An Algebraic Generalization of the Chomsky-Sch utzenberger-Theorem - - PowerPoint PPT Presentation

An Algebraic Generalization of the Chomsky-Sch¨ utzenberger-Theorem

Hans Leiß leiss@cis.uni-muenchen.de extending work by/joint with Mark Hopkins 2017 retired from: Universit¨ at M¨ unchen Centrum f¨ ur Informations- und Sprachverarbeitung Oberseminar Theoretische Informatik, LMU, 10.5. 2019 Oberseminar Mathematische Logik, LMU, 22.5. 2019

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Plan

◮ Algebraic generalization of formal language theory: itempotent

semirings D with sums U of suitable subsets U ⊆ D: RD = regular subsets, CD = context-free subsets, etc.

◮ The Chomsky-Sch¨

utzenberger-Theorem: how to obtain CX ∗ from R(X ∪ ∆)∗ and a single language Dyck ∈ C(X ∪ ∆)∗

◮ Quotients and tensor products: RM ⊗R R∆∗/ρ ◮ A generalization of the CST for arbitrary monoids

CM = Q(RM) = Z2(RM ⊗R R∆∗/ρ) i.e. CM is an algebraic function of RM.

◮ Useful to name L ∈ CX ∗ by regular expressions over X ∪ ∆.

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Reminder: Formal Language Theory for free monoid X ∗

language family set X Construction Automaton recognizing finite

• f L ⊆ X ∗ by

{w ∈ X ∗ | w ∈ L} Finite FX ∗ ∅, {x}, ∪, · finite dir.acyclic graph Regular RX ∗ ∅, {x}, ∪, ·, ∗ finite automaton Context-free CX ∗ ∅, {x}, ∪, ·, µ push-down automaton Context-sensitive SX ∗ . . . linearly bounded TM

• Rec. enumerable

T X ∗ . . . Turing machine (TM) Arbitrary languages PX ∗ . . . — Construction more precisely:

◮ elementwise product: A · B = {a · b | a ∈ A, b ∈ B} ◮ iteration: A∗ = {An | n ∈ N}, A0 = {1}, An+1 = A · An ◮ recursion: for polynomial p(y, ¯

z) ∈ X ∗[y, ¯ z] and ¯ A ∈ (CX ∗)m, the least solution µyp(¯ A) of y ⊇ p(y, ¯ A) in PX ∗ is in CX ∗ Chomsky-hierarchy: FX ∗ ⊂ RX ∗ ⊂ CX ∗ ⊂ SX ∗ ⊂ T X ∗ ⊂ PX ∗

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Algebraization of Formal Language Theory

We work with M the category of monoids (M, ·, 1) and homomorphisms, D the category of dioids (D, +, ·, 0, 1) (= idempotent semirings) and semiring homomorphisms. M≤ the category of partially ordered monoids (M, ·, 1, ≤) and monotone homomorphisms. Each dioid D implicitly has a partial order ≤ defined by d ≤ d′ ⇐ ⇒ d + d′ = d′. The power-set functor P : M → D assigns to a monoid M a dioid PM = (|PM|, ∪, ·, ∅, {1}), where A · B := {a · b | a ∈ A, b ∈ B}, and to each homomorphism f : M → N a dioid-homomorphism Pf = λA {f (a) | a ∈ A} : PM → PN.

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A monadic opertor A (Hopkins 2008) is a subfunctor of the power-set functor P : M → D that satisfies, for each monoid M, A0 AM is a set of subsets of M: AM ⊆ PM, A1 AM contains each finite subset of M: FM ⊆ AM, A2 AM is closed under product (hence a monoid), A3 AM is closed under union of sets from AAM (hence a dioid), A4 A preserves homomorphisms: if f : M → N is a homo- morphism, so is Af := λU {f (u) | u ∈ U} : AM → AN. We say A ≤ A′ iff AM ⊆ A′M for each monoid M.

Theorem (Hopkins 2008)

F ≤ R ≤ C ≤ T ≤ P are monadic operators. (S is not: A4)

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Remark

A monadic operator A is the left adjoint of an adjunction (A, A, η, ǫ) : M → DA between M and a category DA ⊆ D, where

◮

• A : DA → M is the forgetful functor, and if M ∈ M, D ∈ DA,

◮ ηM : M → AM is m → {m}, ǫD : AD → D is U → U.

This adjunction gives rise to a monad TA = ( A ◦ A, η, µ) in M, an endofunctor T = A ◦ A : M → M, where

◮ the unit η : I → T maps m ∈ M to {m} ∈ AM, ◮ the product µ : TT → T maps U ∈ AAM to U ∈ AM.

• A is called a monadic functor in category theory.

DA ≃ MT, the Eilenberg-Moore category of T-algebras (D,

D).

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The category DA ⊆ D of A-dioids

For a partial order M, by x > X means x is an upper bound of X. A map f : M → N between partially ordered monoids M, N is A-continuous, if for all U ∈ AM and n > (Af )(U) there is some m > U with n ≥ f (m). An A-morphism is an A-continuous monotone homomorphism. An A-dioid is a partially ordered monoid M = (M, ·, 1, ≤) which is

◮ A-complete: every U ∈ AM has a supremum U ∈ M, and ◮ A-distributive: for all U, V ∈ AM, (UV ) = ( U)( V ).

Let DA be the category of A-dioids with A-morphisms.

• DF is the category D of dioids (a + b := {a, b}, 0 := ∅).
• DP is the category of (unital) quantales.

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Since every A-dioid (M, ·, 1, ≤) is a (F-) dioid via a + b :=

• {a, b},

0 :=

• ∅,

we often write A-dioids in the dioid-signature: D = (D, +, ·, 0, 1).

Prop.

AX ∗ is the free A-dioid generated by the set X. AM is the free A-dioid extension of the monoid M.

• Prop. (i) For f : M → N between A-dioids M, N:

f is A-continuous iff for all U ∈ AM: f (

• U) =
• (Af )(U)

(ii) An A-complete po-monoid (M, ·, 1, ≤) is A-distributive iff forall a, b ∈ M, U ∈ AM : a(

• U)b =
• aUb.

