Conway and Iteration Semirings Stephen L. Bloom Stevens Institute - - PDF document

Conway and Iteration Semirings

Stephen L. Bloom

Stevens Institute of Technology Hoboken, NJ

International Category Theory Conference 2008

Joint work with Zoltan ´ Esik

Outline

• Some categories with fixed points
• Conway and iteration theories
• Matrix theories and star semirings
• A Kleene type theorem
• Some characterizations of Nrat

A∗ and

N∞rat

A∗

• Open problems

Some categories with fixed point operators

Suppose C is a category with coproducts. A parameterized fixed point operation is a func- tion C(X, X + Y )

†

C(X, Y )

such that f† = X

f

X + Y

f†,1Y

Y.

In product form: C(X×Y, X)

†

C(Y, X) such

that f† = Y

f†,1Y

X × Y

f

X

Pfn(A)

• A is a collection of sets closed under finite

coproducts +

• Pfn(A) is category with objects X ∈ A
• an arrow f : X

Y is partial function

• if f : X

X + Y , f† : X Y is “do

f while value is in X”. f† = X

f

X + Y

f†, 1Y

Y.

Fω(A)

• A is a collection of ω-complete

posets closed under finite products ×

• Fω(A) is category with objects X ∈ A
• an arrow f : X

Y is continuous func-

tion

• if f : X × Y

X, f† : Y X is least

x such that f(x, y) = x. f† = Y

f†,1Y

X × Y

f

X

f†(y) = f(f†(y), y).

Fn(A)

• A a collection of ω-complete

categories closed under finite products

• objects in Fn(A) are categories in A
• an arrow in Fn(A) is ω-continuous functor

f : A

B

• if f : A × B

A, for each b ∈ B,

fb : A

• A is

fb = f(−, b)

• f† : B

A on b is initial fb-algebra

fb(f†(b))

• f†(b).

Trees Σ⊥TR

• Σ a ranked set ; ⊥ ∈ Σ0.
• a morphism 1

p is node labeled Σ-

tree with internal nodes with n-successors labeled by letter in Σn; leaves labeled either by letter in Σ0 or “variable” x1, . . . , xp

• a morphism n

p is n-tuple of trees 1 p.

• composition: tree substition
• if f : 1

1 + p, f† : 1 p is unique

tree such that f† = 1

f

1 + p

f†,1p

p.

Matrices The subsets of words on an alphabet X form a semiring 2X∗ where A + B = A ∪ B and A · B = {uv : u ∈ A, v ∈ B}. Matrices over this semiring form a category Mat(2X∗):

• an arrow f : n

p is n × p matrix

• f · g matrix product
• if f = [a b] : n

n + p, f† = [a∗b], where

a∗ = 1n + a + a2 + . . . .

Theories with fixed points Each of the above examples are categories T with

• finite coproducts or products
• a parameterized fixed point operation
• In Pfn, Mat, Fω, f† is a least fixed point
• In Σ⊥TR, f† is a unique fixed point
• In Fn, f† is an initial fixed point

Main Results

• (Bloom, ´

Esik) A large number of the com- putationally interesting structures have one

• f two equational theories IT, IT0.
• IT := Id(Fω(A)) = Id(Fn(A)) = Id(Σ⊥TR)

IT is the iteration theory identities.

• IT0 = IT + {x = y : x, y : 1 → 0} =

Id(Pfn(A)) = Id(Mat(2X∗)),

• (Simpson, Plotkin) Every consistent itera-

tion theory is either IT or IT0.

Axiomatization Axioms for iteration theory identities fall into two groups.

• The “Conway axioms”:

fixed point, pa- rameter, simple composition and double- dagger

• The group axioms.

Representing f†

Y f = f X Y X X

Parameter identity

= f g f g

(f · (1X ⊕ g))† = f† · g f : X

X + Y , g : Y Z.

Another formulation of the parameter identity C(X, X + Z) C(X, Z)

†

• C(X, X + Y )

C(X, X + Z)

C(1X,1X⊕g)

• C(X, X + Y )

C(X, Y )

†

C(X, Y )

C(X, Z)

C(1X,g)

• commutes, for any g : Y

Z.

Simple Composition identity

= f g f g f

(f · g)† = f · (g · f)† f : X

Y , g : Y X.

Double dagger Identity

f:X −> X+X+Y = f f

f†† = (f · (1X, 1X ⊕ 1Y ))†

Traced monoidal categories Conway theories are essentially the same as traced monoidal categories of Joyal, Street and Verity. (Feedback replaces dagger.)

