midterm fall 2000 solutions. question 1 from sentence to class dictionary abstraction in this case...

Midterm Fall 2000

solutions

Question 1

• From sentence to class dictionary abstraction

• In this case like from object graph to class graph abstraction because in XML the sentences look like object graphs.

• sentences: – <xyz:PurchaseOrder>– <PurchaseOrder>

• Need an optional part

Question 1

• sentences: – <xyz:PurchaseOrder>– <PurchaseOrder>

• Class dictionary:– “<“ [<first> ObjectId “:”] “PurchaseOrder>” …– ObjectId = Ident.– “<“ [“xyz:”] “PurchaseOrder>” … does not work

because absent part not represented in object.

Question 1

• sentence: – <CustomerName> “Henry Ford” </CustomerName>

• class dictionary:– “<CustomerName>” String “</CustomerName>”

Question 1

• sentence: <Order>

<Product> “Apple” </Product>

<Price> 1.50 </Price>

<Order>

<Order>

<Product> “Peach” </Product>

// no Price

<Order>

Price is optional inclass dictionary:Order = “<Order>” Product [Price]“</Order>”.

Question 2:

• Where should we stop the traversal?

• Where can we find out whether an order has a price? At an Order-object.

• The traversal strategy:

• from PurchaseOrders to Order

Question 2:

• Would the following traversal strategy work?– from PurchaseOrders to *

• Yes, but it would be inefficient.

• We should not traverse beyond Order-objects because visitor will not be interested in any of those objects. Only before method of visitor will be at Order.

Rule to follow

Traverse only to please the visitors and not just for fun.

Question 3

• Meaning of strategies

Strategy: From C1 to T

o1:C1 o2:C2e

go down e iff C1 <=.C C3 (=> C2 (<=.C.=>)*.<=) Tgo down e iff C1 EI* EC C3 (EA* C2 (EI* EC EA*)* EI*) Tif flat:go down e iff C1 EC C3 (EA* C2 (EC EA*)* EI*) Tif flat and single inheritance:go down e iff C1 EC C3 (EA* C2 (EC EA*)* EI*) T

declared type of o2 is C3=>C2

Example 1strategy:{A -> B B -> C}

A

B

C

X

xx

b

c

class graph

A B C

Strategy s t

c

BOpt

Empty

Object graph

:A

c2:C

x1:X

:R

x2:X

c1:C

c3:C

e1:Empty SR

go down e iff C1 <=.C C3 (=>.<=.C.=>)*.<=) T

So far: Remarks about traversals

• Traversals are opportunistic: As long as there is a possibility for success (i.e., getting to the target), the branch is taken.

• Traversals do not look ahead. Visitors must delay action appropriately.

A simple view of traversals

• When a traversal reaches a target node in the object graph, the path traversed from the source, with suitable substitution of subclasses by superclasses, must be an expansion of an s-t path in the strategy graph. s is the source and t is the target of the strategy.

Special case

• strategy: {A -> B B -> C}• class graph: A : B. B : C. C = .• object graph: C• Follow rule: each edge in the strategy graph

corresponds to at least one edge in the object graph.

A simple view of traversals

• When a traversal reaches a final node in the object graph without being at a target, the path traversed from the source, with suitable substitution of subclasses by superclasses, must be a prefix of an expansion of an s-t path in the strategy graph. The prefix is the longest prefix such that there is still a possibility of success as determined by the class graph. Assume class graph is flat: subclass and construction edges only.

Definition

• POSS(Class c1, Class t, Object o1) = those edges e outgoing from o1 s.t. there is an object graph O (consistent with the class graph C), containing the object o1 of class c1, an object o2 of a class that is a subclass of t, and a path in O from o1 to o2 such that the first edge in the path is e.

• POSS: possibility of success

Back to the Class Graph

• POSS(Class c1, Class t) = – { c3 | c1 <=.C c3 (=>.(<=C=>)*.<=) t}

• POSS: possibility of success

A1 (=>.(<=C=>)*.<=) E2

A1

A2

K1

B1

B2 D1c1

D2

K2

E1

E2

A1 (=>.(<=C=>)*.<=) E2single inheritance

A1

A2

K1

c1

K2

E1

E2

A1 (=>.(<=C=>)*.<=) E2single inheritance

flat

A1

A2

K1

c1

K2

E1

E2

Definition: Class Graph

• A class graph is a directed graph with a partial order on the nodes. We write c1 -> c2 for the edges (construction edges) and c1 <= c2 for the order (inheritance edges -- c1 is a subclass of c2).