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Theorem

Hopkins 2008: DR is the category of ∗-continuous Kleene algebras. HL 2018: DC is the category of µ-continuous Chomsky algebras. Kleene-Algebra (Kozen 1990): right/left-linearly closed dioid x ≥ ax + b and x ≥ xa + b have least solutions a∗b resp. ba∗, for all values a, b.

∗-continuity: a · c∗ · b = {a · cn · b | n ∈ N}, for all a, b, c ∈ M.

Chomsky-Algebra (Grathwohl e.a. 2015): algebraically closed dioid every polynomial system x1 ≥ p1(¯ x, ¯ y), . . . , xn ≥ pn(¯ x, ¯ y) has a least solution in ¯ x = x1, . . . xn, for each value of ¯ y. µ-continuity: a · µxp · b = {a · pn(0) · b | n ∈ N}, all p ∈ M[x].

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Theorem (Hopkins 2008)

For all monadic operators A ≤ B there is an adjunction QB

A : DA ⇄ DB : QA B

where QB

A(K) is a completion of K by order-ideals of B-subsets of

K, and QA

B is the forgetful functor (resp. restriction of ).

For monoids M, CM = QC

R(RM) is the algebraic closure of RM.

Problem

Can we provide an algebraic construction of QC

R?

Intended advantage: algebraic expressions for context-free languages, instead of µ-terms.

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The classical CST for free monoids

X ∗ = (X ∗, ·, 1) the free monoid generated by the fin.set X M[∆] = the free extension of the monoid M by the set ∆ = all interleaved sequences of elements of M and ∆∗

Theorem (Chomsky/Sch¨ utzenberger 1963)

Let

◮ ∆ = {b, d, p, q} consist of two pairs b, d and p, q of brackets, ◮ h : X ∗[∆] → X ∗ be the “bracket-erasing” homomorphism, ◮ D ∈ C(X ∗[∆]) be the Dyck-language, the least S ⊆ X ∗[∆] s.t.

S ≥ 1 + X + bSd + pSq + SS Then: CX ∗ = {h(R ∩ D) | R ∈ R(X ∗[∆])}. This is not yet CX ∗ = QC

R(RX ∗), but a first step towards our goal.

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The polycyclic monoid P′

n and R-dioid C ′ n

We are looking for an algebra in which h(R ∩ D) can be performed. An inverse semigroup is a semigroup (M, ·) where each element p has a “generalized inverse” p−1, i.e. a q such that p = pqp and q = qpq. Example: partial bijections p ⊆ X × X of X under composition. Let ∆n = Pn ˙ ∪ Qn, for Pn = {p0, . . . , pn−1}, Qn = {q0, . . . , qn−1}, and (∆∗

n)0 the extension of ∆∗ n by an annihilating element 0.

The polycyclic monoid P′

n is the inverse monoid (∆∗ n)0/ρn where

ρn = {piqi = 1 | i < n} ∪ {piqj = 0 | i, j < n, i = j}. Here, pi and qi are generalized inverses of each other, as piqi = 1. Note: the Dyck-language over ∆n is Dn = {w ∈ ∆∗

n | w/ρn = 1}.

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The Cayley-graph of P′

n

The Cayley-graph of P′

n is the graph (P′ n, p0

− → , . . . ,

qn−1

− → ) with u/ρn

pi

− → upi/ρn, u/ρn

qi

− → uqi/ρn. We have ∆∗

n

= ∆∗

nPnQn∆∗ n

˙ ∪ Q∗

nP∗ n.

Hence every w ∈ ∆∗

n has a normal form nfρn(w) in

{0} ∪ Q∗

nP∗ n.

The normal forms represent the elements of P′

n, and 1 w

− → nfρn(w). The monoid P′

n ≃ (Q∗ nP∗ n ∪ {0}, ·′, 1) with u ·′ v = nfρn(uv) is the

partial monoid Q∗

nP∗ n if u ·′ v = 0 is read as “u ·′ v is undefined”.

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The Cayley-graph of P′

2 (without 0 and edges related to 0):

q0 q1

✻ ✻ ✠

• p0
• q0
• ✒

p1

❄

q1

■ ❅ ❅ ❅ q0 ❅ ❅ ❅

• q1
• ✒

p0

❄

q0

❅ ❅ ❅

p1 ❅

❅ ❅ ❘ ■ ❅ ❅ ❅ q1 ❅ ❅ ❅

q0p0p0q1 = 0 q0p1 1 q1p0 q1p1

✻

. . . p1

❄

q1

■ ❅ ❅ ❅ q1 ❅ ❅ ❅ ✠

• p0
• q0
• ✒ ❅

❅ ❅

p1 ❅

❅ ❅ ❘ ■ ❅ ❅ ❅ q1 ❅ ❅ ❅

. . . q0p0p1 p0 p1

✠

• p0
• q0
• ✒ ❅

❅ ❅

p1 ❅

❅ ❅ ❘ ■ ❅ ❅ ❅ q1 ❅ ❅ ❅

. . . D2 = p0p0 p0p1 {w | 1

w

− → 1} If we could restrict

pi

− → ,

qj

− → to P∗

2 ⊆ P′ 2, we had a stack!

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The polycyclic R-dioid C ′

n is R∆∗ n/ρn, viewing ρn as semiring

equations {pi}{qi} = {1}, {pi}{qj} = ∅, (i = j). This is essentially the R-dioid RPn obtained from the polycyclic monoid P′

n ≃ (Q∗ nP∗ n ∪ {0}, ·′, 1) with u ·′ v = nfρn(uv); we only

have to remove the “non-existing” monoid-element 0:

• Prop. C ′

n ≃ RP′ n/({0} = ∅).

[Quotients in DR: later] We can code n ≥ 2 bracket pairs pi, qi by two, i.e. in ∆∗ = ∆∗

2 by

pi := bpi, qi := qid (i < m).