Iteration Theories An iteration theory is a Conway theory satisfying all group identities. For the two element group G2:

= f f f

12 · f · ρ1, f · ρ2† = f††. The G2-identity

Star semirings A semiring, or rig, S consists of

• a commutative monoid (S, +, 0), and
• a monoid (S, ·, 1), such that
• multiplication distributes over addition

x(y + z) = xy + xz (y + z)x = yx + zx 0 · x = x · 0 = 0.

• A star semiring is semiring with ∗ : S → S.

Conway and iteration matrix theories Mat(S), the theory of matrices over a semiring S, is a Conway theory iff S is a star semiring satisfying

• the sum star identity:

(x + y)∗ = (x∗y)∗x∗,

• the product star identity:

(xy)∗ = 1 + x(yx)∗y. Special cases: zero and fixed point identity: 0∗ = 1 x∗ = 1 + xx∗ = 1 + x∗x.

• Definition. A Conway semiring is star semiring

satisfying the sum and product star identities. Examples

• Language semirings:

(2X∗, +, ·, 0, 1,∗ ), where X is an alphabet, and for A, B ⊆ X∗, A + B = A ∪ B; 0 = ∅; 1 = {ǫ} A · B = {uv : u ∈ A, v ∈ B} A∗ =

∞

• k=0

Ak.

More Examples

• N∞ = {0, 1, . . . , } ∪ {∞}, with 0∗ = 1, x∗ =

∞, otherwise.

• The boolean semiring B = {0, 1}, with x∗ =

1 = 1 + x.

Which Rings are reducts of Conway semirings?

• The star fixed point identity:

x∗ = 1 + x · x∗ implies x∗ · (1 − x) = 1.

• Letting x = 1:

= 1.

Inductive definition of M∗ If M is n × n, n > 1, write M =

• a

b c d

• with a, d square. The Conway identities imply

that M∗ is M∗ =

• (a + bd∗c)∗

(a + bd∗c)∗bd∗ (d + ca∗b)∗ca∗ (d + ca∗b)∗

Group identities for matrix theories Suppose G is a group with underlying set {1, 2, . . . , n}. The G identity is (x1 + . . . + xn)∗ = e1M∗

Gun

MG is n × n matrix with entries in {x1, . . . , xn} MG[i, j] = xi−1·j i−1 · j is computed in G. e1 = [1 0 . . . 0]. un is n × 1 matrix of 1’s.

Example When G = G2 is the 2-element group, MG =

• x1

x2 x2 x1

• The G2-identity is

(x1 + x2)∗ = (x1 + x2x∗

1x2)∗(1 + x∗ 1x2).

Special case: x1 = 0 and x2 = 1, 1∗ = 1∗(1 + 1) = 1∗ + 1∗.

Iteration semirings

• Suppose S is a Conway semiring. Mat(S)

is an iteration theory iff S satisfies all group identities.

• Definition. An iteration semiring is a Con-

way semiring satisfying all group identities.

Closure Properties If S is Conway or iteration semiring, so is Sn×n for n ≥ 0, the semiring of n × n matrices over S, with matrix product as product, pointwise sum, and star computed using the inductive formula.

Power series semirings If S is Conway or iteration semiring, so is S A∗

• the semiring of formal power series. Elements

are functions A∗

S.

• Notational convention:

f =

• u∈A∗

(f, u)u, where (f, u), the coefficient of u, is f(u).

• Examples. For a ∈ A, s ∈ S:

τa(u) = 1 if u = a, 0 otherwise. σs(u) = s if u = ǫ, 0 otherwise.

• The notational convention implies

a = τa s = σs.

Semiring operations in S A∗

• If

f =

• u∈A∗

(f, u)u g =

• u∈A∗

(g, u)u then f + g :=

• u∈A∗

((f, u) + (g, u))u f · g :=

• u∈A∗

(

• xy=u

(f, x) · (g, y))u.

• Example If f = 2 + 3a, g = 2b + ab,

fg = 4b + 8ab + 3a2b. ab = ǫ · (ab) or a · b.

f∗ in S A∗ ?

• If f(ǫ) = 0, there is unique f∗ satisfying

f∗ = 1 + f · f∗ f∗ = 1 +

• u

(

• xy=u

(f, x)(f∗, y))u.

• If f = s + g, g(ǫ) = 0, the sum star identity

implies f∗ = (s + g)∗ = (s∗g)∗s∗.

Some iteration and Conway semirings

• Each of

B = {0, 1}, N∞, 2X∗

is an iteration semiring.

• The initial Conway semiring S0 has ele-

ments 0, 1, 2, . . . , k(1∗)p, and 1∗∗.

• In any iteration semiring

1∗ + 1∗ = 1∗.

• The initial iteration semiring S1 has ele-

ments 0, 1, 2, . . . , (1∗)p, 1∗∗.

