The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic Weight Calculus of Subversion
Mike Fisher,
West Chester University,
with N. McKay, R. J. Nowakowski, P. Ottaway, and C. Santos
January 26, 2017
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Table of contents
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
The game
Subversion is a partizan self-referential subtraction game played on
pairs of non-negative integers, (a, b), dictated by the following
rules:
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
The game
Subversion is a partizan self-referential subtraction game played on
pairs of non-negative integers, (a, b), dictated by the following
rules:
▸
If a = 0 or b = 0, then the position (a, b) is terminal.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
The game
Subversion is a partizan self-referential subtraction game played on
pairs of non-negative integers, (a, b), dictated by the following
rules:
▸
If a = 0 or b = 0, then the position (a, b) is terminal.
▸
If a > 0 and b > 0, then
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
The game
Subversion is a partizan self-referential subtraction game played on
pairs of non-negative integers, (a, b), dictated by the following
rules:
▸
If a = 0 or b = 0, then the position (a, b) is terminal.
▸
If a > 0 and b > 0, then
▸
if a ≥ b, then Left can move to (a, 0); if a < b, then Left can
move to (a, b − a);
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
The game
Subversion is a partizan self-referential subtraction game played on
pairs of non-negative integers, (a, b), dictated by the following
rules:
▸
If a = 0 or b = 0, then the position (a, b) is terminal.
▸
If a > 0 and b > 0, then
▸
▸
if a ≥ b, then Left can move to (a, 0); if a < b, then Left can
move to (a, b − a);
if b ≥ a, then Right can move to (0, b); if a > b, then Right
can move to (a − b, b).
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
An example
Consider (25, 9):
(25, 9)
b
(25, 0)
b
(16, 9)
b
(16, 0)
b
(7, 2)
(7, 0)
(7, 9)
b
(0, 9)
b
b
(3, 2)
(5, 2)
b
(5, 0)
b
b
b
(3, 0)
b
(1, 1)
(1, 0)
Mike Fisher
(1, 2)
b
b
b
b
b
(0, 2)
(0, 1)
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
An example
Notice that
25
9
has continued fraction representation 2 +
Using the short-hand notation, we write [2, 1, 3, 2].
1+
1
1
3+ 1
2
b
b
b
b
b
b
b
b
b
Mike Fisher
Atomic Weight Calculus of Subversion
.
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
An example
Notice that
25
9
has continued fraction representation 2 +
Using the short-hand notation, we write [2, 1, 3, 2].
1
3+ 1
2
b
(16, 9)
b
(16, 0)
b
(7, 9)
b
(7, 2)
(7, 0)
1
(25, 9)
b
(25, 0)
1+
(0, 9)
b
b
(3, 2)
(5, 2)
b
(5, 0)
b
b
b
(3, 0)
(1, 1)
(1, 0)
Mike Fisher
b
(1, 2)
b
(0, 1)
b
b
b
b
(0, 2)
Atomic Weight Calculus of Subversion
.
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
An example
Adding negatives for left-hand branching, we utilize the more
descriptive notation: (25, 9) = [2, −1, 3, −2].
(25, 9)
b
(25, 0)
b
(16, 9)
b
(16, 0)
b
(7, 2)
(7, 0)
(7, 9)
b
(0, 9)
b
b
(3, 2)
(5, 2)
b
(5, 0)
b
b
b
(3, 0)
b
(1, 1)
(1, 0)
Mike Fisher
(1, 2)
b
b
b
b
b
(0, 2)
(0, 1)
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
An example
The Subversion position (25, 9) has value {0∣∣∣0∣∣ ⇑ ∗∣0}.
