=head1 NAME

Kleene's Algorithm - the theory behind it

brief introduction

=head1 DESCRIPTION

=head2 B<Semi-Rings>

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

=over 4

=item 1)

a)  C<(S, +, 0) is a Semi-Group with neutral element 0>

b)  C<(S, ., 1) is a Semi-Group with neutral element 1>

c)  C<0 . a = a . 0 = 0  for all a in S>

=item 2)

C<"+"> is commutative and B<idempotent>, i.e., C<a + a = a>

=item 3)

Distributivity holds, i.e.,

a)  C<a . ( b + c ) = a . b + a . c  for all a,b,c in S>

b)  C<( a + b ) . c = a . c + b . c  for all a,b,c in S>

=item 4)

C<SUM_{i=0}^{+infinity} ( a[i] )>

exists, is well-defined and unique

C<for all a[i] in S>

and associativity, commutativity and idempotency hold

=item 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] ) )

=back

EXAMPLES:

=over 4

=item *

C<S1 = ({0,1}, |, &, 0, 1)>

Boolean Algebra

See also L<Math::MatrixBool(3)>

=item *

C<S2 = (pos. reals with 0 and +infty, min, +, +infty, 0)>

Positive real numbers including zero and plus infinity

See also L<Math::MatrixReal(3)>

=item *

C<S3 = (Pot(Sigma*), union, concat, {}, {''})>

Formal languages over Sigma (= alphabet)

See also L<DFA::Kleene(3)>

=back

=head2 B<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.

=head2 B<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]

=head2 B<Kleene's Algorithm and Semi-Rings>

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

EXAMPLES:

=over 4

=item *

C<S1 = ({0,1}, |, &, 0, 1)>

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

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

Kleene's algorithm then calculates

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

using

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

(remember C< 0*  =  1*  =  1 >)

=item *

C<S2 = (pos. reals with 0 and +infty, min, +, +infty, 0)>

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

C<m(v[i],v[j])  :=  costs of (v[i],v[j])>

C<for all (v[i],v[j]) in E>

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

C<  ==E<gt>  a* = 0  for all a in S2>

C<  ==E<gt>  C^k[i,j]  =  min( C^k-1[i,j] ,>

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

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

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

=item *

C<S3 = (Pot(Sigma*), union, concat, {}, {''})>

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

Define

C<m(v[i],v[j])  :=>

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

and

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

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

(C<{''}> is the set containing the empty string, whereas C<{}> is the
empty set!)

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

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

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

and

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

(state C<q[1]> is assumed to be the "start" state)

finally being the language recognized by Deterministic Finite Automaton M.

=back

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

Define  C<A[i,j]  :=  m(v[i],v[j])>

C<  ==E<gt>   A*[i,j]  =  c(v[i],v[j])>

Proof:

C<A*  =  SUM_{i=0}^{+infty} A^i>

where  C<A^0  =  E_{n}>

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

and  C<A^(i+1)  =   A . A^i>

Induction over k yields:

C<A^k[i,j]  =  c_{k}(v[i],v[j])>

=over 10

=item C<k = 0:>

C<c_{0}(v[i],v[j])  =  d_{i,j}>

with  C<d_{i,j}  :=  (i = j) ? 1 : 0>

and  C<A^0  =  E_{n}  =  [d_{i,j}]>

=item C<k-1 -E<gt> k:>

C<c_{k}(v[i],v[j])>

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

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

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

=back

qed

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

=head1 SEE ALSO

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

(All contained in the distribution of the "Set::IntegerFast" module)

Dijkstra's algorithm for shortest paths.

=head1 AUTHOR

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

=head1 COPYRIGHT

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