evolutionary computation: term to term operation continuity
TRANSCRIPT
Term to TermOperationContinuity
David Clark Evolutionary Computation:Term to Term Operation Continuity
David M. Clark
Mathematics: SUNY, New Paltz, NY, USA
July 16, 2013
International Journal of Algebra and Computation(IJAC), in press.
Term to TermOperationContinuity
David Clark
A non-trivial finite algebra is primal if every operation on itsunderlying set is a term operation.
Rousseau’s Theorem (1967) A nontrivial finite groupoid isprimal if and only if it has no proper subalgebras and nonon-trivial automorphisms or congruences.
For example, P is primal:
P := 〈{0, 1, 2, 3, 4}, ∗〉
∗ 0 1 2 3 4
0 1 1 0 2 01 4 2 0 3 32 1 0 3 2 43 0 4 3 4 24 2 0 1 4 0
Term to TermOperationContinuity
David Clark
A non-trivial finite algebra is primal if every operation on itsunderlying set is a term operation.
Rousseau’s Theorem (1967) A nontrivial finite groupoid isprimal if and only if it has no proper subalgebras and nonon-trivial automorphisms or congruences.
For example, P is primal:
P := 〈{0, 1, 2, 3, 4}, ∗〉
∗ 0 1 2 3 4
0 1 1 0 2 01 4 2 0 3 32 1 0 3 2 43 0 4 3 4 24 2 0 1 4 0
Term to TermOperationContinuity
David Clark
A non-trivial finite algebra is primal if every operation on itsunderlying set is a term operation.
Rousseau’s Theorem (1967) A nontrivial finite groupoid isprimal if and only if it has no proper subalgebras and nonon-trivial automorphisms or congruences.
For example, P is primal:
P := 〈{0, 1, 2, 3, 4}, ∗〉
∗ 0 1 2 3 4
0 1 1 0 2 01 4 2 0 3 32 1 0 3 2 43 0 4 3 4 24 2 0 1 4 0
Term to TermOperationContinuity
David Clark
Term Generation Problem: Given a finite groupoid G andan operation f on G that is a term operation of G, find a termthat represents f .
For example, find a term t that represents the ternarydiscriminator operation f on P:
f (a, b, c) :=
{c if a = b
a if a 6= b
This is a finitary problem. But F3(P) will have 553 ≈ 1087
elements. Checking 106 terms per second, the expected time tofind a discriminator term will be ≈ 1073 years(!)
Term to TermOperationContinuity
David Clark
Term Generation Problem: Given a finite groupoid G andan operation f on G that is a term operation of G, find a termthat represents f .
For example, find a term t that represents the ternarydiscriminator operation f on P:
f (a, b, c) :=
{c if a = b
a if a 6= b
This is a finitary problem. But F3(P) will have 553 ≈ 1087
elements. Checking 106 terms per second, the expected time tofind a discriminator term will be ≈ 1073 years(!)
Term to TermOperationContinuity
David Clark
Term Generation Problem: Given a finite groupoid G andan operation f on G that is a term operation of G, find a termthat represents f .
For example, find a term t that represents the ternarydiscriminator operation f on P:
f (a, b, c) :=
{c if a = b
a if a 6= b
This is a finitary problem.
But F3(P) will have 553 ≈ 1087
elements. Checking 106 terms per second, the expected time tofind a discriminator term will be ≈ 1073 years(!)
Term to TermOperationContinuity
David Clark
Term Generation Problem: Given a finite groupoid G andan operation f on G that is a term operation of G, find a termthat represents f .
For example, find a term t that represents the ternarydiscriminator operation f on P:
f (a, b, c) :=
{c if a = b
a if a 6= b
This is a finitary problem. But F3(P) will have 553 ≈ 1087
elements.
Checking 106 terms per second, the expected time tofind a discriminator term will be ≈ 1073 years(!)
Term to TermOperationContinuity
David Clark
Term Generation Problem: Given a finite groupoid G andan operation f on G that is a term operation of G, find a termthat represents f .
