Part 13. Tubular Rulesets

1
Part 13.Tubular Rulesets
2
Definition of Tubular Edges

• Def. For a given state s with rule set R, triple
  E is tubular if for every triple T that E depends
  on, T is a tube of E
• E d T ? E ³? T

S
C
Example
P
T
C
P
?
d
P
E
3
Tubular States, Tubular Rulesets

• Def. State s is tubular if all its triples are
  tubular.
• Def. A ruleset is tubular if all its legal states
  are tubular.
• Most example SoP rules given above are tubular.
• There is no known algorithm to determine if
  rulesets are tubular.

4
Constructive and Phantomic Tubular Systems

• Constructive ruleset
• M D (P M C)
• M D (S)

• Phantomic ruleset
• M1 D (P M2 C)
• M2 D (M3)
• M3 D (M4)
• M4 D (M2)

All states are constructive. Example state
Example tubular state (phantom)
dSe
dM4e
dM3e
d
e
dMe
dM2e
fPd
eCg
fPd
eCg
fMg
f
g
fM1g
5
Another Example of a Phantom in a Tubular System

• Ruleset
• M1 D (P M2 C)
• M2 D (M3)
• M3 D (L1 M4)
• M4 D (M2 R1)
• L1 D (L2)
• L2 D (L3 L4)
• L3 D (ID L4)
• L4 D (L1)
• R1 D (R2)
• R2 D (R2)

dIDd


dL3d
eR2e
dL4d
dL2d
dL1d
eR1e
dM4e
dM3e
dM2e
fPd
eCg
fM1g
Legal sub states, which are phantoms.
6
Part 14.Identic Rulesets and Identic States
7
Identic Rulesets Introduction
Vr . dr
vr

• Six productions of SoP ruleset R
• t1 D (t3)
• t2 D (t3)
• t2 D (t4)
• t3 D (t4 t5)
• t4 D (t5 ID)
• t5 D (t3)
• Ruleset R is identic if it has a sub ruleset r
  such that
• vr vr . dr ID
• Example. t3, t4, t5 t3, t4, t5, ID ID
• Surprise Being identic does not imply phantoms!
  But it does imply hidden phantoms.

t5
ID
t4
t3
t2
t1
Notation needs work, as here ID is used as
triples with ID edges??
8
Identic Rulesets Formal Definition

• Def. SoP ruleset R is said to be identic if
  there is sub ruleset r, r ? R, such that
• vr vr . dr - ID
• where vr is rs set of variables.
• That is, R is identic if the set of variables
  depended upon according to r by vr variables are
  exactly vr when ignoring ID constants.

Example
vr . dr
vr
v3
ID
v2
v1
t2
t1
Should give algorithm to test to decide if R is
identic??
Here, ID is a set containing the name ID of
self-loops.
9
Identic States Example

• Suppose state s has these dependencies among
  triples V1 to V5.
• V1 ? (V3)
• V2 ? (V3)
• V2 ? (V4)
• V3 ? (V4, V5)
• V4 ? (V5, ID)
• V5 ? (V3)
• State s is identic because there is a non-empty
  subset of triples t V3, V4, V5 with
  dependencies such that
• t t . dr ID
• where t . dr V3, V4, V5, ID

t . dr
t
V5
ID
V4
V3
V2
V1
Here ID is the set of triples that are self
loops.
10
Identic States Formal Definition

• Def. Consider SoP ruleset R with sub rle set r, r
  ? R, with legal state s with non-empty sub state
  t, t ? s. If
• t t . dr - ID
• we say state s is identic.
• So, state s is identic if the set of triples
  depended upon according to r by triples in t are
  exactly t when ignoring ID constants.

Example
s
t
V3
ID
V2
V1
T2
T1
Here ID is the set of triples that are self
loops.
11
Identic Rules iff Identic Legal States

• Theorem An SoP ruleset is identic iff it has a
  legal state that is identic.
• Proof
• Ruleset R is identic ? Legal identic state s
  consisting of ID triples of form (a vi a) for
  each variable in subset rule r.
• Legal state s is identic ? Productions v D (R1,
  R2, , Rn) for each dependency in s of form
• (x0 v xn) ?
• (x0 R1 x1, x1 R2 x2, , xn-1 Rn xn )

v5
v4
v3
v2
v1
Somewhere did I assume that identic states are
all legal??
12
Identic Rules Sets Permit Trivial Looping States

• Theorem. If a ruleset is identic, it has a legal
  state all of whose edges are ID loops.
• Example.
• T ? T
• U ? U o T

Legal state which is identic
T
a
U
Should say if s is identic phantom then R is
identic ruleset??
13
Part 15.Tubular Phantoms are Identic
14
Recap Tubular Rules Sets Can Have Identic
Phantoms

• Ideally, tubular rules sets should have no
  phantoms
• However, with cyclic patterns of dependencies,
  tubular states and rulesets can permit phantoms.
• We want to be able detect when these phantoms
  exist in a tubular ruleset.
• We will prove what seems obvious, namely that
  tubular states can be phantoms only if they are
  identic.

15
Tubular Phantoms are Identic

• Theorem. Given a tubular state s,
• s is a phantom ? s is identic
• Proof. given below
• Corollary. Given a tubular state s,
• s is non-identic ? s is constructive
• Corollary. Tubular non-identic rulesets are
  constructive.
• There is an obvious algorithm to test if a
  ruleset is identic, but no known algorithm to
  test if a ruleset is tubular.

