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Thirty Years of Coalgebra:
WhatHave we Learned?
Larry Moss
Indiana University, Bloomington
Stanford Logic Seminar
October 27, 2015
1/51
History
During 1984-85,
Peter Aczel visited CSLI here at Stanford.
He was working on his book Non-Well-Founded Sets.
The book can be read in several ways:
◮ a solution to a semantic problem coming from theoretical
computer science
◮ a contribution to the foundational aspects of set theory.
The set-theoretic aspect was by far more original and
memorable.
In fact, Aczel proposed changing the very axioms of set
theory(!),
replacing the Foundation Axiom with the Anti-Foundation
Axiom.2/51
next chapter in the history
Jon Barwise got interested in subject due to connections with
circularity.
He wrote the introduction to Aczel’s book.
Later he recruited me to co-author a book which makes circular
phenomena the central theme.
He wanted the book to be “popular”, so the key decisions were.
◮ to emphasize the Anti-Foundation Axiom.
◮ to take the few category-theoretic ideas in Aczel’s book,
and replace them with more familiar and more concrete
ideas.
3/51
next chapter in the history
Aczel’s hardest technical result were called
The Final Coalgebra Theorems
but we didn’t know what a final coalgebra was.
One of the theorems talked about
Functors preserving weak pullbacks.
We didn’t understand that, either.
Worst of all, we could understand the book and even extend
the ideas without understanding a long list of unfamiliar ideas.
4/51
What happened to me later
The overall issues of circularity are still with me.
I certainly have not learned all that I want to about the subject.
When I would give talks on the book, people would say
◮ Larry, this is nice, but where is the computation?
◮ If this is about circularity, where is the circle?
What about fractals?
And there were mathematical topics related to circularity that I
just couldn’t answer.
5/51
Harsanyi type spaces
Consider two-person games of imperfect information.
Each player has beliefs about the world and about the other.
In fact, the type of each player is “just” a probability measure on
possible ways the world can be × possible types
6/51
Harsanyi type spaces
Consider two-person games of imperfect information.
Each player has beliefs about the world and about the other.
In fact, the type of each player is “just” a probability measure on
possible ways the world can be × possible types
But what are these types?
Dropping the two-player aspects, we seem to be asking to solve
M = measurable space of probability measures on (R ×M)
And if that isn’t hard enough, we want the biggest solution.
6/51
History 3
About the same time as we were working on Vicious Circles,
researchers in Europe got interested in Aczel’s book.
But their interest was in many ways the opposite of ours.
For them, the category-theoretic aspects of the book were
highly suggestive,
mostly because they could see connections to other topics that
they were interested in.
7/51
What happened next
In order for me to continue my work on circularity,
I basically had to join the coalgebra community.
I had to engage with areas of mathematics that I had never
thought about.
Jon Barwise held that work on non-wellfounded sets was
“lovely mathematics,”
Coalgebra has been even prettier, and this has kept my interest.
8/51
twenty coalgebraists
Jirı Adamek
Filippo Bonchi
Corina Cirstea
H. Peter Gumm
Helle Hansen
Ichiro Hasuo
Bart Jacobs
Bartek Klin
Barbara Konig
Dexter Kozen
Alexander Kurz
Paul Blain Levy
Stefan Milius
Daniela Petrisan
Dirk Pattinson
John Power
Jan Rutten
Lutz Schroder
Alexandra Silva
Yde Venema
9/51
The conceptual comparison chartThis chart is my real goal; everything else is secondary
set with algebraic set with transitions
operations and observations
algebra for a functor coalgebra for a functor
initial algebra final coalgebra
least fixed point greatest fixed point
congruence relation bisimulation equivalence rel’n
equational logic modal logic
recursion: map out of corecursion: map into
an initial algebra a final coalgebra
Foundation Axiom Anti-Foundation Axiom
iterative conception of set coiterative conception of set
useful in syntax useful in semantics
bottom-up top-down
10/51
Algebras and coalgebras
LetA be a category, and let F :A→A be a functor.
An F-algebra is (A , a : FA → A).An F-coalgebra is (A , a : A → FA).
In both cases, A is the carrier and a the structure.