For example, find a term t that represents the ternarydiscriminator operation f on P:
f (a, b, c) :=
{c if a = b
a if a 6= b
This is a finitary problem. But F3(P) will have 553 ≈ 1087
elements. Checking 106 terms per second, the expected time tofind a discriminator term will be ≈ 1073 years(!)
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation.
Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Evolutionary computation finds solutions to problems bysimulating biological evolution.
Given a finite groupoid G and a k-ary target operationf : G k → G , we seek a k-ary term t such that tG = f .
The fitness of t is ‖{~d ∈ G k : tG(~d) = f (~d)}‖.
In 2007, Lee Spector and I looked for discriminator terms for3-element primal groupoids.
1 month by exhaustive search at 106 terms per second.
5 minutes be evolutionarily computation. Example
t(x , y , z) = ((((((((x ∗(y ∗x))∗x)∗z)∗(z ∗x))∗((x ∗(z ∗(x ∗(z ∗y))))∗z))∗z)∗z)∗(z∗((((x∗(((z∗z)∗x)∗(z∗x)))∗x)∗y)∗(((y∗(z∗(z∗y)))∗(((y ∗y)∗x)∗z))∗(x∗(((z∗z)∗x)∗(z∗(x∗(z∗y)))))))))
2008 Hummie Award from the ACM.
Term to TermOperationContinuity
David Clark
Discriminator term for P
— found by EC in 53 seconds!
t(x , y , z) := zzzzxz ∗∗∗∗x ∗z ∗z ∗∗xxxxxx ∗∗xx ∗∗∗x ∗∗xxxx ∗∗∗x ∗x ∗xxxx ∗∗∗xxx ∗xx ∗zx ∗∗∗x ∗∗xzx ∗∗xx ∗∗x ∗x ∗xxx ∗x ∗x ∗∗xxxxxxx∗∗∗∗∗∗xzxx∗x∗∗∗xxxx∗∗∗xx∗xx∗∗xxx∗xx∗zx∗∗∗x∗∗xx∗xx∗∗xxx∗xx∗∗x∗∗xxxxx∗∗∗∗zxxx∗∗∗xxx∗∗xx∗∗x∗xxxx∗∗∗x ∗x ∗xxxxx ∗x ∗∗x ∗x ∗∗∗xxxzx ∗∗∗xxx ∗∗∗x ∗∗xx ∗x ∗x ∗x ∗xxxzxxx∗∗∗∗∗x∗∗xxxx∗∗∗xx∗∗xx∗xxz∗x∗xz∗∗x∗∗xxx∗∗xxx∗x∗∗xx∗x∗∗xz∗xx∗∗x∗zx∗x∗x∗xzzxx∗∗x∗∗x∗∗zxx∗∗x∗xxxzxzz∗∗∗∗∗x∗∗yyxx∗∗x∗∗xzyz∗∗xx∗∗∗xxxzzxx∗x∗∗∗∗∗y ∗∗xxx∗xx∗∗∗zzx∗∗xx∗xy ∗∗y ∗zy ∗x∗x∗zxx∗∗zxx∗∗x∗∗∗∗xx∗x∗z∗∗∗∗∗yy∗z∗xx∗y∗∗x∗∗xxxxx∗x∗x∗∗x∗∗∗∗∗x∗xxy∗xx∗∗∗x∗∗xyzx∗xx∗∗∗z∗xx∗∗x∗∗∗yyz∗∗zy∗∗∗∗xzzxxx∗∗zy∗∗∗∗x∗∗∗zy∗x∗y∗y∗x∗x∗∗∗∗∗zx∗xxxx∗∗∗x∗∗∗∗∗zzz∗∗∗yyx∗yx∗∗∗∗xx∗x∗∗∗∗∗xyy∗xx∗∗x∗x∗∗∗∗∗∗∗∗xxx∗zx∗∗x∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
Term to TermOperationContinuity
David Clark
Discriminator term for P
— found by EC in 53 seconds!
