Fixpoints, Categories, and (Co)Algebraic Modeling

Peter Padawitz

TU Dortmund, Germany November 14, 2021

(actual version: https://fldit-www.cs.tu-dortmund.de/∼peter/DialgSlides.pdf)

The central chapters have boldface titles.

1 The Tai Chi of algebraic modeling 9

2 Preliminaries 10

Products 13

Sums 19

Miscellanea 26

3 Relations, ordered sets, and fixpoints 38

4 Categories 59

5 Functors and natural transformations 74

6 Limits and colimits 86

Limits 87

Colimits 96

7 Sorted sets and types 105

Type models 110

Lifting of sorted relations 114

Σ-arrows 119

Derived Σ-arrows 120

Sample constructive signatures 124

Sample destructive signatures 129

9 Σ-algebras 139

Sample algebras 149

Products, sums, invariants and quotients of Σ-algebras 172

Σ-terms and -coterms 184

Sample terms and coterms 188

Term and coterm algebras 194

Term folding (“denotational semantics”) 196

Sample initial algebras 204

Context-free grammars and their models 208

State unfolding (“operational semantics”) 224

Sample final algebras 243

Σ-flowcharts 263

10 Σ-formulas 276

Syntax 278

Derived Σ-formulas 281

Semantics 289

Sequent logic 316

Continuity of step functions and consequences 327

Deduction 339

Rule applicability 342

Resolution and narrowing 343

11 Fixpoint induction and coinduction 348

Incremental induction 353

Incremental coinduction 355

Duality of (co)resolution and (co)induction 363

12 F-algebras and F-coalgebras 365

Invariants and congruences on F-(co)algebras 376

Complete categories and continuous functors 379

Categorical construction of initial F-algebras and final F-coalgebras 383

13 Σ-functors 396

Functors for constructive signatures 396

Functors for destructive signatures 400

Final models of non-polynomial signatures 405

Recursive functions 413

Sample inductive definitions 433

Sample coinductive definitions 458

Sample biinductive definitions 484

From constructors to destructors 496

From destructors to constructors 500

Continuous algebras 504

14 Iterative equations 518

Term equations 519

Flowchart equations 541

Word acceptors 548

Tree acceptors 555

Iterative equations of a CFG 564

15 Algebraic induction and coinduction 572

Invariants and algebraic induction 572

Algebraic induction as fixpoint induction 578

Invariants are monotone 579

Congruences and algebraic coinduction 583

Algebraic coinduction as fixpoint coinduction 589

Quotients are monotone 601

Final models of destructive polynomial signatures 609

Bounded functors 612

17 Adjunctions 621

Sample adjunctions 625

1. Identity functor 625

2. Monoid functor 625

3. Sequence functor 627

4. Behavior functor 629

5. Weighted-set functors 631

6. Box and diamond functors 633

7. Strongly connected graph components 635

8. Reader and writer 637

9. Product 639

10. Coproduct 641

11. Term functors 643

12. Varieties 648

13. Coterm functors 662

14. Covarieties 665

15. Base algebra extensions 684

19 Conservative extensions 705

Constructor extensions 705

Destructor extensions 708

20 Abstraction 711

Abstraction with a least congruence 714

Abstraction with a greatest congruence 717

21 Restriction 720

Restriction with a greatest invariant 723

Restriction with a least invariant 727

22 λ-bialgebras 731

23 Monads and comonads 736

Sample monads 738

The sequence monad 743

Term monads 744

Sample comonads 751

The behavior comonad 753

Coterm comonads 754

24 Recursive functions defined by adjunctions or distributive laws 761

Arithmetic expressions 772

CCS 781

26 Bibliography 798

The Tai Chi of algebraic modeling constructors

& co/Horn clauses destructors

& co/Horn clauses

F(Ini) Ini Fin G(Fin)

unfold A

initial algebra

Ini/~

~ Σ-congruence

inv

Σ-invariant

F(A) algebra F'(Ini')

Ini'

G'(Fin')

Ini Σ'-consistent Fin'

fold' mono

coalgebra G(A) (Σ,AX)

≅

fold ≅

fold'

Ini Σ'-reachable fold' epi

final≅

coalgebra unfold'

nat inc

Fin Σ'-observabel unfold' mono

Fin Σ'-complete unfold' epi

≅

(Σ',AX')

inv = Ini

is the only Σ-invariant inc = id

Fin/~ = ^Fin

~ is the only Σ-congruence nat = id

∀ a,b ∈ Fin :

a=b iff ∃ ^Σ-congruence ~ : a~b coinduction

⇓

∀ B ⊆Ini :

B=Ini iff ∃ ^Σ-invariant inv : inv ⊆ B induction

⇓

L(Ini) A

F(R(B)) R(B)

B

op^#

R(Fin) B

op L(B) coalgebra G(L(B)) algebra

L:K

➛K Set^S

➛

op*

op L:Set^S➛K

R:K➛Set^S R:Set^S

object

abstraction behavior

restriction

recursive corecursive

extends

1 =_def {()}, 2 =_def {0,1}, N+ =_def N\ {0}, [0] = ∅ and for all n > 0, [n] =_def {1, . . . , n}.

Let A, B be sets and I be a nonempty set.

Both A → B and B^A denote the set of functions from A to B

|A| denotes the cardinality of A. |A| < ω means that A is finite.

id_A : A → A denotes the identity on A that maps every element of A to itself.

∆^I_A = {(a)_i∈I | a ∈ A} denotes the I-dimensional diagonal of A.

In particular, ∆_A =_def ∆^[2]_A . Let f, g : A → B and C ⊆ A.

A is the domain of f. B is the range of f.

graph(f) =_def {(a, b) ∈ A ×B | f(a) = b} is called the graph of f. img(f) =_def {f(a) | a ∈ A} is called the image of f.

ker(f) =_def {(a, a⁰) ∈ A² | f(a) = f(a⁰)} is called the kernel of f.

f is surjective if img(f) = B. f is injective if ker(f) = ∆_A. If both conditions hold true, then f is bijective.