Example: deterministic automata
(S , s : S → 2 × SA )
are coalgebras of 2 × XA ,
as are Kripke models for modal logic.
11/51
Morphisms of algebras and coalgebras
Let (A , a : FA → A) and (B , b : FB → B) be algebras.
A morphism is f : A → B in the same underlying category so that
FAa //
Ff��
A
f��
FBb
// B
commutes.
Let (A , a : A → FA) and (B , b : B → FB) be coalgebras.
A morphism is f : A → B in the same underlying category so that
Aa //
f��
FA
Ff��
Bb
// FB
commutes.12/51
Example: FX = 1 + (X × X)?
Iterate it to get a fixed point
µF =⋃
i∈ω
F i(∅)
F1∅ : •
F2∅ : •,• •������
////
//
F3∅ : •,• •������
////
//
, •
• •
������
////
///
������
////
// , •
• •
////
//
�������
������
////
// ,
• • • •
////
///
�������
������
))))
))
������
))))
))
µF ≈ 1 + (µF × µF) = F(µF).
13/51
Example: FX = 1 + (X × X)?
Recursion Principle for Finite Trees
For all sets X , all x ∈ X , all f : X × X → X ,
there is a unique ϕ : µF → X
so that ϕ is
one-point tree • 7→ x
t u
////
///
�������
������
))))
))
������
))))
)) 7→ f(ϕ(t), ϕ(u))
A fixed point is both an algebra and a coalgebra.
13/51
Example: FX = 1 + (X × X)?
Recursion Principle for Finite Trees
For all sets X , all f : 1 + (X × X)→ X ,
there is a unique ϕ : µF → X so that
F(µF)id //
1+(ϕ×ϕ)
��
µF
ϕ
��FX
f// X
commutes, where (ϕ × ϕ)(t , u) = (ϕ(t), ϕ(u)).
13/51
Recursion on N is tantamount to Initiality
Recursion on N: For all sets A , all a ∈ A , and all f : A → A ,
there is a unique ϕ : N → A so that
ϕ(0) = a, and ϕ(n + 1) = f(ϕ(n)) for all n.
Initiality of N: For all (A , a), there is a unique homomorphism
ϕ : (N, ν)→ (A , a):
1 + Nid //
1+ϕ
��
N
ϕ
��1 + A a
// A
Recursion is often, but not always,
about maps out of an initial algebra.
14/51
Review: initial algebras and final coalgebras
initial algebra FAa //
Ff
��
A
f
��FB
b// B
Aa //
f
��
FA
Ff
��B
b// FB final coalgebra
15/51
Initial algebras and final coalgebras
of various set functors
initial algebra
1 + X ∗ // 0 // 1 // 2 // · · ·
1 + (X × X) finite binary trees
{a, b , c} × X ∅
PfinX Vω = hereditarily finite sets
final coalgebra
1 + X ∗ 0oo 1oo 2oo · · ·oo ∞ ff1 + (X × X) finite and infinite binary trees
{a, b , c} × X infinite streams on {a, b , c}
PfinX finitely branching graphs with a ‘top’ point,
modulo bisimulation
16/51
Initial algebras and final coalgebras
of various set functors
initial algebra
1 + X ∗ // 0 // 1 // 2 // · · ·
1 + (X × X) finite binary trees
{a, b , c} × X ∅
PfinX Vω = hereditarily finite sets
final coalgebra
1 + X ∗ 0oo 1oo 2oo · · ·oo ∞ ff1 + (X × X) finite and infinite binary trees
{a, b , c} × X infinite streams on {a, b , c}
PfinX finitely branching graphs with a ‘top’ point,
modulo bisimulation
Query: is there any relation between these charts?
And how do we get these examples, anyway?
16/51
How can we get our hands on µF?Answer: generalize Kleene’s Theorem
Kleene’s Theorem
Let (A ,≤) be poset with a least element 0 and with the property
that every countable chain C ⊆ A has a least upper bound∨
C.
Let F : A → A be monotone and ω-continuous.
Then the least upper bound of
0 ≤ F0 ≤ F20 ≤ · · ·
has the property that F(µF) ≤ µF ,
and indeed is the least fixed point of F .