t(x , y , z) := zzzzxz ∗∗∗∗x ∗z ∗z ∗∗xxxxxx ∗∗xx ∗∗∗x ∗∗xxxx ∗∗∗x ∗x ∗xxxx ∗∗∗xxx ∗xx ∗zx ∗∗∗x ∗∗xzx ∗∗xx ∗∗x ∗x ∗xxx ∗x ∗x ∗∗xxxxxxx∗∗∗∗∗∗xzxx∗x∗∗∗xxxx∗∗∗xx∗xx∗∗xxx∗xx∗zx∗∗∗x∗∗xx∗xx∗∗xxx∗xx∗∗x∗∗xxxxx∗∗∗∗zxxx∗∗∗xxx∗∗xx∗∗x∗xxxx∗∗∗x ∗x ∗xxxxx ∗x ∗∗x ∗x ∗∗∗xxxzx ∗∗∗xxx ∗∗∗x ∗∗xx ∗x ∗x ∗x ∗xxxzxxx∗∗∗∗∗x∗∗xxxx∗∗∗xx∗∗xx∗xxz∗x∗xz∗∗x∗∗xxx∗∗xxx∗x∗∗xx∗x∗∗xz∗xx∗∗x∗zx∗x∗x∗xzzxx∗∗x∗∗x∗∗zxx∗∗x∗xxxzxzz∗∗∗∗∗x∗∗yyxx∗∗x∗∗xzyz∗∗xx∗∗∗xxxzzxx∗x∗∗∗∗∗y ∗∗xxx∗xx∗∗∗zzx∗∗xx∗xy ∗∗y ∗zy ∗x∗x∗zxx∗∗zxx∗∗x∗∗∗∗xx∗x∗z∗∗∗∗∗yy∗z∗xx∗y∗∗x∗∗xxxxx∗x∗x∗∗x∗∗∗∗∗x∗xxy∗xx∗∗∗x∗∗xyzx∗xx∗∗∗z∗xx∗∗x∗∗∗yyz∗∗zy∗∗∗∗xzzxxx∗∗zy∗∗∗∗x∗∗∗zy∗x∗y∗y∗x∗x∗∗∗∗∗zx∗xxxx∗∗∗x∗∗∗∗∗zzz∗∗∗yyx∗yx∗∗∗∗xx∗x∗∗∗∗∗xyy∗xx∗∗x∗x∗∗∗∗∗∗∗∗xxx∗zx∗∗x∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
Term to TermOperationContinuity
David Clark
Discriminator term for P — found by EC in 53 seconds!
t(x , y , z) := zzzzxz ∗∗∗∗x ∗z ∗z ∗∗xxxxxx ∗∗xx ∗∗∗x ∗∗xxxx ∗∗∗x ∗x ∗xxxx ∗∗∗xxx ∗xx ∗zx ∗∗∗x ∗∗xzx ∗∗xx ∗∗x ∗x ∗xxx ∗x ∗x ∗∗xxxxxxx∗∗∗∗∗∗xzxx∗x∗∗∗xxxx∗∗∗xx∗xx∗∗xxx∗xx∗zx∗∗∗x∗∗xx∗xx∗∗xxx∗xx∗∗x∗∗xxxxx∗∗∗∗zxxx∗∗∗xxx∗∗xx∗∗x∗xxxx∗∗∗x ∗x ∗xxxxx ∗x ∗∗x ∗x ∗∗∗xxxzx ∗∗∗xxx ∗∗∗x ∗∗xx ∗x ∗x ∗x ∗xxxzxxx∗∗∗∗∗x∗∗xxxx∗∗∗xx∗∗xx∗xxz∗x∗xz∗∗x∗∗xxx∗∗xxx∗x∗∗xx∗x∗∗xz∗xx∗∗x∗zx∗x∗x∗xzzxx∗∗x∗∗x∗∗zxx∗∗x∗xxxzxzz∗∗∗∗∗x∗∗yyxx∗∗x∗∗xzyz∗∗xx∗∗∗xxxzzxx∗x∗∗∗∗∗y ∗∗xxx∗xx∗∗∗zzx∗∗xx∗xy ∗∗y ∗zy ∗x∗x∗zxx∗∗zxx∗∗x∗∗∗∗xx∗x∗z∗∗∗∗∗yy∗z∗xx∗y∗∗x∗∗xxxxx∗x∗x∗∗x∗∗∗∗∗x∗xxy∗xx∗∗∗x∗∗xyzx∗xx∗∗∗z∗xx∗∗x∗∗∗yyz∗∗zy∗∗∗∗xzzxxx∗∗zy∗∗∗∗x∗∗∗zy∗x∗y∗y∗x∗x∗∗∗∗∗zx∗xxxx∗∗∗x∗∗∗∗∗zzz∗∗∗yyx∗yx∗∗∗∗xx∗x∗∗∗∗∗xyy∗xx∗∗x∗x∗∗∗∗∗∗∗∗xxx∗zx∗∗x∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗
Term to TermOperationContinuity
David Clark For which groupoids will EC solve the Term GenerationProblem?
