User:Ganaram inukshuk/Notes: Difference between revisions

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# Replacement ruleset 1 (where L - s > s)

#* L -> Ls

#* s-> s

#* s -> s

# Replacement ruleset 2 (where L - s < s)

#* L -> sL

#* s -> L

It should be noted that if the order of L's and s's is reversed, the rulesets are still valid. The numbering of rulesets is also arbitrary. For ~~simplicity~~, rulesets 1 and 2 are denoted as though they were sisters of one another ~~(that is, an additional "zeroth"~~ ruleset ~~is applied where~~ L->s ~~and~~ s-> L).

It should be noted that if the order of L's and s's is reversed (for example, L->sL and s->s for ruleset 1), the rulesets are still valid. The numbering of rulesets is also arbitrary. For explanation purposes, rulesets 1 and 2 and successive pairs of rulesets are denoted as though they were sisters of one another; this [[Operations on MOSes|sistering process]] can be described with its own ruleset:

* L->s

* s->L

Applying ruleset 1 to itself n times produces ruleset 3, where L produces an L followed by n s's:

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* s->s

~~Applying~~ ruleset 1 to itself n times ~~produces a variant of ruleset 3 where~~ L ~~produces~~ an L and n-1 s's. If ruleset 2 is applied to this, it creates ruleset 4, where L produces an s followed by n L's, the sister of ruleset 3:

As such, applying ruleset 1 to itself n-1 times will result in L producing an L and n-1 s's. If ruleset 2 is applied to this, it creates ruleset 4, where L produces an s followed by n L's, the sister of ruleset 3:

* L->sLL...LL (n L's)

* s->L

Reversing the L's and s's of ruleset 2 produces this intermediate ruleset:

Reversing the order of L's and s's of ruleset 2 produces this intermediate ruleset:

* L->Ls

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The final rulesets are as follows:

# ~~Ruleset~~ 1

# Replacement ruleset 1

#* L -> Ls

#* s-> s

# ~~Ruleset~~ 2

# Replacement ruleset 2

#* L -> sL

#* s -> L

# ~~Ruleset~~ 3

# Replacement ruleset 3

#* L->Lss...ss (n s's)

#* s->s

# ~~Ruleset~~ 4

# Replacement ruleset 4

#* L->sLL...LL (n L's)

#* s->L

# ~~Ruleset~~ 5

# Replacement ruleset 5

#* L->Lss...ss (n+1 s's)

#* s->Lss...s (n s's)

# ~~Ruleset~~ 6

# Replacement ruleset 6

#* L->sLL...LL (n+1 L's)

#* s->sLL...L (n L's)

The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on ~~any arbitrary~~ scale ~~of L's and s's reduces~~ the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:

The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on a valid moment-of-symmetry scale will reduce the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:

* Reduction ruleset 5

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** sLL...L (n L's) -> s

However, it may be the case that the reduced scale has only one L or one s, or that the scale started out ~~that~~ way. In either case, rulesets 3 and 4 can be used instead:

However, it may be the case that the reduced scale has only one L or one s, or that that the scale started out this way. In either case, rulesets 5 and 6 cannot be used, but rulesets 3 and 4 can be used instead:

* Reduction ruleset 3

@@ Line 8: / Line 8: @@
 # Replacement ruleset 1 (where L - s > s)
 #* L -> Ls
-#* s-> s
+#* s -> s
 # Replacement ruleset 2 (where L - s < s)
 #* L -> sL
 #* s -> L
-It should be noted that if the order of L's and s's is reversed, the rulesets are still valid. The numbering of rulesets is also arbitrary. For simplicity, rulesets 1 and 2 are denoted as though they were sisters of one another (that is, an additional "zeroth" ruleset is applied where L->s and s-> L).
+It should be noted that if the order of L's and s's is reversed (for example, L->sL and s->s for ruleset 1), the rulesets are still valid. The numbering of rulesets is also arbitrary. For explanation purposes, rulesets 1 and 2 and successive pairs of rulesets are denoted as though they were sisters of one another; this [[Operations on MOSes|sistering process]] can be described with its own ruleset:
+* L->s
+* s->L
 Applying ruleset 1 to itself n times produces ruleset 3, where L produces an L followed by n s's:
@@ Line 20: / Line 23: @@
 * s->s
-Applying ruleset 1 to itself n times produces a variant of ruleset 3 where L produces an L and n-1 s's. If ruleset 2 is applied to this, it creates ruleset 4, where L produces an s followed by n L's, the sister of ruleset 3:
+As such, applying ruleset 1 to itself n-1 times will result in L producing an L and n-1 s's. If ruleset 2 is applied to this, it creates ruleset 4, where L produces an s followed by n L's, the sister of ruleset 3:
 * L->sLL...LL (n L's)
 * s->L
-Reversing the L's and s's of ruleset 2 produces this intermediate ruleset:
+Reversing the order of L's and s's of ruleset 2 produces this intermediate ruleset:
 * L->Ls
@@ Line 42: / Line 45: @@
 The final rulesets are as follows:
-# Ruleset 1
+# Replacement ruleset 1
 #* L -> Ls
 #* s-> s
-# Ruleset 2
+# Replacement ruleset 2
 #* L -> sL
 #* s -> L
-# Ruleset 3
+# Replacement ruleset 3
 #* L->Lss...ss (n s's)
 #* s->s
-# Ruleset 4
+# Replacement ruleset 4
 #* L->sLL...LL (n L's)
 #* s->L
-# Ruleset 5
+# Replacement ruleset 5
 #* L->Lss...ss (n+1 s's)
 #* s->Lss...s (n s's)
-# Ruleset 6
+# Replacement ruleset 6
 #* L->sLL...LL (n+1 L's)
 #* s->sLL...L (n L's)
-The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on any arbitrary scale of L's and s's reduces the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:
+The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on a valid moment-of-symmetry scale will reduce the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:
 * Reduction ruleset 5
@@ Line 70: / Line 73: @@
 ** sLL...L (n L's) -> s
-However, it may be the case that the reduced scale has only one L or one s, or that the scale started out that way. In either case, rulesets 3 and 4 can be used instead:
+However, it may be the case that the reduced scale has only one L or one s, or that that the scale started out this way. In either case, rulesets 5 and 6 cannot be used, but rulesets 3 and 4 can be used instead:
 * Reduction ruleset 3