Algorithm - the theory behind it

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:

) 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
) "+" is commutative and idempotent, i.e., a + a = a
) 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
) SUM_{i=0}^{+infinity} ( a[i] )
exists, is well-defined and unique
for all a[i] in S
and associativity, commutativity and idempotency hold

) 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 MatrixBool(3)
S2 = (pos. reals with 0 and +infty, min, +, +infty, 0)
Positive real numbers including zero and plus infinity
See also MatrixReal(3)
S3 = (Pot(Sigma*), union, concat, {}, {''})
Formal languages over Sigma (= alphabet)
See also 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 vortices 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 vortices 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.
```
  ==E<gt>  a* = 0  for all a in S2
```
```
  ==E<gt>  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])

  ==E<gt>   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 -gt 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!

AUTHOR

This document is based on lecture notes and has been put into POD format by Steffen Beyer .