Preliminary Definition. A finite groupoid is term continuousif, on average, small changes in a term result in small changesin its term operation.
That is, the k-ary term to k-ary term operation function
G : T → GG kt 7→ tG
is in some sense continuous.
Term to TermOperationContinuity
David Clark For which groupoids will EC solve the Term GenerationProblem?
Preliminary Definition. A finite groupoid is term continuousif, on average, small changes in a term result in small changesin its term operation.
That is, the k-ary term to k-ary term operation function
G : T → GG kt 7→ tG
is in some sense continuous.
Term to TermOperationContinuity
David Clark For which groupoids will EC solve the Term GenerationProblem?
Preliminary Definition. A finite groupoid is term continuousif, on average, small changes in a term result in small changesin its term operation.
That is, the k-ary term to k-ary term operation function
G : T → GG kt 7→ tG
is in some sense continuous.
Term to TermOperationContinuity
David Clark For which groupoids will EC solve the Term GenerationProblem?
Preliminary Definition. A finite groupoid is term continuousif, on average, small changes in a term result in small changesin its term operation.
That is, the k-ary term to k-ary term operation function
G : T → GG kt 7→ tG
is in some sense continuous.
Term to TermOperationContinuity
David Clark For which groupoids will EC solve the Term GenerationProblem?
Preliminary Definition. A finite groupoid is term continuousif, on average, small changes in a term result in small changesin its term operation.
That is, the k-ary term to k-ary term operation function
G : T → GG kt 7→ tG
is in some sense continuous.
Term to TermOperationContinuity
David Clark For which groupoids will EC solve the Term GenerationProblem?
Preliminary Definition. A finite groupoid is term continuousif, on average, small changes in a term result in small changesin its term operation.
That is, the k-ary term to k-ary term operation function
G : T → GG kt 7→ tG
is in some sense continuous.
Term to TermOperationContinuity
David Clark
GG k:
For f , g ∈ GG k, the Hamming Distance is
HD(f , g) := ‖{~d ∈ G k | f (~d) 6= g(~d)}‖.
Under HD, the space GG kis a metric space.
Term to TermOperationContinuity
David Clark
GG k:
For f , g ∈ GG k, the Hamming Distance is
HD(f , g) := ‖{~d ∈ G k | f (~d) 6= g(~d)}‖.
Under HD, the space GG kis a metric space.
Term to TermOperationContinuity
David Clark
GG k:
For f , g ∈ GG k, the Hamming Distance is
HD(f , g) := ‖{~d ∈ G k | f (~d) 6= g(~d)}‖.
Under HD, the space GG kis a metric space.
Term to TermOperationContinuity
David Clark
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Figure: Distance measure on the term space T.
Term to TermOperationContinuity
David Clark
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Figure: Distance measure on the term space T.
Term to TermOperationContinuity
David Clark
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t(~x) = (x1x0)(((x3x3)x3)(x3x2)) t ′(~x) = (x1x0)((x1((x2x0)x0))(x3x2))
u(~x) v(~x)
Figure: Distance measure on the term space T.
Term to TermOperationContinuity
David Clark
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t(~x) = (x1x0)(((x3x3)x3)(x3x2)) t ′(~x) = (x1x0)((x1((x2x0)x0))(x3x2))
u(~x) v(~x)
Figure: Distance measure on the term space T.