17/51
The category-theoretic generalization“Preorders are the poor person’s category”
order-theoretic concept categorification
preorder (A ,≤) categoryA
x ≤ y and y ≤ x A and B are isomorphic objects
least element 0 initial object 0
monotone F : A → A functor F :A→A
pre-fixed point: Fx ≤ x F-algebra: f : FA → A
countable chain functor from (ω,<) to A
F is ω-continuous F preserves ω-colimits
least pre-fixed point: Fx ≤ x initial F-algebra: f : FA → A
18/51
A generalization of Kleene’s Theorem
Kleene’s Theorem
Let (A ,≤) be poset with a least element 0 and with the property
that every countable chain C ⊆ A has a least upper bound∨
C.
Let F : A → A be monotone and ω-continuous.
Then the least upper bound of
0 ≤ F0 ≤ F20 ≤ · · ·
has the property that F(µF) ≤ µF ,
and indeed is the least fixed point of F .
Adamek 1974
LetA be a category with initial object 0
and with the property that every ω-chain in A has a colimit.
Let F : A→A preserve ω-colimits,
let µF be the colimit of the initial sequence of F :
0! // F0
F! // F20F2! // · · · Fn−1! // Fn0
Fn! // Fn+10Fn+1! // · · ·
There is a canonical m : F(µF)→ µF
and (µF ,m) is an initial F-algebra.
19/51
Where does it apply?
FX initial algebra µF
1 + (X × X) finite binary trees
PfinX HF
Pκ Hκ
1 + X natural numbers
signature functor terms on the signature
bag functor finite unordered trees
Adamek’s generalization of Kleene’s Theorem is not the only
way to get an initial algebra, but it is the most common.
20/51
Lambek’s Lemma
Lambek’s Lemma
Let P be a poset.
Let f : P→ P be a monotone function.
And let x be a minimal pre-fixed point:
fx ≤ x, and x minimal like this.
Then: fx = x.
Proof.
Note that f(fx) ≤ fx.
And so x ≤ fx, too. �
21/51
Lambek’s Lemma
Categorification of this fact:
Lambek’s Lemma
The structure morphisms of initial algebras
are always categorical isomorphisms.
21/51
Lambek’s Lemma
Categorification of this fact:
Lambek’s Lemma
The structure morphisms of initial algebras
are always categorical isomorphisms.
In particular, P has no initial algebra on Set,
as we see from Cantor’s Theorem.
21/51
Review/Example
initial algebra 1 + Nt //
id1+ϕ
��
N
ϕ
��1 + A a
// A
N is an initial algebra of 1 + X on Set
Coalgebras of 2 × XA are deterministic automata
Ss //
ϕ
��
2 × SA
id2×ϕA
��L
l// 2 × LA final coalgebra
The set L of formal languages is a final coalgebra.
The map ϕ takes a state to the language accepted there.
22/51
Where we are
set with algebraic set with transitions
operations and observations
algebra for a functor coalgebra for a functor
initial algebra final coalgebra
useful in syntax useful in semantics
23/51
Where we are
set with algebraic set with transitions
operations and observations
algebra for a functor coalgebra for a functor
initial algebra final coalgebra
useful in syntax useful in semantics
In some ways, the mathematics of transitions and observations
is less familiar than that of sets and operations.
Coalgebra is trying to be the general mathematics of transitions
and observations.
23/51
Final Coalgebras: why and what?
Final coalgebras are like the most abstract collections of
“transitions” or “observations”.
Again, given F , does the initial algebra exist?
Does the final coalgebra?
How can we get our hands on them?
24/51
The main existence theorem for initial
algebras/final coalgebras
Adamek 1974
Assume thatA has enough structure
to take the colimit µF of the initial sequence
0! // F0
F! // F20F2! // · · · Fn−1! // Fn0
Fn! // · · ·
and that F : A→ A preserves this ω-colimit.
There is a canonical morphism m : F(µF)→ F such that
(µF ,m)
is an initial F-algebra.
25/51
The main existence theorem for initial
algebras/final coalgebras
Barr 1993
Assume thatA has enough structure
to take the limit νF of the initial sequence
1 F1!oo F21
F!oo F2!oo · · · Fn1Fn−1!oo · · ·Fn!oo
and that F : A→ A preserves this ω-limit.