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Question

Can we perform h(R ∩ Dn(M)) algebraically, for R ∈ R(M[∆n])? Assume we can move the monoid elements aside, i.e. take M = 1. To go from R ∈ R∆∗

n to R ∩ Dn, let p = pn+1, q = qn+1. Then ◮ w ∈ Dn ⇐

⇒ w/ρn = 1 ⇐ ⇒ pwq/ρn+1 = 1,

◮ w /

∈ Dn ⇐ ⇒ w/ρn = 1 ⇐ ⇒ pwq/ρn+1 = 0. (Q∗

nP∗ n \ {1})

Since w ∈ ∆∗

n → (w, w/ρn+1) ∈ ∆∗ n × P′ n+1 is a homomorphism,

R′ = {(w, w/ρn+1) | w ∈ R} ∈ R(∆∗

n × P′ n+1).

Therefore, multiplying R′ by {(1, p/ρn+1)} and {(1, q/ρn+1)},

◮ S′ := {(w, pwq/ρn+1) | w ∈ R} ∈ R(∆∗ n × P′ n+1), ◮ S′ ⊆ ∆∗ n × {0, 1}.

Then, with a suitable product · and over the regular set S′,

• {w · c | (w, c) ∈ S′} =
• {w · 1 | (w, 1) ∈ S′} = R ∩ Dn.

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To make the final parts precise, we will have to do three things:

• I. show that DR has quotients and tensor products, such that

◮ there is an R-dioid R∆∗ n/ρn = C ′ n ◮ for R-dioids D1 and D2, there is an R-dioid D1 ⊗R D2 ◮ there is an infinite sum : R(D1 ⊗R D2) → D1 ⊗R D2 ◮ elements of D1 and D2 commute in D1 ⊗ D2 ◮ R(M1 × M2) ≃ RM1 ⊗R RM2, for monoids M1, M2

• II. use R(M × P′

n+1) ≃ RM ⊗ RP′ n+1 to get ◮ w · c by {w} ⊗ {c}, ◮ S := {{w} ⊗ {c} | (w, c) ∈ S′} ∈ R(RM ⊗R RP′ n+1) ◮ {w · c | (w, c) ∈ S′} by S ∈ RM ⊗R RP′ n+1

• III. replace RP′

n+1 by C ′ n+1 = RP′ n+1/({0} = ∅) in II. to get ◮ w · 0 = {w} ⊗ {0} = {w} ⊗ ∅ = 0 (and so eliminate R ∩ Dn).

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For the general case of monoid M instead of ∆∗

n, we also have to

• IV. get (finitely generated) M “out of the way” by

◮ going from M[∆n] to M × ∆∗ n and intersect R ∈ R(M × ∆∗ n)

with M × Dn ∈ C(M × ∆∗

n) to get

h(R ∩ (M × Dn)) =

• {m · pwq/ρn+1 | (m, w) ∈ R}
• V. for L ∈ CM, find n and R ∈ R(M[∆n]) with L = h(R ∩ Dn(M))

◮ by a proof of (CST) CX ∗ ⊆ {h(R ∩ Dn(X)) | R ∈ R(X ∗[∆n])}

Therefore, we should try to prove CM ⊆ RM ⊗R R∆∗

n/ρn = RM ⊗R C ′ n

and read off which elements of RM ⊗R C ′

n belong to CM.

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Application: algebraic terms for L ∈ CX ∗, not µ-terms

Context-free language over X are named by regular expressions evaluated in RX ∗ ⊗R C ′

2:

r, s := x | 0 | 1 | (r + s) | (r · s) | r∗ | 0| | |0 | 1| | |1 Example: X = {a, b}, L = {anbn | n ∈ N}. Write a for its value {a} ⊗ 1 in RX ∗ ⊗R C ′

2, 0| for {1} ⊗ {0|} etc.

0|(a1|)∗(|1b)∗|0 = 0|(

• n

(a1|)n)(

• m

(|1b)m)|0 =

• n,m

0|(a1|)n(|1b)m|0 (R-distrib.) =

• n,m

anbm0|1|n|1m|0 (rel.comm.) =

• n

anbn (0|1|n|1m|0 = δn,m)

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Plan for the remaining parts

We now proceed as follows:

• 1. give a proof of the CST for free monoids X ∗ and monoids M
• 2. sketch why DA has quotients D/ρ
• 3. sketch why DA has tensor products D1 ⊗A D2
• 4. prove CM ⊂

∼ RM ⊗R C ′ 2 and notice CM ⊂ ∼ ZC ′

2(RM ⊗R C ′

2)

• 5. prove ZC ′

2(RM ⊗R C ′

2) ⊂ ∼ CM for monoids M

Outlook for algebraic closure of R-dioids:

• 6. Sketch why QC

RD ≃ ZC ′

2(D ⊗R C ′

2) might follow, for D ∈ DR.

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The CST for free monoids X ∗

Of the intended CX ∗ = {h(R ∩ D2(X)) | R ∈ R(X ∗[∆2])}, the inclusion ⊇ is by well-known facts:

• R ∈ RY ∗, D ∈ CY ∗ =

⇒ R ∩ D ∈ CY ∗,

• h : Y ∗ → X ∗ a homomorphism, L ∈ CY ∗ =

⇒ h(L) ∈ CX ∗. For ⊆, via coding n brackets by two it is sufficient to show:

Theorem (Chomsky/Sch¨ utzenberger 1963)

For every L ∈ CX ∗ there is n ∈ N and R ∈ R(X ∗[∆n]) such that L = h(R ∩ Dn(X)), for ∆n-erasure h : X ∗[∆n] → X ∗.