Term to TermOperationContinuity
David Clark
T:
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[t(~x), t ′(~x)] = 1D = 1
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t(~x) = (x1x0)(((x3x3)x3)(x3x2)) t ′(~x) = (x1x0)((x1((x2x0)x0))(x3x2))
u(~x) v(~x)
Figure: Distance measure on the term space T.
Term to TermOperationContinuity
David Clark
k ,H – positive integers
~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variable
TH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most H
T – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~x
T1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
k ,H – positive integers~x = (x0, x1, . . . , xk−1) – standard variables, ♦ – a new variableTH – all terms in ~x of height at most HT – all terms in ~xT1 ⊆ V ⊆ T – a fixed finite set of mutation terms in ~x
A ♦-term is a term t(~x ,♦) with a single occurrence of ♦.
A mutation is a triple M = (t(~x ,♦), u(~x), v(~x)) where t(~x ,♦)is a ♦-term, u(~x) and v(~x) are terms, and v(~x) ∈ V.
t(~x , u(~x))M−→ t(~x , v(~x))
MH := {M | t(~x , u(~x)) ∈ TH}, a finite set
Term to TermOperationContinuity
David Clark
If D is the depth of ♦ in t(~x ,♦), then the change in the termt(~x , u(~x)) resulting from applying M = 〈t(~x ,♦), u(~x), v(~x)〉 is
‖M‖ := [t(~x , u(~x)), t(~x , v(~x))]
≤ 1
D.
The resulting change in the term operation on G is
HD(M) := HD(t(~x , u(~x))G, t(~x , v(~x))G).
The average change in term operations of mutations in MH is
µHD(H) :=1
‖MH‖∑{HD(M) | M ∈MH}.
Term to TermOperationContinuity
David Clark
If D is the depth of ♦ in t(~x ,♦), then the change in the termt(~x , u(~x)) resulting from applying M = 〈t(~x ,♦), u(~x), v(~x)〉 is
‖M‖ := [t(~x , u(~x)), t(~x , v(~x))] ≤ 1
D.
The resulting change in the term operation on G is
HD(M) := HD(t(~x , u(~x))G, t(~x , v(~x))G).
The average change in term operations of mutations in MH is
µHD(H) :=1
‖MH‖∑{HD(M) | M ∈MH}.
Term to TermOperationContinuity
David Clark
If D is the depth of ♦ in t(~x ,♦), then the change in the termt(~x , u(~x)) resulting from applying M = 〈t(~x ,♦), u(~x), v(~x)〉 is
‖M‖ := [t(~x , u(~x)), t(~x , v(~x))] ≤ 1
D.
The resulting change in the term operation on G is
HD(M) := HD(t(~x , u(~x))G, t(~x , v(~x))G).
The average change in term operations of mutations in MH is
µHD(H) :=1
‖MH‖∑{HD(M) | M ∈MH}.
Term to TermOperationContinuity
David Clark
If D is the depth of ♦ in t(~x ,♦), then the change in the termt(~x , u(~x)) resulting from applying M = 〈t(~x ,♦), u(~x), v(~x)〉 is
‖M‖ := [t(~x , u(~x)), t(~x , v(~x))] ≤ 1
D.
The resulting change in the term operation on G is
HD(M) := HD(t(~x , u(~x))G, t(~x , v(~x))G).
The average change in term operations of mutations in MH is
µHD(H) :=1
‖MH‖∑{HD(M) | M ∈MH}.
Term to TermOperationContinuity
David Clark
If D is the depth of ♦ in t(~x ,♦), then the change in the termt(~x , u(~x)) resulting from applying M = 〈t(~x ,♦), u(~x), v(~x)〉 is
‖M‖ := [t(~x , u(~x)), t(~x , v(~x))] ≤ 1
D.
The resulting change in the term operation on G is
HD(M) := HD(t(~x , u(~x))G, t(~x , v(~x))G).
The average change in term operations of mutations in MH is
µHD(H) :=1
‖MH‖∑{HD(M) | M ∈MH}.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark
Definition. A finite groupoid G is term continuous if
limH→∞
µHD(H) = 0.