There is a canonical morphism m : F → F(νF) such that
(νF ,m)
is a final F-coalgebra.
25/51
Example: streams over 2 = {0, 1}
Here our functor is FX = 2 × X .
1 is any one point set, say {∗}.
So F1 = 2 × 1 = {(0, ∗), (1, ∗)}.
F21 = 2 × F1 = {(0, (0, ∗)), (0, (1, ∗)), (1, (0, ∗)), (1, (1, ∗))}.
1 F1!oo F21
F!oo · · · Fn1 Fn+11Fn!oo · · ·
L
ln
OO
26/51
Example: streams over 2 = {0, 1}
Here our functor is FX = 2 × X .
1 F1!oo F21
F!oo · · · Fn1 Fn+11Fn!oo · · ·
L
ln
OO
A representation that you have seen:
take L = 2N,
ln : L → Fn1 is f 7→ (f(0), (f(1), (f(2), . . . f(n))))).
The coalgebra structure m : 2N → 2 × 2N is a little easier:
m(f) = (f(0), n 7→ f(n + 1)).
26/51
Corecursion on streams is tantamount to finality
Corecursion on 2N: For all sets A , all f : A → 2 × A ,
there is a unique ϕ : A → 2N so that
ϕ(x) = (head f(x), ϕ(tail(f(x))))
Finality of 2N: For all (A , a), there is a unique homomorphism
ϕ : (N, ν)→ (A , a):
Af //
ϕ
��
2 × A
id2×ϕ
��2N
〈head,tail〉// 2 × 2N
Corecursion is about maps into a final algebra.
27/51
Streams: illustration
Consider a coalgebra (A , a : A → R × A), where
A = {α, β, γ, δ}, and
a(α) = (0, β)a(β) = (1, γ)
a(γ) = (0, δ)a(δ) = (−1, α)
What is the map h below?
Aa //
h��
R × A
Fh��
RN
ϕ//R ×RN
28/51
Streams: illustration
Consider a coalgebra (A , a : A → R × A), where
A = {α, β, γ, δ}, and
a(α) = (0, β)a(β) = (1, γ)
a(γ) = (0, δ)a(δ) = (−1, α)
What is the map h below?
Aa //
h��
R × A
Fh��
RN
ϕ//R ×RN
h(α) = (0, h(β)) = (0, 1, 0,−1, 0, 1, . . .)h(β) = (1, h(γ)) = (1, 0,−1, 0, 1, 0, . . .)h(γ) = (0, h(δ)) = (0,−1, 0, 1, 0, 1, . . .)h(δ) = (−1, h(α)) = (−1, 0, 1, 0, 1, 0, . . .)
28/51
Where does the theorem apply?
Finitary Iteration
LetA be a category with final object 1
and with the property that every ωop-chain in A has a limit.
Let F : A→A preserve ωop-limits,
and consider the final ωop-chain of F :
1 F1!oo F21
F!oo · · · Fn1 Fn+11Fn!oo · · ·
Let νF be its limit, and let m : νF → F(νF) be the factorizing
morphism.
Then (νF ,m) is a final F-coalgebra.
We don’t really need all limits, only the one shown.
And this is the only limit we need F to preserve.
29/51
Where does the theorem apply?
Finitary iteration gives final coalgebras for all functors on Set
built from
◮ the identity functor
◮ constant functors
and using
◮ +, ×, FA for fixed sets A
◮ composition
But it doesn’t work for Pfin or its relatives Pκ.
It doesn’t work for the discrete measure functor ∆,
either.
30/51
Where does the theorem apply?
Finitary iteration gives final coalgebras for all functors on
compact Hausdorff spaces built from
◮ the identity functor
◮ constant functors
◮ the Vietoris functor V.
V(X) is the hyperspace of X ,
the set of compact subsets of X , with a certain topology.
For f : X → Y and A ∈ VX ,
(Vf)A = f [A ].
◮ the Borel measure functor B. For f : X → Y and A ∈ VX ,
((Bf)µ)A = µ(f−1(A))
and using
◮ +, ×
◮ composition31/51
Where does the theorem apply?