16
Part 1 Proof Tubular Phantoms Are
Identic.Phantoms Have Cycle of Dependency
Any phantom state s is recursive, and has a cycle
of dependencies T0 D (E0 T1 F0) T1 D
(E1 T2 F1) Ti D (Ei Ti1
Fi) TN-1 D (EN-1 TN FN-1) TN D (EN
T0 FN) where we are using the convention that Ei
and Fi are sequences (really, paths) of triples.
Note that if (a V b) and (c W d) are successive
triples in a path of triples, then necessarily
nodes b and c are identical b c.
17
Part 2 Proof Tubular Phantoms Are
Identic.Triples in basic recursion have same
lengths
Since state s is a phantom, there exists a
non-empty sequence of triples (T0 T1 TN) such
that T0 d T1 d TN d T0 Recall that if tubular
triple V depends on triple W (if V d W) then
Len(V) Len(W) so Len(T0) Len(T1)
Len(TN) Len(T0) Hence Len(T0) Len(T1)
Len(TN) In other words, since s is tubular
All triples, T0 to TN, have the same length.
18
Part 3 Proof Tubular Phantoms Are
Identic.Triples depended on by base recursion
are IDs
We have this pattern of dependency among tuples
Ti D (Ei Ti1 Fi) which is short for Ti
D (Ei,1 Ei,2 Ti1 Fi,1 Fi,2 ) Each triple
Ti depends on a sequence of triples consisting of
(1) the triples in Ei then (2) triple Ti1 and
finally (3) the triples in Fi. Recall that if
tubular triple V depends on tuple sequence (W1 W2
... Wk) then Len(V) Len(W1) Len(W2)
Len(Wk), so Len(Ti) Len(Ei,1)Len(Ei,2)
Len(Ti1) Len(Fi,1) Len(Fi,2) Since we
have already determined that Len(Ti) Len(Ti1),
it follows that for all i, j Len(Ei,j) 0
and Len(Fi,j) 0
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Part 4 Proof Tubular Phantoms Are
Identic.Triples depended on by Ei and Fi are IDs.
If tubular triple V has length zero and V depends
on triple W directly or directly, then the length
of W must also be zero Len(V) 0 ? V d W
? Len(W) 0 Since for all i and j, Len(Ei,j)
Len(Fi,j) 0, it follows that All triples
that Ei,j or Fi,j depend on transitively have
length zero and hence are ID triples.
20
Part 5 Proof Tubular Phantoms Are Identic.
Construction of Sub-States si and Sub-Rulesets ri
In the sequence (T0 T1 TN) each Ti depends on
Ti1 (where TN1 is T0) Ti DR (Ei Ti1
Fi) with corresponding production from R
ti DR (ei ti1 fi) Def. s0 T0, T1, TN
r0 Set of productions for Ti from R
ti DR (ei ti1 fi) Def. si1 si . dri
(Compute sis targets) ri1 union of
ri and set of productions
corresponding to dependencies for T DR
? for each T in si1 - si
Example
Note si ? si1
Since T is legal there must exist tuple sequence
? in s such that T DR ?
21
Part 6 Proof Tubular Phantoms Are
Identic.Triples depended on recursively by Ti
are IDs.
Example
We previously showed that all triples in Ei and
Fi and all triples transitively depended upon by
them are limited to be ID triples. We observe
that each Ti can depend (according to r0) only on
Ti1 or on triples in Ei and Fi. Therefore
Any constant triple depended upon transitively by
any Ti is necessarily an ID constant.
22
Part 7 Proof Tubular Phantoms Are
Identic.Induction Hypothesis
We define a hypothesis Hi, i ³ 0, as follows
Hi def si ? si . dri We will show that
Hi is true for all i ³ 0. We start by proving
that H0 is true. By definition s0 T0, T1,
, TN We conclude that s0 ? s0 . dri because
every triple in s0 is depended upon by a triple
in s0 . For example T1 depends on T2. Hence H0
is true.
23
Part 8 Proof Tubular Phantoms Are
Identic.Inductive Proof of Hypothesis Hi
We will prove that, for all i ³ 0, Hi is true,
i.e., si ? si . dri We have already shown
that H0 is true. We will use induction to prove
that Hi is true for all i gt 0, by assuming Hi is
true and showing that consequently Hi1 must also
be true. Assuming Hi is true, then every triple
in si is depended upon according to ri by at
least one other triple in si. Now consider any
triple that si depended upon according to ri by a
triple in si1, but is not in si. Any such
triple is clearly depended upon by a triple in
si. Since si1 consists of such triples along
with triples already in si, it follows that every
triple in si1 is depended upon according to ri
by at least one triple in si1. Hence For all
i ³ 0, Hi is true.
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Part 9 Proof Tubular Phantoms Are
Identic.Conclusion of proof
Both series s0, s1, and r0, r1, are
monotonically increasing in size. Both are
limited in size, si by s and ri by R. It follows
there is a limiting value, call it L, after which
all states, sL, si1, and all rulesets ri, ri1
, are identical. Hence, for i ³ L, there
remain no triples outside of si that are depended
upon according to ri by triples in si, but are
not members of si. Hence, sL is the same sL. dr
except for constant triples in sL. dr so sL
sL. dr - CONST We claim that for i ³ L si
si. dr - ID This must be true because we have
previously established that every constant triple
depended upon directly or indirectly by any Ei,
Fi or Ti is an ID constant. Since si is a subset
of state s, it follows that phantom tubular state
s is identic. QED