Let a ∈ A and b ∈ B. The update f[b/a] : A → B of f at a by b, the restriction f|_C : C → B of f to C and the restricted equality =_C ⊆ B^A × B^A are defined as follows: For all a⁰ ∈ A and c ∈ C,

f[b/a](a⁰) =

b if a⁰ = a, f(a⁰) otherwise, f|_C(c) = f(c),

f =_C g ⇔ f|_A\C = g|_A\C.

inc_C : C → A denotes the inclusion function that maps every element of C to itself.

set2fun : P(A) → 2^A maps C ⊆ A to the indicator or characteristic function of C that maps all elements of C to 1 and all elements of A \C to 0.

set2fun is bijective.

rel2fun : P(A × B) → P(B)^A transforms binary relations into multivalued functions:

For all a ∈ A and R ⊆ A× B, rel2fun(R) maps a to {b ∈ B | (a, b) ∈ R}.

rel2fun is bijective.

Since A × 1 ∼= A and P(1) ∼= 2, rel2fun with B = 1 yields set2fun. Let A = B. Then f⁰ =_def id_A and for all n > 0, fⁿ⁺¹ =_def fⁿ ◦ f.

For all i ∈ I, let A_i be a set.

X

X

i∈I

A_i | ∀ i ∈ I : f(i) ∈ A_i}.

An element f ∈

X

For all n > 0, A₁ × · · · ×A_n =_def

X

X

i∈I A_i denotes the disjoint union of all A_i: ]

i∈I

A_i =_def {(a, i) | a ∈ A_i, i ∈ I}.

(a, i) ∈ U

i∈I A_i is also written as i(a).

For all n > 0, A₁ +· · · +A_n =_def Un

i=1A_i =_def U

i∈[n]A_i.

The universal properties of products and sums provide a good introduction into categor(y- theoret)ical thinking, i.e., reasoning in terms of equations between morphisms, here:

functions.

Every data model is a product, a sum, a subset, intuitively: a restriction, of a product or a quotient, intuitively: an abstraction, of a sum (see, e.g., chapter 6).

Products

Let A = (A_i)_i∈I be a tuple of sets, P be a set and π = (π_i : P → A_i)_i∈I be a tuple of functions. Since all functions have the same domain, such a tuple is called a cone.

The pair (P, π), also written as Q

i∈I A_i, is called a product of A if for all (f_i : B → A_i)_i∈I there is a unique function f : B → P such that for all i ∈ I,

π_i ◦ f = f_i. (1)

π_i is called the i-th projection of P and f the product extension or range tupling of (f_i)_i∈I to P. Since f is determined by (f_i)_i∈I, we write hf_ii_i∈I for f.

Consequently, for all f, g : B → P,

(∀ i ∈ I : π_i ◦ f = π_i ◦g) ⇒ f = g. (2)

All products of A are isomorphic to each other:

Let (P, π) and (P⁰, π⁰) be products ofA with tupling constructors h i and h i⁰, respectively.

Then hπ_i⁰i_i∈I : P⁰ → P is bijective with inverse hπ_ii⁰_i∈I.

Proof. For all i ∈ I, let f_i = π_i⁰ ◦ hπ_ii⁰_i∈I : P → A_i and f_i⁰ = π_i ◦ hπ_i⁰i_i∈I : P⁰ → A_i. Since for all i ∈ I, f = hπ_i⁰i_i∈I ◦ hπ_ii⁰_i∈I : P → P and f = id_P satisfy (1), both functions agree with each other, i.e.,

hπ_i⁰i_i∈I ◦ hπ_ii⁰_i∈I = id_P. (3) Since for all i ∈ I, f = hπ_ii⁰_i∈I ◦ hπ_i⁰i_i∈I : P⁰ → P⁰ and f = id_P⁰ satisfy π_i⁰ ◦ f = f_i⁰, both functions agree with each other, i.e.,

hπ_ii⁰_i∈I ◦ hπ_i⁰i_i∈I = id_P⁰. (4) By (3) and (4), hπ_ii⁰_i∈I : P → P⁰ is bijective with inverse hπ_i⁰i_i∈I : P⁰ → P. o

All sets that are isomorphic to a product of A are products of A.

Proof. Let (P, π) be a product of A, P⁰ be a set, h : P⁰ → P be a bijection and π⁰ = (π_i ◦h)_i∈I. Let f = (f_i : B → A_i)_i∈I and g = h⁻¹ ◦ hf_ii_i∈I : B → P⁰.

Then for all i ∈ I,

π⁰_i ◦ g = π_i ◦ h◦ g = π_i ◦h ◦h⁻¹ ◦ hf_ii_i∈I = π_i ◦ hf_ii_i∈I = f_i. g is unique: Let g⁰ : B → P⁰ satisfy π_i⁰ ◦ g⁰ = f_i for all i ∈ I. Then

π_i ◦ h◦ g = π_i⁰ ◦g = f_i = π_i⁰ ◦ g⁰ = π_i ◦ h◦ g⁰

and thus by (2), h◦ g = h◦ g⁰. Hence g = h⁻¹ ◦h ◦g = h⁻¹ ◦ h◦ g⁰ = g⁰. o

The Cartesian product

X

The projections and range tuplings for

X

• For all i ∈ I and f ∈

X

• For all (f_i : B → A_i)_i∈I, b ∈ B and i ∈ I, hf_ii_i∈I(b)(i) =_def f_i(b).

For all f = (f_i : A_i → B_i)_i∈I,

Y

i∈I

f_i =_def hf_i ◦ π_ii_i∈I is called the product of f.

For all f : A → B, nonempty sets I, n > 0 and (f_i : A_i → B_i)ⁿ_i=1, f^I =_def Q

i∈I f, f₁ × · · · × f_n =_def Q

i∈[n]f_i. For all functions f : Q

i∈I B_i → B, sets A and g = (g_i : A → B_i)_i∈I, lift^A(f)(g) =_def f ◦ hg_ii_i∈I : Y

i∈I

B_i^A → B^A.