(Countably) complete semirings S is (countably) complete if for any (count- able) set I,

i∈I si exists, and has the usual

• properties. We may define

s∗ := 1 + s + s2 + . . . and the resulting star semiring is an iteration semiring. Example: N∞ is countably complete. In N∞, x∗ =

  

1 x = 0 ∞

• therwise.

For each n ≥ 1,

• Sn×n

is a Conway, but not an iteration semiring, and

• For each alphabet A, S0

A∗ is a Conway, but not an iteration semiring.

Rational power series For a Conway semiring S and alphabet A,

• Srat

A∗ , the rational series, is the least sub star semiring of S A∗ containing each series σs, and τa, for s ∈ S, a ∈ A.

Example: regular sets on A The series in Brat A∗ may be identified with least collection of subsets of A∗ containing the singletons {a}, for a ∈ A, the empty set, the set {ǫ}, closed under binary union, product and star, i.e., the regular sets.

Automata via matrices We may model finite automata as matrices:

c

1 2

a,b b,c d

α = [1 0], M =

• a + b

b + c c d

• , β =
• 1
• .

α, β are matrices over B; M is matrix whose entries are linear combinations of letters. The behavior of (α, M, β) is α · M∗ · β in B A∗ .

S-weighted Automata For a Conway semiring S, and alphabet A, SA is the set of all finite series in S A∗

• f the

form s1a1 + . . . + snan, with si ∈ S, ai ∈ A. An automaton is triple (α, M, β), where

• α is 1 × n matrix over S
• M is n × n matrix over SA
• β is n × 1 matrix over S.

The behavior of (α, M, β) is the series in S A∗ : |(α, M, β)| = α · M∗ · β.

Recognizable series

• A series f is recognizable if

f = |(α, M, β)| for some automaton (α, M, β).

• The collection of all recognizable series is

denoted Srec A∗ .

Examples

• s = σs ∈ S is |(s, 0, 1)| = s · 0∗ · 1
• τa is |α, M, β| when

α = (1 0) , M =

• a
• , β =
• 1
• αM∗β

= (1 0) ·

• 1

a 1

• ·
• 1
• =

a

A Kleene theorem for Conway semirings Theorem(Bloom, ´ Esik, Kuich) Srat A∗

• =

Srec A∗ .

Some free semirings

• Brat

A∗ is freely generated by A in class

• f iteration semirings satisfying 1∗ = 1.

[Krob]

• Brat

A∗ is isomorphic to star semiring of regular subsets of A∗; thus the axioms of iteration semirings together with 1∗ = 1 give an equational axiomatization of the regular sets.

Partial Conway semirings An ideal I in a semiring S is a subset I con- taining 0, satisfying S · I ∪ I · S ∪ I + I ⊆ I A partial Conway semiring is a semiring with a partial star operation defined on an ideal I = I(S) in S such that the Conway identities hold when restricted to elements in I: (x + y)∗ = (x∗y)∗x∗ (xy)∗ = 1 + x(yx)∗y, for x, y ∈ I. The Kleene theorem also holds for partial Con- way semirings.

Examples

• For any semiring S, S

A∗ is a partial Con- way semiring, with f∗ defined on all series f such that f(ǫ) = 0. In fact, for such series, f∗ is the unique solution to ξ = 1 + f · ξ.

• N is partial iteration semiring, with I(N) =

{0}. (If n > 0, what is n∗ = 1 + n · n∗?)

A characterization of Nrat A∗

• Theorem.(Bloom, ´

Esik) Nrat A∗ is freely gen- erated by A in the class of all partial iteration semirings, i.e., for any partial iteration semi- ring S and any function h : A → I(S), there is a unique semiring morphism h# : Nrat A∗ → S such that if (f, ǫ) = 0, then h#(f) ∈ I(S) and h#(f∗) = (h#(f))∗.

We will outline the proof, which uses the Kleene theorem: rational series are the behaviors of automata. Corollary. If S is an iteration semiring, any function h : A → S extends uniquely to a star semiring morphism Nrat A∗ → S.

The extension

• Given a partial iteration semiring S and a

function f : A → I(S), we define the image

• f the automaton (α, M, β) in N

A∗ in the

• bvious way: e.g., if M[i, j] = 3a+4b, then

fM[i, j] = 3f(a) + 4f(b).

• If s is the behavior of (α, M, β) define f#(s) =

α(fM)∗β.

• Must show if two automata have the same

behavior, their image is the same.

Simulations In the classical case, Brat A∗ , if two automata have the same behavior, their accessible and coaccessible parts have a common homomor- phic image. A similar fact holds for Nrat A∗ . Suppose A = (α, M, β) and B = (α′, M′, β′) A simulation ρ : A → B is: n × m matrix over

N such that

1

α

• 1

α′

• m

m

M′

• n

m

ρ

• n

n

M

n

m

ρ

• 1

β

• 1
• β′

Let ∼ be the least equivalence on automata such that if ρ : A → B is a simulation, then

A ∼ B.

