domatic 3-partition.
Proposition
There is a highly computable graph with domatic number n, butno computable splittable domatic (n− 1)-partition.
![Page 73: Non-Computability in Graphs - Oscar Levinmath.oscarlevin.com/talks/UW2013talk.pdf · Computability Theory Interested in the nature of computable functions Alternatively: recursive](https://reader033.vdocuments.mx/reader033/viewer/2022050123/5f53298582234652116ddf99/html5/thumbnails/73.jpg)
Partial results towards and away from the conjecture
Proposition
Let k ≥ 3. For any non-computable c.e. set A, there is anA-computable graph G = (V,E) such that G has a domatick-partition, but no computable domatic 3-partition.
Proposition
There is a highly computable graph with domatic number n, butno computable splittable domatic (n− 1)-partition.