The cube of Kleene algebras and
the triangular prism of
multirelations
Koki Nishizawa (Tohoku Univ.)
With
Norihiro Tsumagari (Kagoshima Univ.)
Hitoshi Furusawa (Kagoshima Univ.)
Contents
1.
2.
3.
4.
Background
Overview of our main result
Details of the result
Future work
Background
Multirelation
Def.
A multirelation on A is a subset of A×P(A).
(P(A) is the power set of A)

An ordinary binary relation on A is a subset
of A×A
e.g. Multirelation for game
[Venema 03]…
Given
 A … the states of a game board
 P⊆A×A … possible transition by player P
 Q⊆A×A … possible transition by player Q
 W⊆A … the winnning sets of player P
Def. Multirelation R for player P
R={(a,X) | ∃b. aPb and ∀c. (bQc ⇒ c∈X)}
Prop.
If (a,W)∈R, then player P can win at the next turn of a)
(a,W)∈R+RR+RRR+…? Iteration of multirelation ?
Kleene algebra (KA)
algebraic structure (K,+,0,・,1,*)
for regular languages
e.g.



K … the binary relations on A
R* is the reflexive transitive closure of R
It is used to represent properties of the
reflexsive transitive closure of a multirelation.
Previous results
Lazy KAs[Moller 04]
Probabilistic KAs
[McIver et al. 05]
KAs
The set of up-closed
multirelations on A
The set of finitary
total up-closed
multirelations on A
The set of binary
relations on A
Our new result is an extension of these results.
Overview of our main result
Approach to extend this figure
Lazy KAs
Probabilistic KAs
KAs
Axioms on lazy KAs
Lazy KAs
Probabilistic
KAs
Axiom “0”
Axiom “D”
Axioms on lazy KAs
Lazy KAs
Axiom “+”
KAs
Axiom “0”
Axiom “D”
Conditions on multirelations
Sets of Multirelations
“closed type”
Lazy KAs
+
0
D
Conditions on multirelations
Sets of Multirelations
Lazy KAs
“affine type”
+
0
D
Conditions on multirelations
Sets of Multirelations
Lazy KAs
+
0
“total type”
D
Conditions on multirelations
Sets of Multirelations
Lazy KAs
+
0
“finite type”
D
Conditions on multirelations
Sets of Multirelations
Lazy KAs
These results contain our previous results.
Remark1: this figure is “cube”
Lazy KAs
+
0,+
0
D,+
0,D,+
D
0,D
0,+
0
0,D,+
0,D
+
Lazy
KAs
D,+
D
Remark2: An intermediate cube
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
Details of the three cubes
1. Cube of lazy KAs
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
1. Cube of lazy KAs
IL-semirings
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
Def. IL-semiring is (K,+,0,・,1)
such that

(K,+,0) is an idempotent commutative
monoid

(K,・,1) is a monoid

a≦b and a’≦b’ imply a・a’≦b・b’



a≦b ⇔ a+b=b
0・a=0
(a+b)・c=a・c+b・c
(Left distributivity)
(Left distributivity)
Def. Lazy KA is (K,+,0,・,1,*)
such that

(K,+,0,・,1) is an IL-semiring

a*・b = min{ c | b+a・c≦c}

a* is a reflexive transitive closure of a.
Def. Conditions on lazy KAs
1.
2.
3.
(The axiom
(The axiom
(The axiom
a・(b+1)≦a
Lazy KAs
“0”) a・0=0
“+”) a・(b+c)=a・b+a・c
“D”)
implies a・b*≦a
+
0
D
Def. Conditions on lazy KAs
1.
2.
3.
(The axiom
(The axiom
(The axiom
a・(b+1)≦a
“0”) a・0=0
“+”) a・(b+c)=a・b+a・c
“D”)
implies a・b*≦a
Prop.
Lazy KAs satisfying “0”,”+”, and ”D”
=KAs
2. Cube of complete IL-semirings
IL-semirings
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
Def. Complete IL-semiring is
(K,+,0,・,1,∨)
such that