• Theorem. (B´

eal, Lombardy, Sakarovitch). Two automata A,

B have the same behavior iff A ∼ B.

Where are the group axioms? They are used to prove that if m m

M′

• n

m

ρ

• n

n

M

n

m

ρ

• commutes, then

m m

(M′)∗

• n

m

ρ

• n

n

M∗

n

m

ρ

• does also.

A characterization of N∞rat A∗

• If S is a countably complete star semiring,

and s∗ = ∞

i=0 si, then S satisfies

1∗ · 1∗ = 1∗; x · 1∗ = 1∗ · x; 1∗ · (1∗ · x)∗ = 1∗ · x∗.

• N∞ is initial in V.
• Theorem (Bloom, ´

Esik) N∞rat A∗ is freely generated by A in the variety V of all iter- ation semirings satisfying these three extra identities.

An A-term t t = 0 | 1 | a | t + t | t · t | t∗. for a ∈ A. The series denoted by t is |t|.

Outline of proof In V, each A-term is equivalent to one of form tc + tI + 1∗s, where

• tc ∈ N,
• tI is “ideal term”;

tI = 0 | a | tI + tI | tI · tI | tI · t∗

I

• s is any A-term
• tc = 0 =

⇒ (|s|, ǫ) = 0.

Claim: |s| = |t| iff s = t holds in V. Fact. If f = |t|, for an ideal term t, then (f, ǫ) = 0 and (f, u) ∈ N, otherwise. Also, f ∈ Nrat A∗ .

First two steps

• If |s0 + sI + 1∗s| = |t0 + tI + 1∗t|, then

|s0| = |t0|, |sI| = |tI|, |1∗s| = |1∗t|.

• |s0| = |t0| are in N; N∞ initial in V implies

s0 = t0 in V.

• |sI| = |tI| belong to Nrat

A∗ , which is free in all partial iteration semirings. Thus, sI = tI in V.

Last step is to show that if |1∗s| = |1∗t|, then 1∗s = 1∗t in V.

• Lemma. Let W be variety of iteration star

semirings satisfying 1∗ = 1. An identity 1∗s = 1∗t holds in V iff it holds in W.

• Brat

A∗ is freely generated by A in W [Krob].

• |1∗s| is a rational series all of whose nonzero

coefficients are 1∗. Thus 1∗s = 1∗t in Brat A∗ , and hence in W by Krob. Thus, by the Lemma, 1∗s = 1∗t in V.

Inductive semirings An ordered semiring is semiring equipped with a partial order ≤ preserved by +, ·. A star semiring is inductive if aa∗ + 1 ≤ a∗ ax + b ≤ x = ⇒ a∗b ≤ x xa + b ≤ x = ⇒ ba∗ ≤ x. Fact:(´ Esik, Kuich) Any inductive semiring is an iteration semiring satisfying 1∗ = 1∗∗.

Examples of inductive semirings

• Mat(2X∗) where A ≤ B ⇐

⇒ A ⊆ B

• N∞ where n < n + 1 < ∞
• N∞rat

A∗ is inductive semiring, when or- dered by sum order: f ≤ g ⇐ ⇒ g = f + h, for some h ∈ N∞rat A∗

Non-examples of inductive semrings

• S0, the initial Conway semiring
• S1, the initial iteration semiring
• S0

A∗ , S1 A∗ .

A second characterization Theorem: (Bloom, ´ Esik) N∞rat A∗ is freely generated by A in class of inductive semirings satisfying x · 1∗ ≤ 1∗ · x.

Two Open problems

• Is there an algorithm to decide, given star

semiring terms r, r′ whether r = r′ in all Conway semirings?

• What is a concrete description of the free

Conway and iteration semirings?

References for Conway and iteration semirings S.L. Bloom and Z. ´

• Esik. Axiomatizing rational

power series, to appear. S.L. Bloom, Z. ´ Esik and W. Kuich. Partial Conway and iteration semirings, Fundamenta Informaticae, to appear. J.C. Conway. Regular Algebra and Finite Ma- chines, Chapman and Hall, London, 1971.

• Z. ´

Esik and W. Kuich. Inductive *-semirings. Theoretical Computer Science 324(2004),3-33.

References for Conway and iteration theories S.L. Bloom and Z. ´ Esik. Iteration Theories: The Equational Logic of Iterative Processes, EATCS Monographs on Theoretical Computer Science, Springer–Verlag, 1993. A.K. Simpson and G. Plotkin. Complete Axioms for categorical fixed-point

• perators.

Proc. Fifteenth Symp.

• n Logic in Com-

puter Science, 2000, 30-41, Washington, IEEE Computer Society Press.