Product equations

For all f : A → B, (f_i : B → B_i)_i∈I, (g_i : A_i → B_i)_i∈I, k ∈ I and (h_i : B_i → A_i)_i∈I, hπ_ii_i∈I = id^Q

i∈IA_i, (5)

hf_ii_i∈I ◦ f = hf_i ◦fi_i∈I, (6) π_k ◦ Y

i∈I

g_i = g_k ◦ π_k, (7)

Y

i∈I

h_i ◦ hf_ii_i∈I = hh_i ◦f_ii_i∈I, (8) ker(hf_ii_i∈I) = \

i∈I

ker(f_i). (9)

Product characterizations Let d = (d_i : P → A_i)_i∈I.

(P, d) is a product ofA iff for all a, b ∈ P, a, b ∈ P are equal iff for alli ∈ I, d_i(a) = d_i(b).

Proof. “⇒”: Let (P, d) be a product of A, f = λx.a : 1 → P and g = λx.b : 1 → P. Then

∀ i ∈ I : d_i(a) = d_i(b) ⇒ ∀ i ∈ I : d_i ◦ f = d_i ◦g ⇒⁽²⁾ f = g ⇒ a = f() = g() = b.

“⇐”: Suppose that for all a, b ∈ P, a, b ∈ P are equal iff for all i ∈ I d_i(a) = d_i(b).

Let (f_i : B → A_i)_i∈I. Then g : B → P with d_i(g(b)) = f_i(b) for all i ∈ I and b ∈ B_i is a product extension of (f_i)_i∈I because every f : B → P that satisfies (1) agrees with g.o (P, d) is a product of (A_i)_i∈I iff hd_ii_i∈I : P → Q

i∈I A_i is iso.

Proof. The “⇒”-direction is shown above.

“⇐”: Let hd_ii_i∈I be iso and (f_i : B → A_i)_i∈I. Then for all i ∈ I,

d_i ◦ hd_ii⁻¹_i∈I ◦ hf_ii_i∈I = π_i ◦ hd_ii_i∈I ◦ hd_ii⁻¹_i∈I ◦ hf_ii_i∈I = π_i ◦ hf_ii_i∈I = f_i. Hence f =_def hd_ii⁻¹_i∈I ◦ hf_ii_i∈I : B → P satisfies d_i ◦f = f_i.

Moreover, f is unique w.r.t. this property: Suppose that f, g : B → P satisfy d_i ◦ f = f_i = d_i ◦ g for all i ∈ I. Then

π_i ◦ hd_ii_i∈I ◦ f = d_i ◦ f = d_i ◦g = π_i ◦ hd_ii_i∈I ◦ g

and thus hd_ii_i∈I ◦f = hd_ii_i∈I ◦ g. Hence f = g because hd_ii_i∈I is iso. o (P, d) is a product of (A_i)_i∈I iff for all (f_i : B → A_i)_i∈I there is a function hf_ii_i∈I : B → P such that for all i ∈ I and f : A → P the following conditions hold true:

d_i ◦ hf_ii_i∈I = f_i, (10)

hd_i ◦ fi_i∈I = f. (11) Proof. “⇐”: Let (f_i : B → A_i)_i∈I and suppose that some hf_ii_i∈I : B → P satisfies (10) and (11). Let f, g : A → P satisfy d_i ◦ f = f_i = d_i ◦ g for all i ∈ I. Then

f ⁽¹¹⁾= hd_i ◦ fi_i∈I = hd_i ◦ gi_i∈I ⁽¹¹⁾= g.

Hence hf_ii_i∈I is unique w.r.t. (10), i.e., (P, d) is a product of (A_i)_i∈I.

“⇒”: Let (P, d) be a product of (A_i)_i∈I. Then (10) holds true. Moreover, for all f : A → P,

hd_i ◦ fi_i∈I ⁽⁶⁾= hd_ii_i∈I ◦ f ⁽⁵⁾= id_P ◦f = f,

i.e., (11) holds true. o

Sums

Let A = (A_i)_i∈I be a tuple of sets, S be a set and ι = (ι_i : A_i → S)_i∈I be a tuple of functions. Since all functions have the same range, such a tuple is called a cocone.

The pair (S, ι), also written as `

i∈I A_i, is called a sum or coproduct of A if for all (f_i : A_i → B)_i∈I there is a unique function f : S → B such that for all i ∈ I,

f ◦ ι_i = f_i. (12)

ι_i is called the i-th injection of S and f the sum extension or source tupling of (f_i)_i∈I to S. Since f is determined by (f_i)_i∈I, we write [f_i]_i∈I for f.

Consequently, for all f, g : S → B,

(∀ i ∈ I : f ◦ι_i = g ◦ι_i) ⇒ f = g. (13)

All sums of A are isomorphic to each other:

Let (S, ι) and (S⁰, ι⁰) be sums of A with tupling constructors [ ] and [ ]⁰, respectively.

Then [ι⁰_i]_i∈I : S → S⁰ is bijective with inverse [ι_i]⁰_i∈I.

Proof. For all i ∈ I, let f_i = [ι_i]⁰_i∈I ◦ ι⁰_i : A_i → S and f_i⁰ = [ι⁰_i]_i∈I ◦ ι_i : A_i → S⁰.

Since for all i ∈ I, f = [ι_i]⁰_i∈I ◦ [ι⁰_i]_i∈I : S → S and f = id_S satisfy (12), both functions agree with each other, i.e.,

[ι_i]⁰_i∈I ◦[ι⁰_i]_i∈I = id_S. (14) Since for all i ∈ I, f = [ι⁰_i]_i∈I ◦ [ι_i]⁰_i∈I : S⁰ → S⁰ and f = id_S⁰ satisfy f ◦ ι⁰_i = f_i⁰, both functions agree with each other, i.e.,

[ι⁰_i]_i∈I ◦ [ι_i]⁰_i∈I = id_S⁰. (15) By (14) and (15), [ι⁰_i]_i∈I : S → S⁰ is bijective with inverse [ι_i]⁰_i∈I : S⁰ → S. o

All sets that are isomorphic to a sum of A are sums of A.