(25, 9)
b
(25, 0)
b
(16, 9)
b
(16, 0)
b
(7, 2)
(7, 0)
(7, 9)
b
(0, 9)
b
b
(3, 2)
(5, 2)
b
(5, 0)
b
b
b
(3, 0)
b
(1, 1)
(1, 0)
Mike Fisher
(1, 2)
b
b
b
b
b
(0, 2)
(0, 1)
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Hackenbush variation
The Subversion position (25, 9) = [2, −1, 3, −2] can be thought of
as the following Hackenbush variation.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Hereditary structure of Subversion
The class of Subversion forms is constructed as follows:
▸
▸
0 = { ∣ } is the only Subversion game born on day 0;
The Subversion games born on day n + 1 have the possible
forms {0 ∣ G R } or {H L ∣ 0} where G R and H L are Subversion
games born on day n.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Motivation/History
▸
An impartial self-referential subtraction game is a
subtraction game where the sizes of the (non-empty) piles of
tokens in a given position form the subtraction set. The
subtraction set is dynamic, changing with every move.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Motivation/History
▸
▸
An impartial self-referential subtraction game is a
subtraction game where the sizes of the (non-empty) piles of
tokens in a given position form the subtraction set. The
subtraction set is dynamic, changing with every move.
Impartial Susen is an example of such a game; from a position
of heaps with sizes 1, 3, and 4, one can remove 1, 3, or 4
tokens. For instance, removing one token from heap two leaves
the position {1, 2, 4}. The new subtraction set is now {1, 2, 4}.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Motivation/History
▸
In Partizan Susen, we are given two sets SL = {a1 , a2 , . . . } and
SR = {b1 , b2 , . . . }. For any i, Left can remove ai tokens from
any bj provided that ai ≤ bj . Right’s moves are similar.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Motivation/History
▸
▸
In Partizan Susen, we are given two sets SL = {a1 , a2 , . . . } and
SR = {b1 , b2 , . . . }. For any i, Left can remove ai tokens from
any bj provided that ai ≤ bj . Right’s moves are similar.
R. J. Nowakowski proposed the ruleset Subversion as a variant
of Partizan Susen while he was trying to generate many
self-referential games.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight
Recall that if G is an infinitesimal, then
n⋅ ↓< G < n⋅ ↑
for some sufficiently large n. Hence, one can use the sequence
↑, ⇑, ↑ 3, . . .
as a scale to calibrate the size of any infinitesimal.
The atomic weight of G describes precisely where G should be
placed on this scale.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Remote stars
Consider the following flower garden.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Remote stars
In the presence of a sufficiently remote nimber (∗5 is the “smallest”
such nimber here), the position becomes a Left win. This position
has atomic weight 1.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Remote stars
As it turns out, if ∗k is remote for a given position, then so is any
∗m, where m ≥ k. It is convenient to adopt the symbol ☆ as
shorthand for a sufficiently remote nimber. We generally refer to
☆ as a remote star (or far star).
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight criterion
aw (G ) = g if and only if ↓ ☆ < G − g ⋅ ↑<↑ ☆.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Two ahead rule
1. If aw (G ) ⩾ 2, then G > 0.
2. If aw (G ) ⩾ 1, then G ⊳ 0.
3. If aw (G ) ⩽ −2, then G < 0.
4. If aw (G ) ⩽ −1, then G ⊲ 0.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight calculus
Let G be dicotic. Consider the form {aw (G L ) − 2 ∣ aw (G R ) + 2}
and let w (G ) be
▸
▸
w (G ) = {aw (G L ) − 2 ∣ aw (G R ) + 2} if the form is not a integer.
⎧
min(I ) if G < ☆
⎪
⎪
⎪
w (G ) = ⎨ max(I ) if G > ☆
if the form is a integer.
⎪
⎪
⎪
0
otherwise
⎩
where I = {x ∈ Z ∣ aw (G L ) − 2 ⊲ x ⊲ aw (G R ) + 2}. Then,
aw (G ) = w (G ).
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Why not just work out canonical forms for Subversion positions?
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Why not just work out canonical forms for Subversion positions?
Consider a disjunctive sum like (211, 155) + (5, 4) + (13, 17).
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Why not just work out canonical forms for Subversion positions?
Consider a disjunctive sum like (211, 155) + (5, 4) + (13, 17).
After simplifying this position with help of CGSuite, we get the
following “interesting” form:
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Why not just work out canonical forms for Subversion positions?
Consider a disjunctive sum like (211, 155) + (5, 4) + (13, 17).
