1

Part 11.Extending Concept of Dependency, as Defined by Permission

2

Review. In a ruleset R, if variable (type) t has a product 1 o 2 … o n

as in t … 1 o 2 … o n …

then we say type t depends on sequence according to R We write this as a production

t R (1 2 … n)or as

t R We also say t depends on each symbol of the sequence

t R i

Review: A type (variable) depends on a sequence of types

1 2 i nt … …

The R “subscript” on and is omitted when obvious from context

3

Suppose SoP ruleset R has production

t R (1 2 … n)

Suppose state s contains triple T = (x0 t xn)

as well as this path (sequence of triples)

= (1 2 … n) = (x0 1 x1, x1 2 x2, …, xn-1 n xn)

then we say T depends on written asT R

We also say triple T depends on each triple i written as

T R i

Extension: A triple (tuple) depends on a sequence of triples

1 2 i nT … …

4

Suppose SoP ruleset R has this productionRi R (V1 V2 … Vm)

Consider these two type sequences:1 = (R1 R2 … Ri … Rn )

2 = (R1 R2 … V1 V2 … Vm … Rn)

In 1, Ri is replaced by (V1 V2 … Vm) to form 2

We say sequence 1 depends on sequence 2, written as:

1 R 2

We can apply transitive closure to : 1 R

+ 2 means there exist a1, a2, … , an such that

1 R a1 R a2 … R an R 2

Extend type dependency so: a type sequence depends on a type sequence

R2 V1 V2 Vm Rn…R1

……

R2 Ri Rn…R1

…

5

Suppose SoP ruleset has productionRi R (V1 V2 … Vm)

Suppose state s contains these triple sequences (paths):1 = (x0 R1 x1 R2 x2 … xi-1 Ri Xi … xn-1 Rn xn )

2 = (x0 R1 x1 R2 x2 … xi-1 V1 y1 V2 y1 … ym-1 Vm Xi … xn-1 Rn xn )

then we say 1 depends on 2, written as:

1 R 2

We can apply transitive closure to : 1 R

+ 2 means there exist A1, A2, … , An such that

1 R A1 R A2 … R An R 2

Extend tuple dependency so: a tuple sequence depends on a tuple sequence

Note analogy to CFG productions and sentences

6

Tuple Length Equals Length of Its Lifted Path of TubesTheorem. In a tubular state, if triple T depends on a

sequence of triples (1 2 … n), that is, if

T (1 2 … n) (Note that each i is a tube of T)

thenLen(T) = Len(1) + Len(2) + … + Len(n)

…

T

1

2

n

Len(x t y) =def min number of P and C edges from x to y

7

Useless Productions

Def. Suppose SoP ruleset has productionR (V1 V2 … Vn)

If for every triple T = (x0 R xn) in every legal state there is no triple sequence = (x0 V1 x1, x1 V2 x2, …, xn-1 Vn xn )

such thatT

then we say the production is useless.

Omit this definition. Instead “useless” applies to constructive states??

8

Legality as DependencyTheorem. State s is legal iff for every triple T in s

there exists triple sequence such thatT

Proof. To prove s is legal, we need to show• Lf(s) is true. We will re-state this successively:• s f(s)• for every T in s, there is W in f(s) such that

V=W• for every T in s, there is such that W is the

composition of the triples in • for every T in s, there is such that T . QED

Recall similar result: s is legal iff every triple of s is legal

9

Def. For a given SoP ruleset, type (variable) t is type recursive, or simply, recursive, if t transitively depends on itself, that is if

t t

Example. t t v o P

v tTypes t and v are both recursive

Recursive Types

10

Def. For a given SoP ruleset, type (variable) t is (type) multi-recursive, if t transitively depends on a sequence that includes more than one instance of t:

t ( … t … t …)If type t is recursive, but not multi-recursive, we say it is

linearly recursive.Example.

t (t P v P)

v (t P)

So, types t and v are both recursive, e.g.,

t (t P t P P)

Multi-Recursive Types

11

Def. For a given SoP ruleset and a given state s, triple T is tuple recursive, or triple recursive, or simply recursive, if T T

Example. t tv v o t

In state s, triple T = (a v a) is recursive because (a v a) a v a)

Triple V = (a v a) is also recursive.

Recursive Triples

t

v

a

Legal state sa v a

a t a

12

Part 12.Phantoms and Recursive Dependency

13

Phantoms Have Recursive Dependency

Theorem. Every phantom state contains a recursive triple T:

T + T

Proof. Assume the contrary (assume a phantom p has no -recursion). Then p can be de-constructed (and then re-constructed) one triple at a time. So p is constructive, which is a contradiction.

14

Properties of Tuple Recursion

Where there is tuple recursion, there exists a cycle of N triples: T0 = (a0 t0 b0),

T1 = (a1 t1 b1),

… TN = (aN tN bN),

that depend successively on the next triple in the cycleT0 T1 … TN T0

15

Properties of Tuple Recursion

When state s is tuple recursive, there exists a triple T0 = (a0 t0 b0) (node a0 is connected by type t0 to node b0) and path of triples such that T0 transitively depends on path T0 +

where T0 is an element of , that is,

T0 + ( … T0 … )

16

Properties of Tuple RecursionWhen state is recursive we have this cycle of dependencies:

T0 E0 T1 F0T1 E1 T2 F1…

Ti Ei Ti+1 Fi…

TN-1 EN-1 TN FN-1TN 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.

EN-1 TN FN-1

EN T0 FN

E0 T1 F0

…