Proof. Let (S, ι) be a sum of A, S⁰ be a set, h : S → S⁰ be a bijection and ι⁰ = (h◦ι_i)_i∈I. Let f = (f_i : A_i → B)_i∈I and g = [f_i]_i∈I ◦h⁻¹ : S⁰ → B. Then for all i ∈ I,

g ◦ ι⁰_i = g ◦ h◦ ι_i = [f_i]_i∈I ◦ h⁻¹ ◦ h◦ ι_i = [f_i]_i∈I ◦ ι_i = f_i.

g is unique: Let g⁰ : S⁰ → B satisfy g⁰ ◦ ι_i = f_i for all i ∈ I. Then g ◦ h◦ ι_i = g ◦ ι⁰_i = f_i = g⁰ ◦ι⁰_i = g⁰ ◦ h◦ ι_i

and thus by (12), g ◦h = g⁰ ◦ h. Hence g = g ◦h ◦h⁻¹ = g⁰ ◦h ◦h⁻¹ = g⁰. o The disjoint union U

i∈I A_i is a sum of A.

The injections and source tuplings for U

i∈I A_i are defined as follows:

• For all i ∈ I and a ∈ A_i, ι_i(a) =_def (a, i).

• For all (f_i : A_i → B)_i∈I, i ∈ I and a ∈ A_i, [f_i]_i∈I(a, i) =_def f_i(a).

For all (f_i : A_i → B_i)_i∈I,

a

i∈I

f_i =_def [ι_i ◦f_i]_i∈I is called the sum or coproduct of f.

For all nonempty sets I, f : A → B, n > 0 and (f_i : A_i → B_i)ⁿ_i=1, f × I =_def `

i∈I f, f₁ +· · · +f_n =_def `

i∈[n]f_i.

For all functions f : B → `

i∈I B_i, sets A and g = (g_i : B_i → A)_i∈I, lift_A(f)(g) =_def [g_i]_i∈I ◦f : Y

i∈I

A^Bi → A^B.

Sum equations

For all (f_i : A_i → A)_i∈I, f : A → B, (g_i : A_i → B_i)_i∈I, k ∈ I and (h_i : B_i → A_i)_i∈I, [ι_i]_i∈I = id^`

i∈IA_i, (16)

f ◦ [f_i]_i∈I = [f ◦ f_i]_i∈I, (17) a

i∈I

g_i ◦ ι_k = ι_k ◦g_k, (18)

[f_i]_i∈I ◦a

i∈I

h_i = [f_i ◦ h_i]_i∈I, (19) img([f_i]_i∈I) = [

i∈I

img(f_i). (20)

Sum characterizations Let (c_i : A_i → S)_i∈I.

(S, c) is a sum of (A_i)_i∈I iff for all a ∈ S there are unique i ∈ I and b ∈ A_i with c_i(b) = a.

Proof. “⇒”: Let (S, c) be a sum of (A_i)_i∈I. Assume that there is a ∈ S\S

i∈I c_i(A_i). Let f, g : S → 2 be defined as follows: f = λx.0 and g = (λx.if x ∈ S \ {a} then 0 else 1).

Then for all i ∈ I and b ∈ A_i,

(f ◦c_i)(b) = f(c_i(b)) = 0 = g(c_i(b)) = (g ◦c_i)(b), and thus by (13), f = g. Hence S = S

i∈I c_i(A_i).

For all i ∈ I, define f_i : A_i → U

i∈I A_i as follows: For all a ∈ A_i, f_i(a) = (a, i). Then for all i, j ∈ I, a ∈ A_i and b ∈ A_j, c_i(a) = c_j(b) implies

(a, i) = f_i(a) = [f_i]_i∈I(ι_i(a)) = [f_i]_i∈I(ι_j(b)) = f_j(b) = (b, j).

“⇐”: Suppose that for all a ∈ S there are unique i ∈ I and b ∈ A_i with ι_i(b) = a.

Let (f_i : A_i → B)_i∈I. Then g : S → B with g(c_i(b)) = f_i(b) for all i ∈ I and b ∈ A_i is a sum extension of (f_i)_i∈I because every f : S → B that satisfies (12) agrees with g. o (S, c) is a sum of (A_i)_i∈I iff [c_i]_i∈I : `

i∈I A_i → S is iso.

Proof. The “⇒”-direction is shown above.

“⇐”: Let [c_i]_i∈I be iso and (f_i : A_i → B)_i∈I. Then for all i ∈ I,

[f_i]_i∈I ◦ [c_i]⁻¹_i∈I ◦ c_i = [f_i]_i∈I ◦ [c_i]⁻¹_i∈I ◦ [c_i]_i∈I ◦ι_i = [f_i]_i∈I ◦ι_i = f_i. Hence f =_def [f_i]_i∈I ◦[c_i]⁻¹_i∈I : S → B satisfies f ◦c_i = f_i = g ◦c_i. Then

f ◦ [c_i]_i∈I ◦ι_i = f ◦ c_i = g ◦ c_i = g ◦[c_i]_i∈I ◦ι_i

and thus f ◦ [c_i]_i∈I = g ◦ [c_i]_i∈I. Hence f = g because [c_i]_i∈I is iso. o (S, c) is a sum of (A_i)_i∈I iff for all (f_i : A_i → B)_i∈I there is [f_i]_i∈I : S → B such that for all i ∈ I and f : S → A the following conditions hold true:

[f_i]_i∈I ◦c_i = f_i, (21)

[f ◦ c_i]_i∈I = f. (22)

Proof. “⇐”: Let (f_i : A_i → B)_i∈I and suppose that some [f_i]_i∈I : S → B satisfies (21) and (22). Let f, g : S → A satisfy f ◦ c_i = f_i = g ◦ c_i for all i ∈ I. Then

f ⁽²²⁾= [f ◦ c_i]_i∈I = [g ◦c_i]_i∈I ⁽²²⁾= g.

Hence [f_i]_i∈I is unique w.r.t. (21), i.e., (S, c) is a sum of (A_i)_i∈I.

