Incremental, Systematic Acquiring of Knowledge

(using Closure Algorithms)

Roger L. CostelloFebruary 1, 2014

Example #1

Problem Statement

• Problem: What cities can you get to from Boston?

<Flights> <Flight> <From>Boston</From> <To>Miami</To> </Flight> <Flight> <From>Miami</From> <To>Houston</To> </Flight> <Flight> <From>Cleveland</From> <To>Akron</To> </Flight> <Flight> <From>Boston</From> <To>Denver</To> </Flight></Flights>

It is easy to eyeball this sample XML and see that from Boston you can get to Miami, Houston (via Miami), and Denver.

But if the XML document had thousands of flights, it would be impossible to eyeball it to get the answer.

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Need a Strategy

• We need a systematic, incremental approach to obtaining the answer.

• That’s what closure algorithms give us.

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Closure algorithms

• A closure algorithm enables the incremental, systematic acquiring of knowledge.

• Closure algorithms are characterized by two components:– An initialization, which is an assessment of what we

know initially. – An inference rule, which is a rule telling how knowledge

from several places is to be combined. • The inference rule(s) are repeated until nothing

changes any more.

Initial KnowledgeThis is what we know initially:

– All the flights– From Boston we can get to Miami and Denver since there are direct flights to those

cities.<Flights> <Flight> <From>Boston</From> <To>Miami</To> </Flight> <Flight> <From>Miami</From> <To>Houston</To> </Flight> <Flight> <From>Cleveland</From> <To>Akron</To> </Flight> <Flight> <From>Boston</From> <To>Denver</To> </Flight></Flights> 6

Inference Rule

• If we can get to the city identified in <From>, then we can get to the city identified in <To>.

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Initial knowledge

Flight Can get there from Boston?

<Flight><From>Boston</From><To>Miami</To></Flight> Yes

<Flight><From>Miami</From><To>Houston</To></Flight>

<Flight><From>Cleveland</From><To>Akron</To></Flight>

<Flight><From>Boston</From><To>Denver</To></Flight> Yes

• Go through the XML and for each <Flight> with a direct flight from Boston, mark it as “Can get there from Boston”.

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Build on top of our knowledge

• Apply the inference rule to gain more knowledge. • This is round two (round one was marking the initial

knowledge).Flight Can get there from

Boston?<Flight><From>Boston</From><To>Miami</To></Flight> Yes

<Flight><From>Miami</From><To>Houston</To></Flight> Yes (since we can get to Miami)

<Flight><From>Cleveland</From><To>Akron</To></Flight>

<Flight><From>Boston</From><To>Denver</To></Flight> Yes

Round Three

• A third round yields no additional cities.

Flight Can get there from Boston?

<Flight><From>Boston</From><To>Miami</To></Flight> Yes

<Flight><From>Miami</From><To>Houston</To></Flight> Yes

<Flight><From>Cleveland</From><To>Akron</To></Flight>

<Flight><From>Boston</From><To>Denver</To></Flight> Yes

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Problem/Answer

• Problem: What cities can you get to from Boston?

• Answer: We can get to these cities:{Miami, Denver, Houston}

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Example #2

Clean XML Schema

• Problem: Are there any element declarations in the adjacent XML Schema that, if used, do not result in a valid XML instance document (e.g., perhaps because they loop infinitely)? In other words, are there any non-productive element declarations?

<xs:schema xmlns:xs="http://www.w3.org/2001/XMLSchema"> <xs:element name="Document"> <xs:complexType> <xs:choice> <xs:sequence> <xs:element ref="A" /> <xs:element ref="B" /> </xs:sequence> <xs:sequence> <xs:element ref="D" /> <xs:element ref="E" /> </xs:sequence> </xs:choice> </xs:complexType> </xs:element> <xs:element name="A" type="xs:string" /> <xs:element name="B"> <xs:complexType mixed="true"> <xs:sequence> <xs:element ref="C" /> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="C" type="xs:string" /> <xs:element name="D"> <xs:complexType mixed="true"> <xs:sequence> <xs:element ref="F" /> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="E" type="xs:string" /> <xs:element name="F"> <xs:complexType mixed="true"> <xs:sequence> <xs:element ref="D" /> </xs:sequence> </xs:complexType> </xs:element>

</xs:schema>13

Concise Notation

• For brevity, let’s depict the element declarations this way:

Document → A B | D EA → stringB → string CC → stringD → d FE → stringF → string D

A → string means the value of element A is a string.

