The cube of Kleene algebras and the triangular
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) n An ordinary binary relation on A is a subset of A×A
e. g. Multirelation for game [Venema 03]… Given n A … the states of a game board n P⊆A×A … possible transition by player P n Q⊆A×A … possible transition by player Q n W⊆A … the winnning sets of player P Def. Multirelation R for player P R={(a, X) | ∃b. a. Pb and ∀c. (b. Qc ⇒ 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. n n n 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 [Mc. Iver 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
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.
Remark 1: this figure is “cube” Lazy KAs + 0, + 0 D, + 0, D, + D 0, + 0 0, D, + 0, D + Lazy KAs D, + D
Remark 2: 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 n (K, +, 0) is an idempotent commutative monoid n (K, ・, 1) is a monoid n a≦b and a’≦b’ imply a・a’≦b・b’ n n n a≦b ⇔ a+b=b 0・a=0 (a+b)・c=a・c+b・c (Left distributivity)
Def. Lazy KA is (K, +, 0, ・, 1, *) such that n (K, +, 0, ・, 1) is an IL-semiring n a*・b = min{ c | b+a・c≦c} n a* is a reflexive transitive closure of a.
Def. Conditions on lazy KAs (The axiom “ 0”) a・ 0=0 2. (The axiom “+”) a・(b+c)=a・b+a ・c 3. (The axiom “D”) a・(b+1)≦a implies a・b*≦a Lazy KAs + 1. 0 D
Def. Conditions on lazy KAs (The axiom “ 0”) a・ 0=0 2. (The axiom “+”) a・(b+c)=a・b+a ・c 3. (The axiom “D”) a・(b+1)≦a implies a・b*≦a Prop. Lazy KAs satisfying “ 0”, ”+”, and ”D” =KAs 1.
2. Cube of complete IL-semirings Lazy KAs Complete IL-semirings Sets of Multirelations
Def. Complete IL-semiring is (K, +, 0, ・, 1, ∨) such that n (K, +, 0, ・, 1) is an IL-semiring n ∨S is the least upper bound of S n (∨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 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 n e. g. Ordinary up-closed multirelation (when T(A)=P(A)) n 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. n n ∀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 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 n To consider infinite streams with respect to multirelations
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