“⇐”: Let (S, c) be a sum of (A_i)_i∈I. Then (21) holds true. Moreover, for all f : S → A, [f ◦ c_i]_i∈I ⁽¹⁷⁾= f ◦[c_i]_i∈I ⁽¹⁶⁾= f ◦ id_S = f,

i.e., (22) holds true. o

Miscellanea

The sets of nonempty or all words or finite lists over A are defined as follows:

A⁺ =_def S

n>0Aⁿ, A^∗ =_def A⁺ ∪ {}.

(a₁, . . . , a_n) ∈ Aⁿ is often written as a₁. . . a_n.

denotes the empty word and is supposed to different from other symbols that may occur in the same context.

|| =_def 0. For all n > 0 and w ∈ Aⁿ, |w| =_def n.

A^N is called the set of streams or infinite lists over A.

A^∞ =_def A^∗ ∪ A^N is called the set of colists over A.

For all sets A, B, Ω : A → B + 1 denotes the nowhere-defined function, i.e., for all a ∈ A, Ω(a) = ().

Functions and relations on words and streams

Let A be a set. For all v = (a₁, . . . , a_m), w = (b₁, . . . , b_n) ∈ A⁺ and f ∈ A^N, head() =_def (),

head(v) =_def a₁, head(f) =_def f(0),

tail() =_def (),

tail(v) =_def if m = 1 then else (a₂, . . . , a_m), tail(f) =_def λn.f(n+ 1),

· =_def , w · =_def w, · w =_def w, · f =_def f,

v · w =_def (a₁, . . . , a_m, b₁, . . . , b_n),

v · f =_def λi.if i < m then a_i+1 else f(m− i), ⁻¹ =_def ,

v⁻¹ =_def (a_m, . . . , a₁).

The concatenation operator · binds stronger than other binary word operator and is often omitted.

For all B ⊆ A^∗ and C ⊆ A^∞,

B · C =_def {a · b | a ∈ B, b ∈ C}, B⁻¹ =_def {a⁻¹ | a ∈ B}.

v ∈ A^∗ is a prefix of w ∈ A^∗ if w = v · v⁰ for some v⁰ ∈ A^∗.

The binary prefix relation ≤ on A^∗ is the set of all pairs (v, w) ∈ (A^∗)² such that v is a prefix of w.

For all v, w ∈ A^∗ and a ∈ A, #a(w) denotes the number of occurrences of a in w, v =_bag w ⇔_def v is a permutation of w, i.e., for all a ∈ A, #a(v) = #a(w),

v =_set w ⇔_def for all a ∈ A, #a(v) > 0 ⇔ #a(w) > 0.

Let A, B, C be sets, g : A → B, h : A → A +B, n ∈ N and p : A → 2.

g⁺ : A⁺ → B⁺

(a₁, . . . , a_n) 7→ (g(a₁), . . . , g(a_n)) g^∗ : A^∗ → B^∗

w 7→ if w = then else g⁺(s) hⁿ : A → A+ B

a 7→







a if n = 0

h(a⁰) if n > 0 ∧ a⁰ = hⁿ⁻¹(a) ∈ A b if n > 0 ∧ b = hⁿ⁻¹(a) ∈ B

h^C : A^C → B^C f 7→ h◦ f p? : A → A +A

a 7→ if p(a) = 1 then ι₁(a) else ι₂(a) pair : A → (A × B)^B

a 7→ λb.(a, b)

curry : C^A×B → (C^B)^A

f 7→ λa.λb.f(a, b) = f^B ◦ pair apply : B^A × A → B

(f, a) 7→ f(a) uncurry : (C^B)^A → C^A×B

f 7→ λ(a, b).f(a, b) = apply ◦(f × id_B) flip : (C^B)^A → (C^A)^B

f 7→ = λb.λa.f(a)(b) f : A^N → B^N is causal if for all s, s⁰ ∈ A^N and n ∈ N,

(s(0), . . . , s(n)) = (s⁰(0), . . . , s⁰(n)) implies f(s)(n) = f(s⁰)(n).

The set C(A, B) of causal functions from A^N to B^N is isomorphic to the set of functions from A⁺ to B.

Proof. Let g : B^A+ → C(A, B) and h : C(A, B) → B^A+ be defined as follows:

• For all f : A⁺ → B, s ∈ A^N and n ∈ N, g(f)(s)(n) = f(s(0), . . . , s(n)).

• For all f⁰ ∈ C(A, B), n ∈ N, (a₁, . . . , a_n+1) ∈ A⁺ and s ∈ A^N,

(s(0), . . . , s(n)) = (a₁, . . . , a_n+1) implies h(f⁰)(a₁, . . . , a_n+1) = f⁰(s)(n).

Since f is causal, h is well-defined. Moreover, for all f : A⁺ → B, s ∈ A^N, f⁰ ∈ C(A, B) and n ∈ N,

h(g(f))(s(0), . . . , s(n)) = g(f)(s)(n) = f(s(0), . . . , s(n)), g(h(f⁰))(s)(n) = h(f⁰)(s(0), . . . , s(n)) = f⁰(s)(n).

Hence g is bijective. o

Let h : A → B.

P_ω(A) =_def {C ⊆ A | C is finite}.

P(h) : P(A) → P(B) and P_ω(h) : P_ω(A) → P_ω(B) map every (finite) subset C of A to {h(c) | c ∈ C}.

P(h)(C) and P_ω(h)(C) are often is abbreviated to h(C).

supp(h) = {a ∈ A | h(a) 6∈ {0,∅, }} is called the support of h.

If supp(h) is finite, then h is called a B-weighted set with weights from B.

A →_ω B and B_ω^A denote the set of B-weighted sets with domain A.

N^A is called the set of bags or multisets of elements of A.

N^Aω is called the set of finite bags or finite multisets of elements of A.

2^A_ω ∼= P_ω(A).

Let (M,+,0) be a commutative monoid and C ⊆ M.