After simplifying this position with help of CGSuite, we get the
following “interesting” form:
{{03 ∣ ↓ 3∣∣0∣∣∣ ↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3}, {03 ∣ ↓ 3∣∣0∣∣∣{03 ∣ ↓
3∣∣0∣∣∣{0∣∣∣03 ∣ ↓ 3∣∣0∣∣∣∣0, {03 ∣ ↓ 3∣∣0}}∣0∣∣∣∣02 }}∣{{03 ∣ ↓ 3∣∣0∣∣∣{0∣∣∣03 ∣ ↓
3∣∣0∣∣∣∣0, {03 ∣ ↓ 3∣∣0}}∣0∣∣∣∣02 }∣∣∣∣ ↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3∣∣{∗, ↓
∣∣ ↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3∣∣∣ ↓ 3, {↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓
6}∣ ↓ 3}∣∣∣∣ ↓ 3}∣∣∣02 }, {↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3∣∣∣∣ ↓, {↓, ⇓
∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3∣∣{∗, ↓ ∣∣ ↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3∣∣∣ ↓
3, {↓, {↓, ⇓ ∗∣∣∣ ⇓ ∗, ↓ 3∣∣ ↓ 3∣ ↓ 6}∣ ↓ 3}∣∣∣∣ ↓ 3}∣∣∣02 }}, where {0k ∣ G }
means {0 ∥ {0k−1 ∣ G }}.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Before we present our main result, we need just a tiny bit more
preparation.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Before we present our main result, we need just a tiny bit more
preparation.
▸
We say that G has adorned atomic weight 0L (written as
aw (G ) = 0L ) if
aw (G ) = 0, aw ({0 ∣ G }) = 1, and {G ∣ 0} = ∗.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
Before we present our main result, we need just a tiny bit more
preparation.
▸
We say that G has adorned atomic weight 0L (written as
aw (G ) = 0L ) if
aw (G ) = 0, aw ({0 ∣ G }) = 1, and {G ∣ 0} = ∗.
▸
Similarly, we say that G has adorned atomic weight 0R
(written as aw (G ) = 0R ) if
aw (G ) = 0, aw ({G ∣ 0}) = −1, and {0 ∣ G } = ∗.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Atomic weight of Subversion
The Subversion position (9, 4) is an example of a position with
atomic weight 0L .
02 | ↓ 3
b
0
b
0| ↓ 3
b
0
b
⇓∗
↓
∗
0
b
↓3
b
b
b
b
b
b
b
b
Mike Fisher
0
0
0
0
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Main result
Theorem
If G is a Subversion position, then the following table can be used
to compute the atomic weight of {0 ∣ G }.
aw (G )
k ⩽ −4
−3
−2
− 23
{−2 ∣ k}
aw ({0 ∣ G })
{−2 ∣ k + 2}
− 32
−1
0L
0L
aw (G )
aw ({0 ∣ G })
G =0
{0 ∣ G } = ∗
G =∗
{0 ∣ G } =↑
Mike Fisher
0L
1
−1
If G =↓ then {0 ∣ G } = ∗
If G = {G L ∣ 0} then − 1
If G = {0 ∣ G R } then 0L
1
2
3
2
3
2
3
3
4
{k ∣2}
k +2
0R
{0 ∣ G } = ∗
k ⩾4
k +1
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Main result
Theorem
(Mirror image version) If G is a Subversion position, then the
following table can be used to compute the atomic weight of
{G ∣ 0}.