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Proof of the CST for X ∗ and M

a) The original proof (similar: Kozen 1997, Kanazawa 2010) L ∈ CX ∗ has a context-free grammar G = (Y , X, P, S) with finite set P ⊆ Y × (X ∗[Y ]) of “grammar rules” and S ∈ Y . The w ∈ X ∗ that belong to L = L(G) are those which have a “parse tree” with respect to G. A parse tree t = [A, t1, . . . , tk] with root category A ∈ Y is projected to a string over X ∗[∆m] via τ([A, t1, . . . , tk]) = [A τ(t1) . . . τ(tk) ]A, τ([x]) = x for x ∈ X. From the rules in G one reads off a finite relation I ⊆ (X ∪ ∆m)2 such that the well-bracketed projections of parse trees belong to D2(X) ∩ R, where the regular set R is R = [SX ∗[∆n]]S \ (X ∗[∆n]((X ∪ ∆n)2 \ I)X ∗[∆n]). b) The proof in Harrison 1979: uses push-down automata

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c) New proof. Let G ′ be CFG ¯ Y ≥ ¯ p′( ¯ Y , ¯ x) over X = ¯ x be given, with n variable occurrences in ¯ p′. Let G or ¯ Y ≥ ¯ p( ¯ Y , ¯ x) be G ′ with the i-th variable occurrence surrounded by brackets i| and |i of ∆n, i < n. There is a bijection between the parse trees of G ′ and G, so it is sufficient to find R ∈ R(X ∗[∆n]) with L(G) = R ∩ Dn(X). Let GF be the right-linear approximation of G obtained as follows:

◮ for each recursion variable Yi, add a “continuation variable”

Yi,F and distribute these to the summands in ¯ Y ≥ ¯ p( ¯ Y ) ¯ YF: Y1 ≥ p1( ¯ Y )Y1,F, Y2 ≥ p2( ¯ Y )Y2,F, . . . Ym ≥ pm( ¯ Y )Ym,F,

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◮ state SF → 1 for the continuation of the main variable S and

break the above rules (summands of Y ≥ pY ( ¯ Y )YF) Y → v1i1|Y1|i1 . . . vkik|Yk|ikvk+1YF (vj ∈ X ∗) into “right-linear pieces” Y → v1i1|Y1, Y1,F → |i1v2i2|Y2, . . . Yk,F → |ikvk+1YF Then L(GF) ∈ R(X ∗[∆n]), since GF is right-linear. L(GF) ⊇ L(G), since the Yi,F collect right contexts of all occurrences of Yi in G. To show L(G) = L(GF) ∩ Dn(X), notice L(G) ⊆ Dn(X) and prove by induction on m: for variable Y and w ∈ X ∗[∆n], (i) if Y ⇒m

G w, then Y ⇒∗ GF wYF,

(ii) if Y ⇒m

GF wYF and w ∈ Dn(X), then Y ⇒∗ G w.

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In (ii), we use w ∈ Dn(X) to show that for m > 1 there are i1, . . . , ik, v1, . . . , vk+1 ∈ X ∗ and w1, . . . , wk ∈ D(X) such that w = v1i1|w1|i1 . . . vkik|wk|ikvk+1 and there are GF-derivations A ⇒GF v1i1|A1, Aj ⇒mj

GF wjAj,F ⇒GF wj|ijvj+1ij+1|Aj+1

for j < k, Ak ⇒mk

GF wkAk,F ⇒GF wk|ikvk+1AF.

By induction, Aj ⇒∗

G wj for j < k, and by construction of GF,

there is A → v1i1|A1|i1 . . . vkik|Ak|ikvk+1 in G, so A ⇒∗

G w.

The proof works for any monoid M instead of X ∗, since the monoid product is never used (because of the brackets).

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Example

Source grammar G ′ S ≥ x + yTz, T ≥ SS brackets added: G S ≥ x + y1|T|1z, T ≥ 2|S|23|S|3 with continuation vars S ≥ xSF + y1|T|1zSF, T ≥ 2|S|23|S|3TF right-linearized: GF S ≥ xSF + y1|T, T ≥ 2|S SF ≥ |23|S + |3TF + 1 TF ≥ |1zSF Minimal solution in R(X ∗[∆3]): SF = |3|1zSF + |23|S + 1 = (|3|1z)∗(|23|S + 1) S = xSF + y1|2|S = (y1|2| + x(|3|1z)∗|23|)∗x(|3|1z)∗

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Corollary

Let M be a monoid. For each L ∈ CM there are n ∈ N and R ∈ R(M × ∆∗

n) such that

L = π1(R ∩ (M × Dn)), where π1 : M × ∆∗

n → M is the first projection and Dn ∈ C∆n the

(pure) Dyck-language over ∆n. Proof: Take n, ∆n and R′ ∈ R(M[∆n]) and h : M[∆n] → M as in the CST, such that L = h(R′ ∩ Dn(M)). Let e = (h, h∆) : M[∆n] → M × ∆∗

n be the homomorphism where

h∆ : M[∆n] → ∆∗

n erases elements of M. The claim follows for

R := e(R′) ∈ R(M × ∆∗

n).

Think of (m, t) ∈ R ∩ (M × Dn) as m with a parse tree t of m

• proof of m ∈ L

.

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The category DA has quotients and tensor products

For a dioid D and U, V ⊆ D, put U ≃ V : ⇐ ⇒ U≤ = V ≤, where U≤ := {d ∈ D | d ≤ u for some u ∈ U}. For a dioid-congruence ρ on D, the set D/ρ of congruence classes is a dioid under the operations defined as expected. An A-congruence on an A-dioid D is a dioid-congruence ρ s.th. for all U, V ∈ AD, if U/ρ ≃ V /ρ, then ( U)/ρ = ( V )/ρ. For any E ⊆ D × D, there is a least A-congruence on D above E.

Lemma

Let q : D → Q be an A-morphism between A-dioids D, Q. Then ker(q) := {(a, b) | q(a) = q(b), a, b ∈ D} is an A-congruence on D.

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• Prop. If D is an A-dioid and ρ an A-congruence on D, then D/ρ

is an A-dioid and the canonical map d → d/ρ is an A-morphism.

Proof.

D/ρ is A-complete: Each U′ ∈ A(D/ρ) is U/ρ for some U ∈ AD. Since ρ is an A-congruence,

• U′ := (
• U)/ρ

is well-defined, and an upper bound of U′ = U/ρ. Least: by A3 Let e/ρ be any upper bound of U′. As {U, {e}} ∈ FAD ⊆ AAD, we have U ∪ {e} ∈ AD. By choice of e, (U ∪ {e})/ρ ≃ {e}/ρ, so (e +

• U)/ρ = (
• (U ∪ {e}))/ρ = (
• {e})/ρ = e/ρ.

Hence ( U)/ρ ≤ e/ρ, and U′ is a least upper bound of U′.

