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Math::Kleene.3pm

NAME

Kleene's Algorithm - the theory behind it

brief introduction

DESCRIPTION

Semi-Rings

A Semi-Ring (S, +, ., 0, 1) is characterized by the following properties:

1)

a) "(S, +, 0) is a Semi-Group with neutral element 0"

b) "(S, ., 1) is a Semi-Group with neutral element 1"

c) "0 . a = a . 0 = 0 for all a in S"

2)

"+" is commutative and idempotent, i.e., "a + a = a"

3)

Distributivity holds, i.e.,

a) "a . ( b + c ) = a . b + a . c for all a,b,c in S"

b) "( a + b ) . c = a . c + b . c for all a,b,c in S"

4)

"SUM_{i=0}^{+infinity} ( a[i] )"

exists, is well-defined and unique

"for all a[i] in S"

and associativity, commutativity and idempotency hold

5)

Distributivity for infinite series also holds, i.e.,

   ( SUM_{i=0}^{+infty} a[i] ) . ( SUM_{j=0}^{+infty} b[j] )
   = SUM_{i=0}^{+infty} ( SUM_{j=0}^{+infty} ( a[i] . b[j] ) )
 
 

EXAMPLES:

•

"S1 = ({0,1}, |, &, 0, 1)"

Boolean Algebra

See also Math::MatrixBool(3)

•

"S2 = (pos. reals with 0 and +infty, min, +, +infty, 0)"

Positive real numbers including zero and plus infinity

See also Math::MatrixReal(3)

•

"S3 = (Pot(Sigma*), union, concat, {}, {''})"

Formal languages over Sigma (= alphabet)

See also DFA::Kleene(3)

Operator '*'

(reflexive and transitive closure)

Define an operator called ``*'' as follows:

     a in S   ==>   a*  :=  SUM_{i=0}^{+infty} a^i
 
 

where

     a^0  =  1,   a^(i+1)  =  a . a^i
 
 

Then, also

     a*  =  1 + a . a*,   0*  =  1*  =  1
 
 

hold.

Kleene's Algorithm

In its general form, Kleene's algorithm goes as follows:

     for i := 1 to n do
         for j := 1 to n do
         begin
             C^0[i,j] := m(v[i],v[j]);
             if (i = j) then C^0[i,j] := C^0[i,j] + 1
         end
     for k := 1 to n do
         for i := 1 to n do
             for j := 1 to n do
                 C^k[i,j] := C^k-1[i,j] +
                             C^k-1[i,k] . ( C^k-1[k,k] )* . C^k-1[k,j]
     for i := 1 to n do
         for j := 1 to n do
             c(v[i],v[j]) := C^n[i,j]
 
 

Kleene's Algorithm and Semi-Rings

Kleene's algorithm can be applied to any Semi-Ring having the properties listed previously (above). (!)

EXAMPLES:

•

"S1 = ({0,1}, |, &, 0, 1)"

"G(V,E)" be a graph with set of vertices V and set of edges E:

"m(v[i],v[j]) := ( (v[i],v[j]) in E ) ? 1 : 0"

Kleene's algorithm then calculates

"c^{n}_{i,j} = ( path from v[i] to v[j] exists ) ? 1 : 0"

using

"C^k[i,j] = C^k-1[i,j] | C^k-1[i,k] & C^k-1[k,j]"

(remember " 0* = 1* = 1 ")

•

"S2 = (pos. reals with 0 and +infty, min, +, +infty, 0)"

"G(V,E)" be a graph with set of vertices V and set of edges E, with costs "m(v[i],v[j])" associated with each edge "(v[i],v[j])" in E:

"m(v[i],v[j]) := costs of (v[i],v[j])"

"for all (v[i],v[j]) in E"

Set "m(v[i],v[j]) := +infinity" if an edge (v[i],v[j]) is not in E.

" ==> a* = 0 for all a in S2"

" ==> C^k[i,j] = min( C^k-1[i,j] ,"

" C^k-1[i,k] + C^k-1[k,j] )"

Kleene's algorithm then calculates the costs of the ``shortest'' path from any "v[i]" to any other "v[j]":

"C^n[i,j] = costs of "shortest" path from v[i] to v[j]"

•

"S3 = (Pot(Sigma*), union, concat, {}, {''})"

"M in DFA(Sigma)" be a Deterministic Finite Automaton with a set of states "Q", a subset "F" of "Q" of accepting states and a transition function "delta : Q x Sigma --> Q".

Define

"m(v[i],v[j]) :="

" { a in Sigma | delta( q[i] , a ) = q[j] }"

and

"C^0[i,j] := m(v[i],v[j]);"

"if (i = j) then C^0[i,j] := C^0[i,j] union {''}"

("{''}" is the set containing the empty string, whereas "{}" is the empty set!)

Then Kleene's algorithm calculates the language accepted by Deterministic Finite Automaton M using

"C^k[i,j] = C^k-1[i,j] union"

" C^k-1[i,k] concat ( C^k-1[k,k] )* concat C^k-1[k,j]"

and

"L(M) = UNION_{ q[j] in F } C^n[1,j]"

(state "q[1]" is assumed to be the ``start'' state)

finally being the language recognized by Deterministic Finite Automaton M.

Note that instead of using Kleene's algorithm, you can also use the ``*'' operator on the associated matrix:

Define "A[i,j] := m(v[i],v[j])"

" ==> A*[i,j] = c(v[i],v[j])"

Proof:

"A* = SUM_{i=0}^{+infty} A^i"

where "A^0 = E_{n}"

(matrix with one's in its main diagonal and zero's elsewhere)

and "A^(i+1) = A . A^i"

Induction over k yields:

"A^k[i,j] = c_{k}(v[i],v[j])"

"k = 0:"

"c_{0}(v[i],v[j]) = d_{i,j}"

with "d_{i,j} := (i = j) ? 1 : 0"

and "A^0 = E_{n} = [d_{i,j}]"

"k-1 -> k:"

"c_{k}(v[i],v[j])"

"= SUM_{l=1}^{n} m(v[i],v[l]) . c_{k-1}(v[l],v[j])"

"= SUM_{l=1}^{n} ( a[i,l] . a[l,j] )"

"= [a^{k}_{i,j}] = A^1 . A^(k-1) = A^k"

qed

In other words, the complexity of calculating the closure and doing matrix multiplications is of the same order "O( n^3 )" in Semi-Rings!

SEE ALSO

Math::MatrixBool(3), Math::MatrixReal(3), DFA::Kleene(3).

Dijkstra's algorithm for shortest paths.

AUTHOR

This document is based on lecture notes and has been put into POD format by Steffen Beyer <sb@sdm.de>.

COPYRIGHT

Copyright (c) 1997, 1998, 1999 by Steffen Beyer. All rights reserved.