B → string Cmeans the value of element B is a string and a child element C (i.e., B has mixed content). 14

Terminology

These are rules.Each rule has a left-hand side and a right-hand side (the two sides are separated by an arrow →).Document, A, B, …, F are non-terminal symbols.string is a terminal symbol.

Document → A B | D EA → stringB → string CC → stringD → d FE → stringF → string D

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Find the productive rules

• We find the non-productive rules by finding the productive ones.

• A rule is productive if its right-hand side consists of symbols all of which are productive.

• Symbols that are productive:– Terminal symbols are productive since they

produce values.– A non-terminal is productive if there is a productive

rule for it.

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Initial knowledge

Rule Is it productive?

Document → A BDocument → D EA → string Yes

B → string CC → string Yes

D → string FE → string Yes

F → string D

• Go through the grammar and for each rule for which we know that all its right-hand side members are productive, mark the rule and the non-terminal it defines as “productive”.

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Build on top of our knowledge

• Apply the inference rule to gain more knowledge.

Rule Is it productive?

Document → A BDocument → D EA → string Yes

B → string C Yes (since “string” and C are productive)

C → string Yes

D → string FE → string Yes

F → string D

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Round threeRule Is it productive?

Document → A B Yes (since A and B are productive)

Document → D EA → string Yes

B → string C Yes (since “string” and C are productive)

C → string Yes

D → string FE → string Yes

F → string D

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Round four

• A fourth round yields nothing new.

Rule Is it productive?

Document → A B Yes (since A and B are productive)

Document → D EA → string Yes

B → string C Yes (since “string” and C are productive)

C → string Yes

D → string FE → string Yes

F → string D

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Recap

• We now know that Document, A, B, C, and E are productive and D, F, and the rule Document → D E are not productive.

Rule Is it productive?

Document → A B Yes (since A and B are productive)

Document → D EA → string Yes

B → string C Yes (since C is productive)

C → string Yes

D → string FE → string Yes

F → string D

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Remove non-productive rules• We have pursued all possible avenues for productivity,

and have not found any possibilities for D, F, and the second rule for Document. That means the rules for D, F, and the second rule for Document can be removed from the XML Schema.

The XML Schema after removing non-productive rules

Rule Is it productive?

Document → A B Yes (since A and B are productive)

A → string Yes

B → string C Yes (since C is productive)

C → string Yes

E → string Yes

Cleaned XML Schema

• Problem: Does the XML Schema have any non-productive element declarations?

• Answer: Yes, and the cleaned XML Schema is shown in the adjacent box.

<xs:schema xmlns:xs="http://www.w3.org/2001/XMLSchema"> <xs:element name="Document"> <xs:complexType> <xs:sequence> <xs:element ref="A" /> <xs:element ref="B" /> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="A" type="xs:string" /> <xs:element name="B"> <xs:complexType mixed="true"> <xs:sequence> <xs:element ref="C" /> </xs:sequence> </xs:complexType> </xs:element> <xs:element name="C" type="xs:string" /> <xs:element name="E" type="xs:string" /> </xs:schema>

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Bottom-up process

• Removing the non-productive rules is a bottom-up process: only the bottom level, where the terminal symbols live, can we know what is productive.

Why is it called “closure”?• We have seen two examples where new knowledge was

incrementally, systematically acquired using a closure algorithm.• Question: Where does the term “closure” come from?• Answer: Applying the inference rule yields knowledge about

existing symbols, without going outside the universe of discourse. – In the last example, applying the inference rule resulted in new

knowledge about the elements that are productive in the XML Schema. Conversely, say the inference rule were to result in knowledge about elements in a completely different XML Schema, then the algorithm would not be closed.

In a closure algorithm the universe of discourse is self-contained, no information outside of the universe is generated.

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Acknowledgement

• Some of the information in these slides come from this fabulous book:Parsing Techniques by Dick Grune and Ceriel Jacobs.X

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