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In fact, DA has coequalizers, and A lifts coequalizers in M to coequalizers in DA. This can be used to show

Theorem

Let E be a congruence on the monoid M, AE the least A-congruence on AM above {({m}, {m′}) | (m, m′) ∈ E}. Then A(M/E) ≃ AM/AE. Writing ρ′

n for the lifting Rρn of the monoid-congruence

ρn = {piqj = δi,j | i, j < n}, this gives us RP′

n := R((∆∗ n)0/ρn) = R((∆∗ n)0)/ρ′ n, hence

RP′

n/({0} = ∅) = (R((∆∗ n)0)/ρ′ n)/({0} = ∅) = R∆∗ n/ρ′ n =: C ′ n.

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Tensor Product

In M, two morphisms f : M1 → M ← M2 : g are relatively com- muting, if for all m1 ∈ M1, m2 ∈ M2, f (m1)g(m2) = g(m2)f (m1). In a category whose objects have a monoid structure, a tensor product of two objects M1 and M2 is an object M1 ⊗ M2 with two relatively commuting morphisms ⊤1 : M1 → M1 ⊗ M2 ← M2 : ⊤2 such that for any pair f : M1 → M ← M2 : g of relatively com- muting morphisms the diagram M1 ⊤1

✲ M1 ⊗ M2 ✛

⊤2 M2 ˙ ˙ ˙ ˙ ˙ ˙

❅ ❅ ❅

f ❅

❅ ❅ ❘

˙ ˙ ˙ ˙ ˙

❄

hf ,g

✠

• g
• M

can be uniquely completed as shown.

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Intuitively, M1 ⊗ M2 is a free extension of both objects in which elements of one commute with elements of the other.

Example

In M, the tensor product ⊤1 : M1 → M1 ⊗ M2 ← M2 : ⊤2 consists

• f M1 ⊗ M2 := M1 × M2 with T1(m) = (m, 1), T2(m′) = (1, m′).

The induced map hf ,g : M1 × M2 → M is hf ,g(a, b) = f (a)g(b). M1 ⊤1

✲ M1 ⊗ M2 ✛

⊤2 M2 ˙ ˙ ˙ ˙ ˙ ˙

❅ ❅ ❅

f ❅

❅ ❅ ❘

˙ ˙ ˙ ˙ ˙

❄

hf ,g

✠

• g
• M

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Theorem

In the category DA, the tensor product of A-dioids D1, D2 is ⊤1 : D1 → D1 ⊗A D2 ← D2 : ⊤2, where we use

◮ the underlying monoid Mk :=

ADk of Dk, k = 1, 2,

◮ their tensor product ˆ

⊤1 : M1 → M1 ⊗ M2 ← M2 : ˆ ⊤2 in M,

◮ the least A-congruence ≡ on A(M1 ⊗ M2) above

{({(

• A,
• B)}, A × B) | A ∈ AM1, B ∈ AM2},

to put D1 ⊗A D2 := A(M1 ⊗ M2)/≡ and ⊤k(dk) = {ˆ ⊤k(dk)}/≡. The induced map of f : D1 → D ← D2 : g is hf ,g(U/≡) :=

• {f (a)g(b) | (a, b) ∈ U},

U ∈ A(M1 × M2).

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The A-congruence ≡ is needed to make Ti be A-morphisms. Notation: For a ∈ D1, b ∈ D2, we write a ⊗ 1 := T1(a) = {(a, 1)}/≡ 1 ⊗ b := T2(b) = {(1, b)}/≡ For U ∈ A(D1 × D2), U/≡ ∈ D1 ⊗A D2 can be written as U/≡ =

• {a ⊗ b | (a, b) ∈ U} =: [U].
• Prop. AM1 ⊗A AM2 ≃ A(M1 ⊗ M2) for monoids M1, M2.

Proof (Sketch) Recall M1 ⊗ M2 = M1 × M2 and use U/≡ → {( A, B) | (A, B) ∈ U}, U ∈ A(AM1 × AM2) V → {({a}, {b}) | (a, b) ∈ V }/≡, V ∈ A(M1 × M2).

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Algebraic CST for monoids

We claim that the R-dioid RM ⊗R C ′

2 contains a copy of the

C-dioid CM as a subdioid. In RM ⊗R C ′

2, images of m ∈ M and c ∈ C ′ 2 commute:

({m} ⊗ 1)({1} ⊗ c) = {m} ⊗ c = ({1} ⊗ c)({m} ⊗ 1). Let ZC ′

2(RM ⊗R C ′

2) be the centralizer of C ′ 2 in RM ⊗R C ′ 2, i.e.

{e ∈ RM ⊗R C ′

2 | e(1 ⊗ c) = (1 ⊗ c)e for all c ∈ C ′ 2}.

We want to prove that this is the algebraic closure CM of FM. Claim: ZC ′

2(RM ⊗R C ′

2) is an R-dioid.

Proof: Each V ∈ R(ZC ′

2(RM ⊗R C ′

2)) ⊆ R(RM ⊗R C ′ 2) has a

V ∈ RM ⊗R C ′

2, which commutes with C ′ 2 by R-distributivity.

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Theorem (algebraic CST for monoids)

For every monoid M, CM ⊂

∼ ZC ′

2(RM ⊗R C ′

2) via

L → ˆ L :=

• {{m} ⊗ 1 | m ∈ L}.

Analogy: for D := ZC ′

2(RM ⊗R C ′

2),

L =

• {{m} | m ∈ L}
• ∈CCM

∈ CM →

• {{m} ⊗ 1 | m ∈ L}
• ∈CD

∈ D.

• : CCM → CM

→

• : CD → D

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Proof

Suppose L ∈ CM. By the Corollary, there is n ≥ 2 and R ∈ R(M × ∆∗

n) such that

L = {m | (m, d) ∈ R, d ∈ Dn}. The isomorphism R(M × ∆∗

n) ≃ RM ⊗R R∆∗ n maps R to

{{m} ⊗ {d} | (m, d) ∈ R} ∈ RM ⊗R R∆∗

n.

By adding new brackets p, q, going from R to {(1, p)} · R · {(1, q)} ∈ R(M × ∆∗

n+1),

and recoding n + 1 bracket pairs by n pairs, we can assume that in (m, d) ∈ R, d = pd′q where p, q do not occur in d′, so that d ∈ Dn ⇐ ⇒ {d}/ρn = 1, d / ∈ Dn ⇐ ⇒ {d}/ρn = 0.