Finitary iteration gives final coalgebras for all functors on MS
built from
◮ the identity functor
◮ constant functors
◮ εPk , the scaled version of the closed set functor Pk ,
using the Hausdorff distance
d(s, t) = max{supx∈s
infy∈t
d(x, y), supx∈s
infy∈t
d(x, y)}.
The distance from ∅ to any other closed set is 1.
ε < 1 scales distances.
◮ using +, ×, and composition.
van Breugel: Pk without scaling has no final coalgebra
on MS.
32/51
Iteration in CPO⊥-enriched categories
A CPO⊥ is a complete partial order with ⊥.
A is CPO⊥-enriched if its homsetsA(X ,Y)carry the structure of a CPO with ⊥
and composition is strict (preserves the least element) and
continuous (preserves ω-joins) in both variables.
F : A→ A is locally continuous if F⊔
fn =⊔
Ffnfor all ω-chains fn ∈ A(X ,Y).
33/51
Iteration in CPO⊥-enriched categories
A CPO⊥ is a complete partial order with ⊥.
A is CPO⊥-enriched if its homsetsA(X ,Y)carry the structure of a CPO with ⊥
and composition is strict (preserves the least element) and
continuous (preserves ω-joins) in both variables.
F : A→ A is locally continuous if F⊔
fn =⊔
Ffnfor all ω-chains fn ∈ A(X ,Y).
Theorem (Adamek, based on Smyth and Plotkin 1982)
Every locally continuous F : A→A has a canonical fixed point:
there is an initial algebra and it is the inverse of a final
coalgebra.
This result is at the core of Dana Scott’s construction of
D � [D → D]
giving a model of the lambda calculus.33/51
Where does the theorem apply?
SB = standard Borel spaces,
measurable spaces which use the Borel subsets of a Polish
space
∆ : SB→ SB takes M to the set of its Borel probability
measures with σ-algebra generated by
{Bp(E) | p ∈ [0, 1],E ∈ Σ},
where
Bp(E) = {µ ∈ ∆(M) | µ(E) ≥ p}.
One uses the Kolmogorov Consistency Theorem to see
that the functor preserves the limit.
∆ : SB→ SB has a final coalgebra, as does a functor like
FX = ∆(X × [0, 1])
34/51
What about ∆ : Meas→ Meas?
Viglizzo 2005
The functor ∆ : Meas→ Meas does not preserve limits of
ωop-chains.
So it looks like we’re out of luck!
35/51
Pavlovic and Pratt 2002
Consider FX = N × X on Set
Final coalgebra is the Baire space B = irrationals in [0, 1],with structure
〈β, γ〉 : B → N × B
where
β(x) =
⌊
1
x
⌋
− 1 and γ(x) =
(
1
x
)
mod 1
γ is called the Gauss map.
The set Nω of streams on N is also a final coalgebra,
and the isomorphism
Nω
ϕ
��
〈head,tail〉 // N × Nω
N×ϕ
��B
〈β,γ〉// N × B
is given by continued fractions.
36/51
[0, 1] as a final coalgebra
Let BiP be the category of bi-pointed sets.
These are (X ,⊤,⊥) with ⊤,⊥ ∈ X and ⊤ , ⊥.
The bipointed set {⊤,⊥} is initial, but there is no final object.
37/51
[0, 1] as a final coalgebra
Let BiP be the category of bi-pointed sets.
These are (X ,⊤,⊥) with ⊤,⊥ ∈ X and ⊤ , ⊥.
The bipointed set {⊤,⊥} is initial, but there is no final object.
X XX X7→⊤⊥ ⊤⊥
identify ⊤ of left with ⊥ of right
Initial algebras and final coalgebras of F : BiP→ BiP
0 is the two point space {0, 1} with d(0, 1) = 1.
F0 is {0, 12 , 1} with evident distances.
F20 is {0, 14, 1
2, 3
4, 1}.
Initial algebra of F on BiP is dyadic rationals.
37/51
[0, 1] as a final coalgebra
Let BiP be the category of bi-pointed sets.
These are (X ,⊤,⊥) with ⊤,⊥ ∈ X and ⊤ , ⊥.
The bipointed set {⊤,⊥} is initial, but there is no final object.