M_ω^h : M_ω^A → M_ω^B is defined as follows: For all f ∈ M_ω^A and b ∈ B, M_ω^h(f)(b) = X

{f(a) | a ∈ A, h(a) = b}. (1) M_ω^h is well-defined: For all f ∈ M_ω^A and b ∈ B,

b ∈ supp(M_ω^h(f)) ⇒⁽¹⁾ P

{f(a) | h(a) = b} 6= 0 ⇒ ∃ a ∈ A : f(a) 6= 0∧ h(a) = b

⇒ ∃ a ∈ supp(f) : h(a) = b ⇒ b ∈ h(supp(f)).

Hence |supp(M_ω^h(f))| ≤ |h(supp(f))| ≤ |supp(f)| < ω, i.e., M_ω^h(f) has finite support and thus belongs to M_ω^B.

M_C^A =_def {f ∈ M_ω^A | P

a∈Af(a) ∈ C} is called the set of constrained M-weighted sets with domain A and constraint ϕ (see [160], section 7).

M_C^h : M_C^A → M_C^B denotes the restriction of M_ω^h to M_C^A.

M_C^h is well-defined: |supp(M_C^h(f))| < ω can be proved along the lines of the above proof that M_ω^h(f) has finite support. Moreover,

P

b∈B M_C^h(f)(b) ⁽¹⁾= P

b∈B

P

a∈A,h(a)=bf(a) = P

a∈A,h(a)∈B f(a) = P

a∈Af(a) ∈ C.

Hence M_C^h(f) ∈ M_C^B.

Given a f ∈ M_ω^A with supp(f) = {a₁, . . . , a_n} and values m_i = f(a_i) for all 1 ≤ i ≤ m, f is often denoted by the expression

n

X

i=1

m_i · a_i where a₁, . . . , a_n are regarded as variables constants.

This notation also allows us to define M_C^h (and M_ω^h) simply as follows:

For all m₁, . . . , m_n ∈ M,

M_C^h(

n

X

i=1

m_i · a_i) =

n

X

i=1

m_i · h(a_i) (2)

(see [79], Def. 4.1.1).

D_ω(A) =_def (R^≥0)^A_{1} is called the set of (discrete) probability distributions of elements of A.

Since (R^≥0,+,0) is zero-sum free, i.e., if for all r, s ∈ R^≥0, r +s = 0 implies r = 0 and s = 0, D_ω(A) is a subset of [0,1]^A_ω where [0,1] denotes the closed interval of real numbers between 0 and 1.

D_ω(h) : D_ω(A) → D_ω(B) is defined as (R≥0)^h_{1}.

L ⊆ A^∗ is prefix closed if for all w ∈ A^∗ and a ∈ A wa ∈ L implies w ∈ L.

ltr(A, B) denotes the set of labelled trees over (A, B), i.e., functions from A^∗ to B+ 1 with prefix closed domain.

In fact, t ∈ ltr(A, B) represents a tree with edge labels from A and node labels from B such that two different edges each with the same source have different labels. t() is the label of the root and each word w ∈ A^∗ corresponds to a path that emanates from the root and ends in a node labelled with t(w).

If t(w) = () ⇔ w = , then t is identified with t(). Otherwise t may be written as follows:

t(){a → λw.t(aw) | a ∈ A, t(a) 6= ()}.

This notation is inspired by the syntax of Haskell records, i.e., data types with attributes (field names), which come here as edge labels.

For mutually disjoint edge label predicates p₁, . . . , p_n : A → 2, x{a . p₁(a) → t₁, . . . , a . p_n(a) → t_n | a ∈ A} stands for x(

n

[

i=1

{a → t_i | a ∈ A, p_i(a)}).

Further notational conventions

For all b ∈ B + 1 and sets I, b{i → t_i | i ∈ I} is also written as b(t_i)_i∈I. For all sets I, ()(t_i)_i∈I is also written as (t_i)_i∈I.

For all b ∈ B + 1, b{() → t} is also written as b(t).

For all b ∈ B + 1, b() is also written as b.

A leaf, i.e., a tree of the form Ω[b/], is identified with its label b.

t is finite if def (t) is finite. t is infinite if def (t) is infinite.

t is finitely branching if for all w ∈ A^∗, def (t) ∩ w · A is finite.

t is well-founded if there is n ∈ N such that for all w ∈ def (t), |w| ≤ n, intuitively: t has finite depth, i.e., all paths emanating from the root are finite.

t⁰ ∈ ltr(A, B) is a subtree of t ∈ ltr(A, B) if t⁰ = λw.t(vw) for some v ∈ X^∗. t ∈ ltr(A, B) is rational if t has only finitely many subtrees.

If t ∈ ltr(A, B) is rational and finitely branching, then def (t) is a regular subset of A^∗. t is rational iff t can be represented as a finite graph and thus as an iterative equation (see chapter 14). For instance, t = b{a → t} represents the tree t = λw.b with def(t) = {a}^∗.

ftr(A, B), fbtr(A, B), itr(A, B), wtr(A, B) and rtr(A, B) denote the sets of finite, finitely branching, infinite, well-founded and rational labelled trees over (A, B), respec- tively.

Labelled trees are used in chapter 9 as representations of the elements of initial as well as final models. This works mainly because

• functions on sets of well-founded labelled trees can usually be defined by structural induction and

• the values of functions into sets of (even non-well-founded) labelled trees over (A, B) can usually be defined by induction on A^∗.

Inductive definitions of the second kind become coinductive if one reformulates them in terms of non-functional representations of the labelled trees (see chapter 13).

Let A be a set and R be a binary relation on A, i.e., R is a subset of A². R is reflexive if R contains the 2-dimensional diagonal of A.

R is transitive if for all (a, b),(b, c) ∈ R, (a, c) ∈ R.

R is symmetric if for all (a, b) ∈ R, (b, a) ∈ R.

R is an equivalence relation on A if R is reflexive, transitive and symmetric. Then A/R =_def {[a]_R | a ∈ R} is called the quotient (set) of A by R where for all a ∈ A, [a]_R =_def {b ∈ A_s | (a, b) ∈ R_s} is called the equivalence class of A by R. nat_R : A → A/R denotes the natural function that maps every element of A to the equivalence class it belongs to.