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The homomorphism (1RM, ·/ρn) : RM × R∆n → RM × C ′

n gives

S := {({m}, {d}/ρn) | (m, d) ∈ R} ∈ R(RM × C ′

n),

and therefore, since {m} ⊗ d/ρn = {m} ⊗ 0 = 0 for d / ∈ Dn, ˆ L =

• {{m} ⊗ 1 | m ∈ L}

=

• {{m} ⊗ {d}/ρn | (m, d) ∈ R, d ∈ Dn}

=

• {{m} ⊗ {d}/ρn | (m, d) ∈ R} = [S] ∈ RM ⊗R C ′

n.

Claim: ˆ L ∈ ZC ′

2(RM ⊗R C ′

2)

ˆ L is a of a set S⊗ ∈ R(RM ⊗R C ′

2) of elements {m} ⊗ 1 and

{m} ⊗ 0 that commute with C ′

• 2. As RM ⊗R C ′

2 is R-distributive,

c′(

• S⊗) =
• (c′S⊗) =
• (S⊗c′) = (
• S⊗)c′

for each c′ = {1} ⊗ c with c ∈ C ′

2.

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Claim: L → ˆ L is an embedding of CM into ZC ′

n(RM ⊗R C ′

n).

Proof: We show there are relatively commuting R-morphisms f , g RM ⊤1

✲ ZC ′

n(RM ⊗R C ′

n) ✛

⊤2 C ′

n

❅ ❅ ❅ f ❅ ❅ ❅ ❘ ❄

hf ,g

✠

• g
• PM ⊗P MatP∗

n ,P∗ n (B)

≃ MatP∗

n ,P∗ n (PM)

such that the induced R-morphism hf ,g maps ˆ L to a copy of L. Let f be defined by f (A) = A ⊗ I for A ∈ RM and unit matrix I. For g, let h : ∆∗

n → MatP∗

n ,P∗ n (B) map pi, qj to the transition

relations

pi

− → ,

qj

− → on P′

n, restricted to P∗ n × P∗

• n. This hom. h

extends to an R-morphism h∗ : R∆∗

n → MatP∗

n ,P∗ n (B) by

h∗(U) =

• {h(u) | u ∈ U}.

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The semiring equations piqj = δi,j hold in MatP∗

n ,P∗ n (B) under this

interpretation, so h∗ is constant on ρn-congruence classes, and g(U/ρn) = {1} ⊗ h∗(U), for U ∈ R∆∗

n,

defines an R-morphism g : C ′

n → PM ⊗P MatP∗

n ,P∗ n (B).

Obviously, f and g are relatively commuting. For d ∈ Dn, g({d}/ρn) = I, and for d / ∈ Dn, g({d}/ρn) = 0. By the choice of R and since hf ,g is an R-morphism, hf ,g(ˆ L) = hf ,g(

• {{m} ⊗ 1 | m ∈ L})

= hf ,g(

• {{m} ⊗ {d}/ρn | (m, d) ∈ R})

=

• {hf ,g({m} ⊗ {d}/ρn) | (m, d) ∈ R}

=

• {f ({m}) · g({d}/ρn) | (m, d) ∈ R}

=

• {{m} ⊗ I | m ∈ L} = L ⊗P I.

Thus, hf ,g is essentially an inverse to L → ˆ L. ✷

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Algebraic reverse CST for monoids

Theorem (algebraic RCST for monoids)

For every monoid M, ZC ′

2(RM ⊗R C ′

2) ⊂ ∼ CM.

Recall that C ′

n = R∆∗ n/ρn has a representation by R-sets of

ρn-reduced strings, C ′

n ≃ RP′ n/({0} = ∅).

For simplicity, we write C ′

n = R′P′ n = {B \ {0} | B ∈ RP′ n}.

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Write [S] for the congruence class S/≡ ∈ RM ⊗R C ′

2 of S.

Lemma

ZC ′

2(RM ⊗R C ′

2) =

{[S] | S ∈ R(RM × R′P′

2), S ⊆ RM × {∅, {1}}}.

Proof ⊇: For c ∈ C ′

2 and B ∈ {∅, {1}}, cB = Bc, so

(1 ⊗ c)(A ⊗ B) = (A ⊗ B)(1 ⊗ c). Since [S] = {A ⊗ B | (A, B) ∈ S}, by R-distributivity we get (1 ⊗ c)[S] = [S](1 ⊗ c).

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⊆: Suppose [R] ∈ ZC ′

2(RM ⊗R C ′

2) for R ∈ R(RM × C ′ 2). Add a

new pair p, q of brackets and recode C ′

3 in C ′

• 2. By choice of R,

({1} ⊗ {p})[R]({1} ⊗ {q}) = [R]. Since RM ⊗R R′P′

2 is R-distributive, we have

[R] = ({1} ⊗ {p})[R]({1} ⊗ {q}) = ({1} ⊗ {p})(

• {A ⊗ B | (A, B) ∈ R})({1} ⊗ {q})

=

• {A ⊗ {p}B{q}) | (A, B) ∈ R}) = [S],

where S ∈ R(RM × R′P′

2) is

{(A, {p}B{q}) | (A, B) ∈ R} = {({1}, {p})}R{({1}, {q})}. As B ⊆ Q∗

2P∗ 2 and p, q do not occur in B, pBq ⊆ {∅, {1}}.

✷

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Proof of the RCST for monoids M

Take [R] ∈ ZC ′

2(RM ⊗R C ′

2) with R ∈ R(RM × R′P′ 2). By the

lemma, we can assume R ⊆ RM × {∅, {1}}. Put LR :=

• {A | (A, B) ∈ R, 1 ∈ B} ⊆ M.

Then [R] =

• {A ⊗ B | (A, B) ∈ R}

=

• {{w} ⊗ {v} | (A, B) ∈ R, w ∈ A, v ∈ B}

=

• {{w} ⊗ {1} | w ∈ LR}.

It remains to show LR ∈ CM. (Notice that then [R] = LR.)