aw (G )
aw ({G ∣ 0})
k ⩽ −4
k −1
aw (G )
G =0
G =∗
0L
aw ({G ∣ 0})
∗
{G ∣ 0} =↓
∗
−3
−4
−2
−3
− 32
−3
{−2 ∣ k}
k −2
1
If G =↑ then {G ∣ 0} = ∗
If G = {G L ∣ 0} then 0R
If G = {0 ∣ G R } then 1
Mike Fisher
−1
−2
0R
−1
3
2
2
3
{k ∣2}
k ⩾4
0R
1
3
2
0R
{k − 2 ∣ 2}
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
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Further directions
Atomic weight of Subversion
Returning to the Subversion position (9, 4) we see how to apply the
theorem
0L
b
0
b
−3/2
b
0
b
−2
↓
∗
0
b
−3
b
b
b
b
b
b
b
b
Mike Fisher
0
0
0
0
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Algorithm for computing aw(a, b)
Consider E , the possible entries of the table shown in theorem. The
classification of the turning points motivates the partitions
E = L1 ⊔ L2 ⊔ L3 = R1 ⊔ R2 ⊔ R3 , where
▸
▸
L1 = {. . . , −3, −2, −1}, L2 = {− 32 , {−2 ∣ k}, ↓, 0, 0R }, L3 =
{0L , ∗, 1, 32 , 2, . . . , {k ∣ 2}}, and
R1 = {1, 2, 3, . . .}, R2 = { 23 , {k ∣ 2}, ↑, 0, 0L }, R3 =
{{−2 ∣ k}, . . . , −2, − 32 , −1, ∗, 0R }.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Algorithm for computing aw(a, b)
Two functions which we’ll need are the ceiling function
⌈g ⌉ = minn∈Z {n ⩾ g } and the floor function ⌊g ⌋ = maxn∈Z {n ⩽ g }.
Note that for the special case when g = ∗, we define ⌈∗⌉ = ⌊∗⌋ = 0.
We will also need the following two functions when n ⩾ 2:
⎧
n − 2 if g ∈ L1
⎪
⎪
⎪
ψL (n, g ) = ⎨ n − 1 if g ∈ L2 ,
⎪
⎪
⎪
⎩ n + ⌈g ⌉ if g ∈ L3
adorning 0L if the result is 0 and
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Algorithm for computing aw(a, b)
⎧
2 − n if g ∈ R1
⎪
⎪
⎪
ψR (n, g ) = ⎨ 1 − n if g ∈ R2 ,
⎪
⎪
⎪
⎩ ⌊g ⌋ − n if g ∈ R3
adorning 0R if the result is 0.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Algorithm for computing aw(a, b)
Finally, consider the class of objects
Υ = { [n1 , −n2 , n3 , −n4 , . . .] ∶ g ∣ ni ∈ Z+ and g ∈ E }
and the function ξ ∶ Υ → Υ defined by the following rules
1. ξ([ ⋅ ] ∶ g ) = [ ⋅ ] ∶ g
2.
3.
⎧
[n , . . . , −nk−1 ] ∶ aw ({0∣G })† if nk = 1
⎪
⎪
⎪ 1
ξ([n1 , −n2 , . . . , nk ] ∶ g ) = ⎨ [n1 , . . . , −nk−1 ] ∶↑
if nk = 2 and (g = 0 or g = 0R or g =↓)
⎪
⎪
⎪
otherwise
⎩ [n1 , . . . , −nk−1 ] ∶ ψL (nk , g )
⎧
[n , . . . , nk−1 ] ∶ aw ({G ∣0})† if nk = 1
⎪
⎪
⎪ 1
ξ([n1 , −n2 , . . . , −nk ] ∶ g ) = ⎨ [n1 , . . . , nk−1 ] ∶↓
if nk = 2 and (g = 0 or g = 0L or g =↑)
⎪
⎪
⎪
[n
]
,
.
.
.
,
n
∶
ψ
(n
,
g
)
otherwise
k−1
R k
⎩ 1
(Note that † means to use the table in the main theorem.)