The equivalence closure of R, R^eq, is the least equivalence relation that contains R.

R is antisymmetric if for all a, b ∈ A, aRb and bRa implies a = b.

R is a partial order and (A, R) is a partially ordered set or poset if R is reflexive, transitive and antisymmetric.

Let (A, R) be a poset.

R is a total order and (A, R) is a totally ordered set if for all a, b ∈ A, aRb or bRa.

If, in addition, for all a ∈ A, (a, a) ∈/ R, then R is strictly total.

R is well-founded if every nonempty subset of A contains a minimal element w.r.t. R.

If, in addition, R is total, then R is a well-order and, consequently, each nonempty subset of A has a least element w.r.t. R.

A labelled tree t over (A, B) is R-based if ∈ def (t) and for all w ∈ A^∗ and a, b ∈ A, wb ∈ def (t) ∧ (a, b) ∈ R ⇒ wa ∈ def (t).

otr(A, R, B) denotes the set of all R-based labelled trees over (A, B).

C ⊆ A is a chain if the restriction of R to C is a total order.

C ⊆ A is directed if every finite subset of C has a supremum in A.

Let (A,≤) be a poset and λ be an ordinal number, i.e., either 0 or

• a successor ordinal n + 1 = n∪ {n} for some ordinal n or

• a limit ordinal, i.e., the set of all smaller ordinals.

For instance, ω = N is the first limit ordinal.

A subset {a_i | i < λ} of A is a λ-chain or λ-cochain if for all ordinals i < j < λ, a_i ≤ a_j or a_j ≤ a_i, respectively.

A poset (A,≤) is λ-complete or a λ-CPO if A contains a least element ⊥_A w.r.t. ≤ and for each λ-chain B = {a_i | i < λ} of A, A contains the supremum F

B of B, also written as: F

i<λa_i.

A poset (A,≤) is λ-cocomplete or a λ-co-CPO if A has a greatest element > w.r.t.

≤ and for each λ-cochain B of A, A contains the infimum d

B of B, also written as:

d

i<λa_i.

A poset(A,≤) isλ-bicomplete or aλ-bi-CPOif Ais both λ-complete and λ-cocomplete.

Note that ≥ =_def ≤⁻¹ is a partial order iff ≤ is a partial order. However, completeness w.r.t. ≥ need not be the same as cocompleteness w.r.t. ≤ !

Examples

A λ-CPO (A,≤) is flat if for all a, b ∈ A, a ≤ b iff a ∈ {⊥_A, b}.

{⊥_A} and {⊥_A, a}, a ∈ A, are the only λ-chains of A.

⊥_A and a are the suprema of {⊥_A} and {⊥_A, a}, respectively.

For every set A, the powerset of A is a λ-CPO and a λ-co-CPO: The partial order is sub- set inclusion, the least element is the empty set, the greatest element is A, the supremum of a λ-chain is the union of the elements of the chain, and the infimum of a λ-cochain is the intersection of the elements of the chain.

For all sets A, B, the set A → B + 1 is a λ-CPO: The partial order is defined as follows:

For all f, g : A → B + 1,

f ≤ g ⇐⇒ ∀ a ∈ A : f(a) = g(a) ∨ f(a) = ().

The least element is the nowhere-defined function Ω : A → B + 1 and every λ-chain F ⊆ (A → B + 1) has a supremum: For all a ∈ A,

(G

F)(a) =

( f(a) if ∃ f ∈ F : f(a) 6= (), () otherwise.

A product of n λ-CPOs is a λ-CPO. Partial order, least element and suprema are defined componentwise.

The set A → B of functions from a set A to a λ-CPO (B,≤_B) is a λ-CPO. The partial order is defined argumentwise: For all f, g : A → B,

f ≤ g ⇔_def ∀ a ∈ A : f(a) ≤_B g(a). (1) The least element of A → B is λx.⊥_B. Suprema are defined argumentwise: For all λ-chains F ⊆ A → B and a ∈ A,

(G

F)(a) =_def G

f∈F

f(a). o (2)

Proposition DIR ([100], Cor. 1)

Let (A,≤) be λ-CPO. For all directed subsets B of A with |B| ≤ λ, A contains the supremum F

B of B.

Proof. We show the conjecture only for λ = ω and refer to the proof of [100], Thm. 1, for the generalization to arbitrary ordinal numbers.

Let B be a countable directed subset of A. If B is a chain, then F

B exists because A is ω-complete. Otherwise B is infinite: If B were finite, B would contain two different maximal elements w.r.t. R, which contradicts the directedness of B.

Since B is infinite, there is a bijection f : N → B. We define subsets B_i, i ∈ N, of B inductively as follows: B₀ = {f(0)} and B_i+1 = B_i ∪ {f(i), b_i} where i = min(f⁻¹(B \ B_i)) and b_i is an upper bound of f(i) and (all elements of) B_i. b_i exists because B is directed and B_i ∪ {f(i)} is a finite subset of B.

For all i ∈ N, B_i is finite and directed and thus a (countable) chain. Since A is ω- complete, B_i contains the supremum F

B_i of B_i. Since B_i ⊆ B_i+1, {F

B_i | i ∈ N} is also a countable chain and thus has a supremum c in A. c is the supremum of C = ∪_i<ωB_i: For all i ∈ N and b ∈ B_i, b ≤ F

B_i ≤ c. Hence c is an upper bound of C. Let d be an upper bound of C. Then for all i ∈ N, F

B_i ≤ d and thus c ≤ d.

Of course, ∪_i<ωB_i ⊆ B. Conversely, let b ∈ B. Since for all i ∈ N, |B_i| > i, there is k ∈ N with b ∈ B_k. Hence B = C and thus c = F

B. o

Let (A,≤) and (B,≤⁰) be posets and f : A → B. f is monotone if for all a, b ∈ A,

a ≤ b implies f(a) ≤⁰ f(b).

Let A and B have least element ⊥_A and ⊥_B, respectively.

f is strict if f(⊥_A) = ⊥_B.