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By RM ⊗R RP′

2 ≃ R(M × P′ 2), we assume R ∈ R(M × P′ 2) and

LR = {m | (m, 1) ∈ R} to show LR ∈ CM. First, “undo” the separation M[∆2] → M × ∆∗

2.

For R ∈ R(M × P′

2), define R′ ∈ R(M[∆2]) inductively by: ◮ ∅′ = ∅ = {(m, 0)}′, for m ∈ M, ◮ {(m, ¯

q¯ p)}′ = {¯ qm¯ p}, for m ∈ M, ¯ q ∈ Q∗

2, ¯

p ∈ P∗

2, ◮ (R ∪ S)′ = R′ ∪ S′, ◮ (R1 · R2)′ = R′ 1 · R′ 2 ◮ (R∗)′ = (R′)∗

• Claim. LR = hM(R′ ∩ D2(X)) for the erasure hM : M[∆2] → M

and a finite X ⊆ M. (For M = X ∗, it follows that LR ∈ CM.)

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Consider (m, ¯ q¯ p) ∈ R as a relation on P′

2 generating output in M:

from u ∈ P′

2, go to u ·′ ¯

q¯ p ∈ P′

2 and output m.

For R ∈ R(M × P′

2) and R′ ∈ R(M[∆2]), define ternary relations

= ⇒R , − →R′ ⊆ P′

2 × P′ 2 × M by ◮ u m′

= ⇒R v : ⇐ ⇒ ∃(m, ¯ q¯ p) ∈ R[u ·′ ¯ q¯ p = v ∧ m′ = m]

◮ u m′

− →R′ v : ⇐ ⇒ ∃α ∈ R′[nf (h∆(uα)) = v ∧ m′ = hM(α)] where nf : ∆∗

2 → P′ n = Q∗ 2P∗ 2 ∪ {0} is the ρ2-normal form and

h∆ : M[∆2] → ∆∗

2 is the homomorphism erasing elements of M.

• Claim. For R ∈ R(M × P′

2), −

→R′ = = ⇒R . Proof: by induction on the construction of R.

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R = ∅ = R′: there are no m′, u, v with u m′ − →R′ v or u m′ = ⇒R v. R = {(m, ¯ q¯ p)}: Then R′ = {¯ qm¯ p}, and hM(¯ qm¯ p) = m, nf (h∆(u¯ qm¯ p)) = u · ¯ q¯ p, so

m′

− →R′ = ∅ =

m′

= ⇒R for m′ = m and

m

− →R′ = {(u, u · ¯ q¯ p) | u ∈ P′

2} = m

= ⇒R . R = R1 ∪ R2: Then R′ = R′

1 ∪ R′ 2, the claim follows by induction.

R = S∗ = {Sn | n ∈ N}: Then R′ = (S′)∗ = {(S′)n | n ∈ N}. By induction, − →S′ = = ⇒S , and − →(Sn)′ = − →(S′)n = = ⇒Sn by induction from the product case, so = ⇒R =

• { =

⇒Sn | n ∈ N} =

• { −

→(Sn)′ | n ∈ N} =

R′

− →.

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R = R1 · R2: u m′ − →R′

1R′ 2 v

⇐ ⇒ ∃α1 ∈ R′

1∃α2 ∈ R′ 2

[nf (h∆(uα1α2)) = v ∧ m′ = hM(α1α2)] ⇐ ⇒ ∃m1, m2 ∈ M, w ∈ P′

2

[∃α1 ∈ R′

1(nf (h∆(uα1)) = w ∧ m1 = hM(α1)) ∧

∃α2 ∈ R′

2(nf (h∆(wα2)) = v ∧ m2 = hM(α2)) ∧ m′ = m1m2]

⇐ ⇒ ∃m1, m2 ∈ M, w ∈ P′

2 [u m1

− →R′

1 w ∧ w

m2

− →R′

2 v ∧ m′ = m1m2]

⇐ ⇒ ∃m1, m2 ∈ M, w ∈ P′

2 [u m1

= ⇒R1 w ∧ w

m2

= ⇒R2 v ∧ m′ = m1m2] ⇐ ⇒ ∃(m1, ¯ q1¯ p1) ∈ R1, (m1, ¯ q2¯ p2) ∈ R2 [u ·′ (¯ q1¯ p1 ·′ ¯ q2¯ p2) = v ∧ m′ = m1m2] ⇐ ⇒ u m′ = ⇒R1R2 v

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Now let X be the finitely many members of M in the elements of F(M × P′

2) used in the construction of R (and hence R′). Then

1

m

− →R′ 1 ⇐ ⇒ ∃α ∈ R′(nf (h∆(1α)) = 1 ∧ m = hM(α)) ⇐ ⇒ ∃α ∈ R′(α ∈ Dn(X) ∧ m = hM(α)) ⇐ ⇒ m ∈ hM(R′ ∩ Dn(X)). By the previous Claim, the claim on LR follows: LR = {m ∈ M | (m, 1) ∈ R} = {m | (m, ¯ q¯ p) ∈ R, ¯ q¯ p = 1} (⇐: Cayley-graph) = {m | ∃(m′, ¯ q¯ p) ∈ R [1 ·′ ¯ q¯ p = 1 ∧ m = m′]} = {m | 1

m

= ⇒R 1} = {m | 1

m

− →R′ 1} = hM(R′ ∩ Dn(X)). This proves the RCST if R′ ∩ Dn(X) ∈ C(M[∆n]), as for M = X ∗.

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For arbitrary monoid M′, one can’t use automata to show R′ ∈ R(M′), L′ ∈ C(M′) ⇒ R′ ∩ L′ ∈ C(M′), as “inputs” m ∈ M′ need not compose into factors from R′ ∩ L′.

• Claim. For finitely generated monoid M,

R ∈ R(M[∆2]) ⇒ R ∩ D2(M) ∈ C(M[∆2]). Proof: R is a component of the least solution ¯ R of a right-linear system y1 ≥ (w1,1 + . . .)y1 + . . . + (w1,m + . . .)ym + (w1 + . . .) . . . ym ≥ (wm,1 + . . .)y1 + . . . + (wm,m + . . .)ym + (wm + . . .)

