Outer and Inner Fringes of a Knowledge Structure

Returns the outer or inner fringe for each state in a knowledge structure.

getKFringe(K, nstates = nrow(K), nitems = ncol(K), outer = TRUE)

Arguments

K

a state-by-problem indicator matrix representing the knowledge structure. An element is one if the problem is contained in the state, and else zero.

nstates

the number of knowledge states in K.

nitems

the number of items in K.

outer

logical. If TRUE return outer fringe, else return inner fringe.

Details

The outer fringe of a knowledge state is the set of all items that can be learned from that state, such that adding an outer-fringe item to the state results in another state in K,

$$K^O = \{q \notin K | K \cup \{q\} \in \mathcal{K}\}.$$

The inner fringe of a knowledge state is the set of all items that have been learned most recently to reach that state, such that deleting an inner-fringe item from the state results in another state in K,

$$K^I = \{q \in K | K - \{q\} \in \mathcal{K}\}.$$

Value

A state-by-problem indicator matrix representing the outer or inner fringe for each knowledge state in K.

See also

slm, simulate.blim.

Examples

data(DoignonFalmagne7)

## Which items can be learned from each state?
getKFringe(DoignonFalmagne7$K)
#>       a b c d e
#> 00000 1 1 0 0 0
#> 10000 0 1 0 0 0
#> 01000 1 0 0 0 0
#> 11000 0 0 1 1 0
#> 11100 0 0 0 1 1
#> 11010 0 0 1 0 0
#> 11110 0 0 0 0 1
#> 11101 0 0 0 1 0
#> 11111 0 0 0 0 0

## Which items in each state have been recently learned?
getKFringe(DoignonFalmagne7$K, outer = FALSE)
#>       a b c d e
#> 00000 0 0 0 0 0
#> 10000 1 0 0 0 0
#> 01000 0 1 0 0 0
#> 11000 1 1 0 0 0
#> 11100 0 0 1 0 0
#> 11010 0 0 0 1 0
#> 11110 0 0 1 1 0
#> 11101 0 0 0 0 1
#> 11111 0 0 0 1 1