(K,+,0,・,1) is an IL-semiring

∨S is the least upper bound of S

(∨S)・a=∨{x・a | x∈S}
Lemma.
Every complete IL-semiring has an operator /
such that x・b≦c⇔x≦c/b.
Prop. Complete IL-semiring
is a lazy KA.
IL-semirings
Lazy KAs
IL-semirings
Complete IL-semirings
a* is given by
min{ c | 1+a・c≦c}
Proof
It satisfies b+a・a*・b≦a*・b.
And,
b+a・c≦c
⇒b+a・(c/b)・b≦c
⇔1+a・(c/b)≦c/b
⇒a*≦c/b
⇔a*・b≦c
where x・b≦c⇔x≦c/b.
Therefore, a*・b = min{ c | b+a・c≦c}.
□
Def. The axiom “0” on
complete IL-semirings
Lazy KAs
Complete IL-semirings
a・0=0
a・0=0
Trivial
Def. The axiom “+” on
complete IL-semirings
Lazy KAs
Trivial
a・(b+c)=a・b+a・c
Complete IL-semirings
a・(b+c)=a・b+a・c
Def. The axiom “D” on
complete IL-semirings
Complete IL-semirings
Lazy KAs
a・(b+1)≦a
implies a・b*≦a
D
a・(∨S)=
∨{a・x | x∈S}
for directed S
Tarski’s fixed point
theorem
Result
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
3. Cube of multirelations
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
Prop.
Sets of Multirelations
Complete IL-semirings
Sets of up-closed
multirelations of
some closed type
Type
Def. We call a subset of P(A) a type.
Def. An Up-closed multirelation of type T(A) is
R⊆A×T(A) s.t.
(a,X)∈R, Y∈T(A), X⊆Y imply (a,Y)∈R
 e.g. Ordinary up-closed multirelation
(when T(A)=P(A))
 e.g. Ordinary binary relation
(when T(A)={{a} | a∈A})
Closed type
Def. T(A) is called closed if T(A)=φ or
1.
2.


∀a∈A. {a}∈T(A)
I∈T(A), ∀i∈I. Xi∈T(A) imply
∪{Xi | i∈I} ∈ T(A)
E.g. T(A)=P(A) is closed.
E.g. T(A)={{a} | a∈A} is closed.
Prop.
Sets of Multirelations
Complete IL-semirings
Sets of up-closed
multirelations of
some closed type
Prop.
The set of up-closed
Complete
IL-semirings
multirelations
on A
The set of binary
relations on A
The set of finitary
total up-closed
multirelations on A
Sets of Multirelations
Sets of up-closed
multirelations of
some closed type
Conditions on multirelation
Lazy KAs
Complete
IL-semirings
Sets of
Multirelations
Def. Total type
Sets of Multirelations
Complete IL-semirings
a・0=0
Total
Def. T(A) is called total if T(A)=φ implies A=φ
Def. Total type
Sets of Multirelations
Complete IL-semirings
The set of binary
relations on A
a・0=0
The singleton set
Total
The set of finitary
total up-closed
multirelations on A
Def. T(A) is called total if T(A)=φ implies A=φ
Def. Affine type
Sets of Multirelations
Complete IL-semirings
a・(b+c)=a・b+a・c
Affine
Def. T(A) is called affine if ∀X∈T(A). |X|≦1
Def. Affine type
Sets of Multirelations
The set IL-semirings
of binary
Complete
relations on A
The singleton set
a・(b+c)=a・b+a・c
Affine
Def. T(A) is called affine if ∀X∈T(A). |X|≦1
Def. Finite type
Sets of Multirelations
Complete IL-semirings
a・(∨S)=
∨{a・x | x∈S}
for directed S
Finite
Def. T(A) is called finite if ∀X∈T(A). X is finite
Def. Finite type
Sets of Multirelations
Complete IL-semirings
The set of binary
relations
a・(∨S)=on A
Finite
∨{a・x | x∈S}
The singleton set
for directed S
The set of finitary
total up-closed
multirelations
on
A
Def. T(A) is called finite if ∀X∈T(A). X is finite
Affineness implies finiteness.
Sets of Multirelations
Closed
Affine
This area is empty
This area is empty
Total
So, this figure is
not a cube but a triangular prism.
Finite
Main result
Sets of up-closed
multirelations of some type
Closed
Lazy KAs
+
0
Affine
D
Total
Finite
Previous results
The set of up-closed
multirelations on A
The set of binary
relations on A
Closed
Affine
The singleton set
The set of finitary
total up-closed
multirelations on A
Total
Finite
Future Work

To consider infinite streams
with respect to multirelations