LINEAR PROGRAMMING IN CONCEPT SEMANTIC NETWORKS

By Naser Madi

2

Text Comprehension

Comprehension is understanding letters and words, syntactic parsing of sentences, understanding the meaning of words and sentences [1]

Comprehension

SegmentationRecognizing ideas

IntegrationConnecting ideas tobackground knowledge

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Text Comprehension

"Some kids found her upstairs"

"Hasn't been here long, her name's Jennifer Wilson according to her credit cards"

"We're running them now for contact details"

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Segmentation & Integration thresholds

Concept recognition (segmentation) threshold is the individual limit for recognizing concepts [2]

Association recognition (integration) threshold is the individual limit for recognizing associations [2]

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Segmentation & Integration thresholds

a

b

c

d

e

f

e1=5

e2=5e3=3

e4=2

e5=3

e6=2

e7=5e

8=5

e9=3

a

b

c

d

f

a

b

c

d

f

Recognition threshold, α = 10 Association threshold, β= 4

Segmentation Integration

Currently recognized concepts

e1=5

e2=5e3=3

e7=5e

8=5e9=3

e1=5

e2=5

e7=5e

8=5

321 eee

987 eee

654 eee

1e 2e

7e8e

3e

9e

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Optimization

321 eee 654 eee 987 eee

01 e

02 e

03 e

07 e08 e09 e

Linear programmi

ng

,,...,, 321 eee

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Base Semantic Network I

a

b c

d

e

a

b

d

e

a

b c

d

e

a

b c

e

ISN (reader 1)

ISN (reader 2) ISN (reader

3)

BSN

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Mutual Exclusion

The problem of mutual exclusivity may arise between two individuals when given the same background knowledge two or more individuals recognize a different set of concepts

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Mutual Exclusion

The solution is to add a hidden node for each reader indicating the previous knowledge possessed by a reader [2]

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Base Semantic Network II

CXC + CXE + SXC + α+ β

CXC: associations between concepts CXE: concepts discovered at episode SXC: subject recognized concept α: recognition threshold β: association threshold

a

b c

d

e

E1 E2 E3

S1

S2

S3

S4

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Linear Programming

The matrix representation for the equations as a linear programming problem is as follows:

min ƒ*x subject to constraints Ax ≤ b

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Linear Programming

Each inequality is a line (half space)

Each variable is a dimension

If a solution is possible and the inequalities are Satisfiable, then the polygon covers the area of feasible solution [2]

Testing the corner values (intersection points) of the polygon gives us the min & max [4]

simple linear program with two variables and six inequalities

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Sample BSN

Weighted graph. 8534 variables. 87 concepts.

Contains individual association and recognition thresholds (α and β).

CXC + CXE + SXC + α+ β

7*7+7*3+2*7+2*2 = 98

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Linear Programming Example For example, we can maximize: F = 2 α + 3 β

Constraint by: 2 α + 4 β <= 12 α + β <= 4 α >=0 β >=0

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Linear Programming Example Plot: 2 α + 4 β <= 12 α + β <= 4 α >=0 β >=0

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Linear Programming Example Plot: 2 α + 4 β <= 12 α + β <= 4 α >=0 β >=0

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Linear Programming Example Plot: 2 α + 4 β <= 12 α + β <= 4 α >=0 β >=0

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Linear Programming Example Substitute corner values: F = 2 α + 3 β(0,0)=0(4,0)=8(2,2)=10(0,3)=9

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Complexity

In general the computational complexity of current interior point methods [5] is O(N3L) where N is the number of variables and L is the size of data (number of inequalities) [2]

Worst case of simplex method is exponential [4]

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References

[1] W. Kintsch, The construction-integration model of text comprehension and its implications for instruction," Theoretical models and processes of reading, vol. 5, pp. 1270{1328, 2004.

[2] M. Hardas and J. Khan, Concept learning in text comprehension," in Brain Informatics. Springer, 2010, pp. 240{251.

[3] Dantzig, G.B., A. Orden, and P. Wolfe, "Generalized Simplex Method for Minimizing a Linear Form Under Linear Inequality Restraints," Pacific Journal Math., Vol. 5, pp. 183–195, 1955.

[4] http://en.wikipedia.org/wiki/Linear_programming

[5] Khachiyan, Leonid G. "Polynomial algorithms in linear programming." USSR Computational Mathematics and Mathematical Physics 20.1 (1980): 53-72.