Theorem. A non-trivial quasigroup is not term continuous.
Theorem. A finite left semigroup is term continuous.
What about random groupoids?
Continuity Theorem. A finite groupoid is term continuous ifit has no subgroupoid with a separating relation and it isasymptotically complete.
Term to TermOperationContinuity
David Clark Definition. A binary relation σ on G is a separating relationif, for all a, b, c ∈ G ,
1 σ 6= ∅,
2 (a, b) ∈ σ implies (b, a) ∈ σ,
3 (a, a) /∈ σ,
4 (a, b) ∈ σ implies (ac, bc) ∈ σ and (ca, cb) ∈ σ.
Note: 6= is a separating relation on a non-trivial quasigroup.
Test for Separating Relations: Either shows there are noseparating relations or produces one. E.g., P has none.
Term to TermOperationContinuity
David Clark Definition. A binary relation σ on G is a separating relationif, for all a, b, c ∈ G ,
1 σ 6= ∅,
2 (a, b) ∈ σ implies (b, a) ∈ σ,
3 (a, a) /∈ σ,
4 (a, b) ∈ σ implies (ac, bc) ∈ σ and (ca, cb) ∈ σ.
Note: 6= is a separating relation on a non-trivial quasigroup.
Test for Separating Relations: Either shows there are noseparating relations or produces one. E.g., P has none.
Term to TermOperationContinuity
David Clark Definition. A binary relation σ on G is a separating relationif, for all a, b, c ∈ G ,
1 σ 6= ∅,
2 (a, b) ∈ σ implies (b, a) ∈ σ,
3 (a, a) /∈ σ,
4 (a, b) ∈ σ implies (ac, bc) ∈ σ and (ca, cb) ∈ σ.
Note: 6= is a separating relation on a non-trivial quasigroup.
Test for Separating Relations: Either shows there are noseparating relations or produces one.
E.g., P has none.
Term to TermOperationContinuity
David Clark Definition. A binary relation σ on G is a separating relationif, for all a, b, c ∈ G ,
1 σ 6= ∅,
2 (a, b) ∈ σ implies (b, a) ∈ σ,
3 (a, a) /∈ σ,
4 (a, b) ∈ σ implies (ac, bc) ∈ σ and (ca, cb) ∈ σ.
Note: 6= is a separating relation on a non-trivial quasigroup.
Test for Separating Relations: Either shows there are noseparating relations or produces one. E.g., P has none.
Term to TermOperationContinuity
David Clark
Let ~d ∈ G k and let a ∈ G . For H ∈ N define
β~d ,a(H) := Prob〈t(~d) = a | t ∈ TH〉.
Note: If a /∈ sg{d0, d1, . . . , dk−1}, then β~d ,a(H) = 0.
Definition. The groupoid G is asymptotically complete if,for each k ∈ N, each ~d ∈ G k and eacha ∈ sg{d0, d1, . . . , dk−1}, the sequence β~d ,a
is eventuallybounded away from 0.
Term to TermOperationContinuity
David Clark
Let ~d ∈ G k and let a ∈ G . For H ∈ N define
β~d ,a(H) := Prob〈t(~d) = a | t ∈ TH〉.
Note: If a /∈ sg{d0, d1, . . . , dk−1}, then β~d ,a(H) = 0.
Definition. The groupoid G is asymptotically complete if,for each k ∈ N, each ~d ∈ G k and eacha ∈ sg{d0, d1, . . . , dk−1}, the sequence β~d ,a
is eventuallybounded away from 0.
Term to TermOperationContinuity
David Clark
Let ~d ∈ G k and let a ∈ G . For H ∈ N define
β~d ,a(H) := Prob〈t(~d) = a | t ∈ TH〉.
Note: If a /∈ sg{d0, d1, . . . , dk−1}, then β~d ,a(H) = 0.
Definition. The groupoid G is asymptotically complete if,for each k ∈ N, each ~d ∈ G k and eacha ∈ sg{d0, d1, . . . , dk−1}, the sequence β~d ,a
is eventuallybounded away from 0.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Test for Asymptotic Completeness.
