LINEAR PROGRAMMING IN CONCEPT SEMANTIC NETWORKS By Naser Madi

Text Comprehension  Comprehension is understanding letters and words, syntactic parsing of sentences, understanding the meaning of words and sentences [1] 2 Comprehension Segmentation Recognizing ideas Integration Connecting ideas to background knowledge

Text Comprehension 3 "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"

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] 4

Segmentation & Integration thresholds 5 a b c d e f e 1 =5 e 2 =5 e 3 =3 e 4 =2 e 5 =3 e 6 =2 e 7 =5 e 8 =5 e 9 =3 a b c d f a b c d f Recognition threshold, α = 10 Association threshold, β = 4 Segmentation Integration Currently recognized concepts e 1 =5 e 2 =5 e 3 =3 e 7 =5 e 8 =5 e 9 =3 e 1 =5 e 2 =5 e 7 =5 e 8 =5

Optimization 6 Linear programming

Base Semantic Network I 7 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

Mutual Exclusion 8  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

Mutual Exclusion 9  The solution is to add a hidden node for each reader indicating the previous knowledge possessed by a reader [2]

Base Semantic Network II 10  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 E1E2E3 S1 S2 S3 S4

Linear Programming 11  The matrix representation for the equations as a linear programming problem is as follows:  min ƒ*x subject to constraints Ax ≤ b

Linear Programming 12  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

Sample BSN 13  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

Linear Programming Example 14  For example, we can maximize:  F = 2 α + 3 β  Constraint by:  2 α + 4 β <= 12  α + β <= 4  α >=0  β >=0

Linear Programming Example 15  Plot:  2 α + 4 β <= 12  α + β <= 4  α >=0  β >=0

Linear Programming Example 16  Plot:  2 α + 4 β <= 12  α + β <= 4  α >=0  β >=0

Linear Programming Example 17  Plot:  2 α + 4 β <= 12  α + β <= 4  α >=0  β >=0

Linear Programming Example 18  Substitute corner values:  F = 2 α + 3 β (0,0)=0 (4,0)=8 (2,2)=10 (0,3)=9

Complexity 19  In general the computational complexity of current interior point methods [5] is O(N 3 L) 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]

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,  [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,  [4]  [5] Khachiyan, Leonid G. "Polynomial algorithms in linear programming." USSR Computational Mathematics and Mathematical Physics 20.1 (1980):