Kleene's Algorithm - the theory behind it
brief introduction
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 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)
(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.
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 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.
==> 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!
Math::MatrixBool(3), Math::MatrixReal(3), DFA::Kleene(3).
Dijkstra's algorithm for shortest paths.
This document is based on lecture notes and has been put into POD format by Steffen Beyer <sb@sdm.de>.
Copyright (c) 1997 by Steffen Beyer. All rights reserved.