P distributions with k = 10 and ~d = (4, 3, 0, 3, 1, 3, 3, 0, 2, 3).
H β~d ,0(H) β~d ,1
(H) β~d ,2(H) β~d ,3
(H) β~d ,4(H)
1 0.2 0.1 0.1 0.5 0.1
2 0.172727 0.100000 0.118182 0.318182 0.290909
4 0.374467 0.191126 0.193836 0.088773 0.151799
8 0.288474 0.238744 0.171516 0.120603 0.180662
12 0.292753 0.235886 0.173491 0.121582 0.176288
16 0.292574 0.236026 0.173482 0.121482 0.176436
20 0.292581 0.236024 0.173479 0.121485 0.176431
700 0.292581 0.236024 0.173479 0.121485 0.176431
Experimental evidence that P is asymptotically complete.
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
pp
q an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
pp
q bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
pp
q an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
pp
q bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.
qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
pp
q an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
pp
q bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
pp
q an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
pp
q bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
pp
q an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
pp
q bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
pp
q an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
pp
q bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
Proof of the Continuity Theorem.G: a finite asymptotically complete groupoid with subgroupoidshaving no separating relations. Let n := ‖G‖.
Let ~d ∈ G k , K := sg{d0, d1, . . . , dk−1}, a1 6= b1 in K .
• (a1, b1), (b1, a1) ∈ σ• If (a, b) ∈ σ and c ∈ K , then (ca, cb), (ac, bc) ∈ σ.qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qq q a4QQQ
c4
ppq an2
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qq q b4QQQ
c4
ppq bn2=
No separating relationsimplies there is m ∈ Nsuch that am = bm
with m ≤ n2. Thus
an2 = bn2 .
Term to TermOperationContinuity
David Clark
M = (t(~x ,♦), u(~x), v(~x))
qu(~x)
qq qQQQr1(~x) qq qQQQr2(~x) q qq��� r3(~x)qq qQQQr4(~x)
qt(~x , u(~x))
qv(~x)
qq qQQQr1(~x) qq qQQQr2(~x) q qq��� r3(~x)qq qQQQr4(~x)
qt(~x , v(~x))
Term to TermOperationContinuity
David Clark
M = (t(~x ,♦), u(~x), v(~x))
qu(~x)
qq qQQQr1(~x) qq qQQQr2(~x) q qq��� r3(~x)qq qQQQr4(~x)
qt(~x , u(~x))
qv(~x)
qq qQQQr1(~x) qq qQQQr2(~x) q qq��� r3(~x)qq qQQQr4(~x)
qt(~x , v(~x))
Term to TermOperationContinuity
David Clark
M = (t(~x ,♦), u(~x), v(~x))
↓
All values in K.
pp
qv(~d)
qq q a1
QQQ
c1 qq q a2
QQQ
c2 q qq a3
���
c3qqq a4
c4
an2
qt(~d , u(~d))
pp
qv(~d)
qq q b1
QQQ
c1 qq q b2
QQQ
c2 q qq b3
���
c3qqq b4
c4
bn2
qt(~d , v(~d))
Term to TermOperationContinuity
David Clark
Thanks for listening!
— DC
Term to TermOperationContinuity
David Clark
References
1 D. Clark, Evolution of algebraic terms 1: term to termoperation continuity, International Journal of Algebra andComputation, to appear, 31pp.
2 D. Clark, B. Davey, J.Pitkethly, D.Rifqui, Flat unars: theprimal, the semi-primal and the dualisable , AlgebraUniversalis Vol 63, No. 4 (2010), 303-329.
3 D. Clark, M. Keijzer and L. Spector, Evolution of algebraicterms 2: evolutionary algorithms, in preparation.
4 L. Spector, D. Clark, B. Barr, J. Klein and I. Lindsay,Genetic programming for finite algebras, Genetic andEvolutionary Computation Conference (GECCO) 2008Proceedings, Atlanta GA (July 2008), Editor-in-ChiefMaarten Keijzer, Association for Computing Machinery(ACM), ISBN: 978-1-60558-130-9, 1291-1298.