Let A, B be λ-CPOs. f : A → B is λ-continuous if for all λ-chains C of A, f(G

C) = G

f(C).

Let A, B be λ-co-CPOs. f : A → B is λ-cocontinuous if for all λ-cochains C of A, f(l

C) = l

f(C).

Let A, B be λ-bi-CPOs. f : A → B is λ-bicontinuous if f is both λ-continuous and λ-cocontinuous.

Proposition If f is λ-continuous or λ-cocontinuous, then f is monotone. o Proposition MON Let f be monotone.

(1) For all λ-chains C of A, F

f(C) ≤ f(F C).

(2) f is λ-continuous iff for all λ-chains C of A, f(F

C) ≤ F

f(C).

(3) f is λ-cocontinuous iff for all λ-cochains C of A, d

f(C) ≤ f(d C).

(4) f is λ-continuous if A is chain-finite, i.e., all λ-chains of A are finite.

(5) f is λ-cocontinuous if A is cochain-finite, i.e., all λ-cochains of A are finite. o Given λ-CPOs A and B, A →_c B denotes the set of λ-continuous functions from A to B.

Since Ω and suprema of λ-chains of λ-continuous functions are λ-continuous, A →_c B is a λ-CPO.

Let f : A → A.

a ∈ A is f-closed or f-reductive if f(a) ≤ a.

a is f-dense or f-extensive if a ≤ f(a).

a is a fixpoint of f if f(a) = a.

Kleene’s Fixpoint Theorem [86] (also known as Kleene’s first recursion theorem) (1) Let A be an ω-CPO, f : A → A be monotone and the upper Kleene closure f^∞ =_def F

n<ω fⁿ(⊥) be f-closed. Then f^∞ is the least fixpoint of f.

(2) Let A be an ω-co-CPO, f : A → A be monotone and the lower Kleene closure f_∞ =_def d

n<ω fⁿ(>) be f-dense. Then f_∞ is the greatest fixpoint of f. Proof. (1) Since f is monotone, {fⁿ(⊥) | n < ω} is an ω-chain.

Let a be f-closed. Then fⁿ(⊥) ≤ a for all n ∈ N. (3)

We show (3) by induction on n: f⁰(⊥) = ⊥ ≤ a. If fⁿ(⊥) ≤ a, then fⁿ⁺¹(⊥) ≤ f(a) ≤ a because f is monotone and a is f-closed.

Since f^∞ is f-closed, f(f^∞) is f-closed. Hence by (3), f^∞ = F

n<ω fⁿ(⊥) ≤ f(f^∞), i.e., f is also f-dense. We conclude that f^∞ is a fixpoint of f.

Let a be a fixpoint of f. Then a is f-closed and thus by (3), fⁿ(⊥) ≤ a for all n ∈ N. Hence f^∞ ≤ a, i.e., f^∞ is the least fixpoint of f.

(2) Analogously. o

Proposition INF1 Let f : A → A be monotone.

(1) If f^∞ = fⁿ(⊥) for some n ∈ N, then f^∞ is f-closed.

(2) If f is ω-continuous, then f^∞ is f-closed.

(3) If f_∞ = fⁿ(>) for some n ∈ N, then f_∞ is f-dense.

(4) If f is ω-continuous, then f_∞ is f-dense.

Proof. (1) Suppose that f^∞ = fⁿ(⊥) holds true for some n ∈ N. Then f(f^∞) = f(fⁿ(⊥)) = fⁿ⁺¹(⊥) ≤ F

i<ωfⁱ(⊥) = f^∞. (2) Suppose that f is ω-continuous. Then

f(f^∞) = f(F

i<ω fⁱ(⊥)) ≤ F

i<ωf(fⁱ(⊥)) ≤ F

i<ωfⁱ(⊥) = f^∞.

(3) and (4): Analogously. o

Proposition INF2 Let f : A → A be monotone. Then for all n ∈ N,

f^∞ = fⁿ(⊥) ⇔ ∀ i > n : fⁱ(⊥) = fⁿ(⊥), (1) f_∞ = fⁿ(⊥) ⇔ ∀ i > n : fⁱ(>) = fⁿ(>). (2) Proof. (1) “⇒”: Let f^∞ = fⁿ(⊥). Then for all i > n, fⁱ(⊥) ≤ F

k<ω f^k(⊥) = f^∞ = fⁿ(⊥) and thus fⁱ(⊥) = fⁿ(⊥) because fⁿ(⊥) ≤ fⁱ(⊥).

“⇐”: Suppose that for all i > n, fⁱ(⊥) = fⁿ(⊥). Then for all i ∈ N, fⁱ(⊥) ≤ fⁿ(⊥), and thus f^∞ = F

i<ω fⁱ(⊥) ≤ fⁿ(⊥). Hence f^∞ = fⁿ(⊥) because fⁿ(⊥) ≤ F

i<ωfⁱ(⊥) = f^∞.

(2) Analogously. o

Fixpoint Theorem for finite posets

Let A be a finite poset and f : A → A be monotone.

(1) If A has a least element ⊥, then for some n < ω, f^∞ = fⁿ(⊥) is the least fixpoint of f.

(2) If A has a greatest element >, then for some n < ω, f_∞ = fⁿ(>) is the greatest fixpoint of f.

Proof. (1) Since f is monotone, induction on i implies fⁱ(⊥) ≤ fⁱ⁺¹(⊥) for all i ∈ N. Hence there is n ∈ N such that fⁿ(⊥) = fⁿ⁺¹(⊥) because A is finite. Therefore, f(fⁿ(⊥)) = fⁿ⁺¹(⊥) = fⁿ(⊥) ≤ f(fⁿ(⊥)) and thus f(fⁿ(⊥)) = fⁿ(⊥). Induction on i implies fⁱ(⊥) = fⁿ(⊥) for all i > n. Hence by Proposition INF2 (1), f^∞ = fⁿ(⊥), and thus by Proposition INF1 (1), f^∞ is f-closed. We conclude by Kleene’s Fixpoint Theorem that f^∞ is the least fixpoint of f.

(2) Analogously. o