Initial algebras and final coalgebras of F : BiP→ BiP
Freyd 1999:
νF can be taken to be the (set of points in) unit interval [0, 1],with a structure
i : [0, 1]→ F [0, 1]
as shown below.
37/51
The map i : [0, 1]→ F [0, 1]
d 12dd 1
2d→10 10
a < 12
7→ 2a on left12
7→ midpoint
a > 12
7→ 2a on right
Note that i is an isometry.
38/51
Proof of Freyd’s Theorem
Let e : X → FX be any morphism of BiP.
Recall that [0, 1] is a complete metric space.
Regard the set X a (discrete) space.
The space
S = homBiP(X , [0, 1]).
is a closed subspace of homCMS(X , [0, 1]).
F : homCMS(X , [0, 1])→ homCMS(FX ,F [0, 1])
is a contracting map: for f , g : X → [0, 1],
d(Ff ,Fg) ≤1
2d(f , g).
39/51
Proof of Freyd’s Theorem
Recall
i : [0, 1]→ F [0, 1].
This map is a bijection and an isometry.
We have a contracting endofunction ψ : S → S given by
ψ(f) = i−1 · Ff · e.
By the Contraction Mapping Thm., there’s a unique f = ψ(f).
f is exactly a coalgebra morphism (X , e)→ ([0, 1], i).
39/51
The Sierpinski Gasket is a final coalgebra, even
with the metric
With the topology but not the metric,
this was done in the seminal paper of Leinster.
With 3 IU graduate students, we got it with the metric,
using tripointed sets.
There are further connections to be made with metric geometry
and with self-similar groups.
40/51
Finality at work: FX = R × X
RA is the set of function f : R→ R such that
for all n, fn(0) exists, and f agrees with its
Taylor series on an interval containing 0.
The final coalgebra is (RA , ϕ) where
ϕ : RA → R × RA is f 7→ (f(0), f ′)
41/51
Finality at work: FX = R × X
RA is the set of function f : R→ R such that
for all n, fn(0) exists, and f agrees with its
Taylor series on an interval containing 0.
The final coalgebra is (RA , ϕ) where
ϕ : RA → R × RA is f 7→ (f(0), f ′)
Consider a coalgebra (A , a : A → R × A), where
A = {α, β, γ, δ}, and
a(α) = (0, β)a(β) = (1, γ)
a(γ) = (0, δ)a(δ) = (−1, α)
What is the map h below?
Aa //
h��
R × A
Fh��
RA ϕ//R × RA
41/51
Finality at work: FX = R × X
RA is the set of function f : R→ R such that
for all n, fn(0) exists, and f agrees with its
Taylor series on an interval containing 0.
The final coalgebra is (RA , ϕ) where
ϕ : RA → R × RA is f 7→ (f(0), f ′)
Consider a coalgebra (A , a : A → R × A), where
A = {α, β, γ, δ}, and
a(α) = (0, β)a(β) = (1, γ)
a(γ) = (0, δ)a(δ) = (−1, α)
What is the map h below?
Aa //
h��
R × A
Fh��
RA ϕ//R × RA
α 7→ sin x, β 7→ cos x, γ 7→ − sin x, γ 7→ − cos x
Pavlovic and Escardo 1998, “Calculus in Coinductive Form”41/51
Bisimulation and the final coalgebra of Pfin
Let (G,→) be a graph.
A relation R on G is a bisimulation iff the following holds:
whenever xRy,
(zig) If x → x′, then there is some y → y′ such that x′Ry′.
(zag) If y → y′, then there is some x → x′ such that x′Ry′.
42/51
Bisimulation and the final coalgebra of Pfin
Let (G,→) be a graph.
A relation R on G is a bisimulation iff the following holds:
whenever xRy,
(zig) If x → x′, then there is some y → y′ such that x′Ry′.
(zag) If y → y′, then there is some x → x′ such that x′Ry′.
For an example, let’s look at the following graph G:
3b
3a 2aoo
@A
//
2boo
=={{{{{{{{1oo
OO
��
// 3c 3d
2c
=={{{{{{{{
aaCCCCCCCC
The largest bisimulation on our graph G is the relation that
relates 1 to itself,
all 2-points to all 2-points,
and all 3-points to all 3-points.42/51
Bisimulation and the final coalgebra of Pfin
The final coalgebra of Pfin
is the set of finitely braching graphs with the following features:
◮ There is a “top” point p
◮ Every point is reachable from the top.