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Example
Let’s first compute the atomic weight of (25, 9). Recall that the
signed continued fraction representation is [2, −1, 3, −2]. Using the
algorithm presented in the preceding slides, we perform the
following calculations
ξ
ξ
[2, −1, 3, −2] ∶ 0 Ð→ [2, −1, 3] ∶↓Ð→ [2, −1] ∶ ψL (3, ↓) = [2, −1] ∶ 2
ξ
ξ
Ð→ [2] ∶ 1 Ð→ [ ] ∶ ψL (2, 1) = [ ] ∶ 3
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Example
Let’s first compute the atomic weight of (25, 9). Recall that the
signed continued fraction representation is [2, −1, 3, −2]. Using the
algorithm presented in the preceding slides, we perform the
following calculations
ξ
ξ
[2, −1, 3, −2] ∶ 0 Ð→ [2, −1, 3] ∶↓Ð→ [2, −1] ∶ ψL (3, ↓) = [2, −1] ∶ 2
ξ
ξ
Ð→ [2] ∶ 1 Ð→ [ ] ∶ ψL (2, 1) = [ ] ∶ 3
Hence, aw (25, 9) = 3.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Example
Next, we compute the atomic weight of (37, 24). The signed
continued fraction representation is [1, −1, 1, −5, 2]. Using the
algorithm presented in the preceding slides, we perform the
following calculations
ξ
ξ
[1, −1, 1, −5, 2] ∶ 0 Ð→ [1, −1, 1, −5] ∶↑Ð→ [1, −1, 1] ∶ ψR (5, ↑)
ξ
ξ
ξ
= [1, −1, 1] ∶ −4 Ð→ [1, −1] ∶ −2∗ Ð→ [1] ∶ −4 Ð→ [ ] ∶ −2∗
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Example
Next, we compute the atomic weight of (37, 24). The signed
continued fraction representation is [1, −1, 1, −5, 2]. Using the
algorithm presented in the preceding slides, we perform the
following calculations
ξ
ξ
[1, −1, 1, −5, 2] ∶ 0 Ð→ [1, −1, 1, −5] ∶↑Ð→ [1, −1, 1] ∶ ψR (5, ↑)
ξ
ξ
ξ
= [1, −1, 1] ∶ −4 Ð→ [1, −1] ∶ −2∗ Ð→ [1] ∶ −4 Ð→ [ ] ∶ −2∗
Hence, aw (37, 24) = −2∗.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Example
Lastly, we compute the atomic weight of (2017, 26). The signed
continued fraction representation is [77, −1, 1, −2, 1, −3]. Using the
algorithm presented in the preceding slides, we perform the
following calculations
ξ
[77, −1, 1, −2, 1, −3] ∶ 0 Ð→ [77, −1, 1, −2, 1] ∶ ψR (3, 0)
ξ
ξ
= [77, −1, 1, −2, 1] ∶ −2 Ð→ [77, −1, 1, −2] ∶ −1 Ð→ [77, −1, 1] ∶ ψR (2, −1)
ξ
ξ
ξ
= [77, −1, 1] ∶ −3 Ð→ [77, −1] ∶ −3/2 Ð→ [77] ∶ −3 Ð→ [ ] ∶ ψL (77, −3)
= [ ] ∶ 75
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Example
Lastly, we compute the atomic weight of (2017, 26). The signed
continued fraction representation is [77, −1, 1, −2, 1, −3]. Using the
algorithm presented in the preceding slides, we perform the
following calculations
ξ
[77, −1, 1, −2, 1, −3] ∶ 0 Ð→ [77, −1, 1, −2, 1] ∶ ψR (3, 0)
ξ
ξ
= [77, −1, 1, −2, 1] ∶ −2 Ð→ [77, −1, 1, −2] ∶ −1 Ð→ [77, −1, 1] ∶ ψR (2, −1)
ξ
ξ
ξ
= [77, −1, 1] ∶ −3 Ð→ [77, −1] ∶ −3/2 Ð→ [77] ∶ −3 Ð→ [ ] ∶ ψL (77, −3)
= [ ] ∶ 75
Hence, aw (2017, 26) = 75.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Further directions
▸
Given an all-small ruleset, is there any relationship between the
number of eccentric cases in the atomic weight calculus being
finite and the nim dimension of the ruleset?
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
Further directions
▸
▸
Given an all-small ruleset, is there any relationship between the
number of eccentric cases in the atomic weight calculus being
finite and the nim dimension of the ruleset?
Analyze the full version of Subversion. Positions in this
generalization are ordered (n + m)-tuples,
(p1 , p2 , . . . , pn ; q1 , q2 , . . . , qm ).
A Left move is to choose some pi and subtract it from some
qj , etc. Right’s moves are similar.
Mike Fisher
Atomic Weight Calculus of Subversion
The game
An example
Hereditary structure of Subversion
Motivation/History
Atomic weight
Atomic weight of Subversion
Examples
Further directions
The End
Mike Fisher
Atomic Weight Calculus of Subversion