• ver M[∆2], where wi,j, wi ∈ M[∆2]. Assume wi,j ∈ M ∪ ∆2 and

wi ∈ {0, 1} (by splitting y ≥ mbz into y ≥ my′, y′ ≥ bz etc.).

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We can assume that M is generated by the finite set X ⊆ M of elements of M occurring in this right-linear system, and use X ∗. Variables: [y, Z, z], with y, z ∈ {y1, . . . , ym}, Z ∈ M ∪ ∆2 ∪ {D} context-free G for R ∩ D2(M) definition of R contains S ≥ [y, D, z] y as main variable [y, D, z] ≥ m y ≥ mz with m ∈ X [y, d, z] ≥ d y ≥ dz with d ∈ ∆2 [y, D, 1] ≥ 1 y ≥ 1 [y, D, y] ≥ 1 [y, D, z] ≥ [y, D, y′][y′, D, z] [y, D, z] ≥ [y, pi, y′][y′, D, z′][z′, qi, z] Claim: if [y, D, z] ⇒k

G w ∈ M[∆2], then y ⇒∗ R wz, w ∈ D2(X)

by induction on k. So, if S ⇒∗

G w ∈ M[∆2], then w ∈ R ∩ D2(X).

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Claim: if w ∈ R ∩ D2(X), then w ∈ L(G). Proof: Suppose y ⇒k

R wz and w ∈ D2(X). There is a sequence

yi0 ≥ w1yi1, yi1 ≥ w2yi2, . . . , yik−1 ≥ wkyik s.th. y = yi0, w = w1 · · · wk ∈ D2(X) and yik = z. We show [yi0, D, yik] ⇒∗

G w1 · · · wk

by induction on the construction of w ∈ X ∗. If y in y ⇒k

R wz is

the main variable in the definition of R, it follows that S ⇒G [y, D, z] ⇒∗

G w, so w ∈ L(G).

If k = 0, then yi0 ⇒0

R wyi0, so w = 1, and [yi0, D, yi0] ⇒G 1.

If k = 1, then yi0 ≥ wyi1 and w ∈ R ∩ D2(X), hence yi1 ≥ 1 and w ∈ X, hence [yi0, D, yi1] ⇒G w (and [yi1, D, 1] ⇒G 1).

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Suppose k > 1. By the construction of D2(X), for w1 · · · wk ∈ X ∗ there either is j < k with u := w1 · · · wj, v := wj+1 · · · wk ∈ D2(X),

• r u := w2 · · · wk−1 ∈ D2(X) and w = p0uq0 or w = p1uq1.

In the first case, we have [yi0, D, yij] ⇒∗

G u and [yij+1, D, yik] ⇒∗ G v

by induction, hence [yi0, D, yik] ⇒G [yi0, D, yij][yij+1, D, yik] ⇒∗

G uv = w.

In the second case, we have [yi1, D, yik−1] ⇒∗

G u by induction, hence

[yi0, D, yik] ⇒G [yi0, pj, yi1][yi1, D, yik−1][yik−1, qj, yik] ⇒2

G

pj[yi1, D, yik−1]qj ⇒∗

G pjuqj = w.

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Conclusion

• 1. For monoids M, the fixed-point-closure of RM or FM is an

algebraic function of RM: QC

R(RM) = CM = ZC ′

2(RM ⊗R C ′

2).

• 2. All cf-languages L ∈ CX ∗ are values in ZC ′

2(RX ∗ ⊗R C ′

2) of

(certain) regular expressions over X ∪ ∆2.

• 3. For QC

R : DR → DC one can probably show

QC

RD = ZC ′

2(D ⊗R C ′

2)

by considering quotients D = RM/ρ QC

R(D)

= QC

R(RM/ρ) =? QC R(RM)/ρ′

= (ZC ′

2(RM ⊗R C ′

2))/ρ′ =? ZC ′

2(RM/ρ ⊗R C ′

2)

= ZC ′

2(D ⊗R C ′

2)

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Remark

Hopkins proves an RCST ZC2(RM ⊗R C2) ⊆ CM where Cn := R∆∗

n/(ρn ∪ {1 =

• {qipi | i < n}}).

But the additional relation 1 = {qipi | i < n} is not valid in a stack: at the empty stack 1, we cannot perform any qipi. To get a stack, we might add an emptyness test e, i.e. use 1 = e +

• {qipi | i < n},

ee = e, pie = 0 = eqj But these additional relations

◮ are not inherited from monoid equations, ◮ make understanding C2 harder than understanding C ′ 2 ◮ do not fit to the CST CM ⊆ ZC ′

2(RM ⊗R C ′

2) with C ′ 2 ◮ don’t(?) give CM ⊆ ZC ′

2(RM ⊗R C ′

2) ⊆? ZC2(RM ⊗R C2).

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Questions and References

Questions:

◮ Closure under A-transductions: if R ∈ A(M1 × M2) and

A ∈ AM1, is R(A) := {b ∈ M2 | (a, b) ∈ R, a ∈ A} ∈ AM2?

◮ Closure under matrix ring formation: if D ∈ DA, is

Dn×n ∈ DA? Then: Dn×n ≃ D ⊗A Bn×n. (F, R, C, P: yes)

◮ How much follows from results in category theory?

References: Mark Hopkins. The Algebraic Approach I+II: The Algebraization of the Chomsky Hierarchy + Dioids, Quantales and Monads. In Relational Methods in Computer Science/Applications of Kleene Algebra, Springer LNCS 4988, pages 155–190, 2008.

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Mark Hopkins and Hans Leiß. Coequalizers and tensor products for continuous idempotent semirings. In Proc. RAMiCS’17, pp 37–52, Springer LNCS 11194, 2018. Niels Grathwohl, Fritz Henglein, and Dexter Kozen. Infinitary axiomatization of the equational theory of context-free languages. In Fixed Points in Computer Science (FICS 2013), vol. 126 of EPTCS, pp 44–55, 2013. Hans Leiß and Mark Hopkins. C-dioids and µ-continuous Chomsky-algebras. In Proc. RAMiCS’17, pp 21–36, Springer LNCS 11194, 2018.

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