• We write => for the inverse of <=

Relations

• We think of directed graphs as relations.

• Write C(c1,c2) or c1 C c2 for edge from c1 to c2 in C.

• Composition of relations by . E.g., x (R.S) z iff there is a y such that xRy and ySz.

• R* is the reflective transitive closure of a relation R.

Object Graph Slice

• The object graph slice starting with o1 is the slice built by following the edges POSS(Class(o1), t, o1) starting at o1 and continuing until every path terminates (at an object of type t or it terminates prematurely).

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1

A -> TT -> D

A = [“x” X] [“r” R].B = [“b” B] D.R = S.S = [“t” T] CC = D.X = B.T = R.D = .

:A

:R :S

:C :D

classgraph

strategyclass dictionary

object graph “r”

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1

A -> TT -> D


a1:A

r1:R s1:S

:C :D

classgraph



POSS(A,T,a1) = 1 edgePOSS(R,T,r1) = 1 edgePOSS(S,T,s1) = 0 edges

answer: R S

before visitor

object graph slice

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1

A -> TT -> D

a1:A

r1:R

s1:Sc1:C

:D

classgraph

strategyPOSS(A,T,a1) = 1 edgePOSS(R,T,r1) = 1 edgePOSS(S,T,s1) = 1 edgePOSS(T,D,t1) = 1 edgePOSS(R,D,r2) = 1 edge

t1:T

r2:R

c2:C

d2:D

s2:S

answer: R S T R S C

object graph “r t”

before visitor

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1

A -> TT -> D

a1:A

x:X

b1:B

classgraph

strategyPOSS(A,T,a1) = 0 edges

b2:B

d1:D

d2:D

object graph “x b”

not needed for answer

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1


a1:A

r1:R s1:S

:C :D

classgraph

class dictionary


POSS(A,T,a1) = 1 edgePOSS(R,T,r1) = 1 edgePOSS(S,T,s1) = 0 edges

answer: R S

before visitor

object graph slice

equivalent strategies

A -> T bypassing Tfrom A to-stop T

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1

a1:A

r1:R

s1:Sc1:C

:D

classgraph

equivalent strategiesPOSS(A,T,a1) = 1 edgePOSS(R,T,r1) = 1 edgePOSS(S,T,s1) = 1 edgePOSS(T,T,t1) = 0 edges

t1:T

r2:R

c2:C

d2:D

s2:S

answer: R S T

object graph “r t”

A -> T bypassing Tfrom A to-stop T

Example

A

R

BX

S

D

0..1

0..1

0..1

C

T0..1

A -> BB -> DA -> CC -> D


classgraph


Path set regular expressionARS(TRS)*CD|AXBB*D =A(RS(TRS)*CD|XBB*D)

From Regular Expression toClass Dictionary

Path set regular expressionA(RS(TRS)*CD|XBB*D)RegExp = A E1.E1 : E2 | E3.E2 = R S List(E4) C D.E3 = X B List(B) D.E4 = T R S. List(S) ~ {S}.A = “A”. R = “R”. S = “S”. T = “T”. etc.

Strategy: From C1 to T

o1:C1 o2:C2e

go down e iff C1 <=.C C3 =>.(<=.C.=>)*.<=) T

declared type of o2 is C3=>C2

X2

X1

X3

S Y2

Y1

Y3

Z2

Z1

Z3

T

x

y

z

t

x31:X3 y31:Y3 z31:Z3 t1:Ts1:Sx y z t

go down e iff S <=.C C3 =>.(<=.C.=>)*.<=) T

StrategyS -> T

X2

X1

X3

S Y2

Y1

Y3

Z2

Z1

Z3

T

x

y

z

t

x31:X3 y31:Y3 z31:Z3 t1:Ts1:Sx y z t

go down e iff S <=.C X1 =>.(<=.C.=>)*.<=) T

StrategyS -> T

X2

X1

X3

S Y2

Y1

Y3

Z2

Z1

Z3

T

x

y

z

t

x31:X3 y31:Y3 z31:Z3 t1:Ts1:Sx y z t

go down e iff S <=.C X1 =>.(<=.C.=><=.C.=><=.C=>).<=) T

StrategyS -> T

<=,=>notused

midterm fall 2000 solutions. question 1 from sentence to class dictionary abstraction in this case...

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