◮ Every bisimulation on the graph is a subset of the diagonal.
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Induction and bisimulation
Induction
Every subalgebra of an initial algebra is invertible.
Coinduction
Every bisimulation relation on a final coalgebra
is a subset of the diagonal.
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arithmetic : induction ::
analysis : co-inductionThe proof here is due to Jan Rutten
For example, let’s prove that
sin(x + y) = sin x cos y + cos x sin y.
Fix a real a. Write b for sin a and c for cos a.
Consider the set of pairs of real functions, where
(sin(x + a), c sin x + b cos x)(cos(x + a),−b sin x + c cos x)(− sin(x + a),−c sin x − b cos x)(− cos(x + a), b sin x − c cos x)
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arithmetic : induction ::
analysis : co-inductionThe proof here is due to Jan Rutten
For example, let’s prove that
sin(x + y) = sin x cos y + cos x sin y.
Fix a real a. Write b for sin a and c for cos a.
We get a relation on RA .
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arithmetic : induction ::
analysis : co-inductionThe proof here is due to Jan Rutten
(sin(x + a), c sin x + b cos x)(cos(x + a),−b sin x + c cos x)(− sin(x + a),−c sin x − b cos x)(− cos(x + a), b sin x − c cos x)
It’s a set of pairs of real functions.
In each pair, the value of the two functions at 0 is the same.
And for (f , g) in the list, (f ′, g′) is also in the list.
So we have a bisimulation.
And thus, by coinduction, the entries in each pair are the same.
So for all a,
sin(x + a) = cos a sin x + sin a cos x
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arithmetic : induction ::
analysis : co-inductionThe proof here is due to Jan Rutten
(sin(x + a), c sin x + b cos x)(cos(x + a),−b sin x + c cos x)(− sin(x + a),−c sin x − b cos x)(− cos(x + a), b sin x − c cos x)
It’s a set of pairs of real functions.
In each pair, the value of the two functions at 0 is the same.
And for (f , g) in the list, (f ′, g′) is also in the list.
So we have a bisimulation.
And thus, by coinduction, the entries in each pair are the same.
So for all a,
sin(x + a) = cos a sin x + sin a cos x
Other applications: enumerative combinatorics, power series,
continued fractions.46/51
Bisimulation in real-world settings
Let’s watch a video!
Click on this link.
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Another success: coalgebraic generalizations of
modal logic
Modal logic
???=
the functor K (a) = P(a) × P(AtSen)
an arbitrary (?) functor F
The logic ??? should be interpreted on all coalgebras of F .
It should characterize points in (roughly) the sense that
points in a coalgebra have the same L theory
iff they are bisimilar
iff they are mapped to the same point in the final coalgebra
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What about ∆ : Meas→ Meas?
LM and Viglizzo 2006, building on Heifetz and Samet 1998
Every functor F : Meas→ Meas built from the usual stuff and
∆ : Meas→ Meas
has a final coalgebra.
The proof used a version of the probabilistic modal logic,
using the set of all theories of all points in all spaces,
and also using the π-λ Theorem of measure theory.
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The universe of sets
Consider the categoryA of classes.
P : A→ A gives the class of subsets of a given class.
Note that PV = V .
Work in ZF − Foundation
The Foundation Axiom is equivalent to the assertion that
(V , id : PV → V)
is an initial algebra of P.
The Anti-Foundation Axiom is equivalent to the assertion that
(V , id : V → PV)
is a final coalgebra of P.
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The conceptual comparison chartFilling out the details is my goal for coalgebra
set with algebraic set with transitions
operations and observations
algebra for a functor coalgebra for a functor
initial algebra final coalgebra
least fixed point greatest fixed point
congruence relation bisimulation equivalence rel’n
equational logic modal logic
recursion: map out of corecursion: map into
an initial algebra a final coalgebra
Foundation Axiom Anti-Foundation Axiom
iterative conception of set coiterative conception of set
useful in syntax useful in semantics
bottom-up top-down
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