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Fibrational View on Breadth-First Search

Thorsten Wißmann

Informatik 8, Erlangen

Oberseminar, May 14th, 2019

Last update: May 20, 2019

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 2 / 8

Assumptions

1 Fibration p : E → C together with a lifted functor ¯

F : E → E.

2 Every fibre EX is countably cocomplete (writing ⊔ and ⊥). 3 For every f : X → FY , the functor EY

¯ FY

− − → EFY

f ∗

− → EX has a left-adjoint − f .

4 Fixed ¯

I ∈ E, s.t. every : p(¯ I) → C has a cocartesian lifting.

Definition: Liftings Fib(C, i, c) of I

i

− → C

c

− → FC

Fib(C, i, c) ⊆ Coalg¯

I( ¯

F) is the fibre above (C, i, c).

Theorem

The initial lifting of I

i

− → C

c

− → FC, i.e. initial object of Fib(C, i, c), is carried by ¯ C :=

• k≥0

− k

c(i∗(¯

I)) in EC.

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Reachability

Subobject fibration p : Sub → C, intersection preserving functor.

Definition: reachable coalgebra

No proper subcoalgebra.

Immediately:

Initial lifted coalgebra = reachable subcoalgebra.

Assumptions

C is/has: arbitrary small coproducts, well-powered (E , M )-factorization system, M ⊆ Mono. Assume that F : C → C preserves (arbitrary) intersections.

Construction Instantiates to known reachability constructions:

Barlocco, Kupke, Rot, 2019 Wißmann, Milius, Katsumata, Dubut, 2019

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Alternative description of Sub if C is a topos

X Y 2

f x

≤

y

In Set: Sub = (Set/(2, ≤))lax

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Alternative description of Sub if C is a topos

X Y 2

f x

≤

y

In Set: Sub = (Set/(2, ≤))lax

Liftings of F : Set → Set to (Set/2) = (Set/(2, =))lax

= (Unary) predicate liftings for F.

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2 R∞

≥0

1 1 . . . ∞ X Y 2

f x

≤

y

֒ → X Y R∞

≥0 f x

≥

y

E := (Set/(R∞

≥0, ≥))lax

FX = P≤ω(R≥0 × X)

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E := (Set/(R∞

≥0, ≥))lax

FX = P≤ω(R∞

≥0 × X)

Lifting of F

FX = P(R∞

≥0 × X) lifts to (Set/R∞ ≥0)lax by

¯ F(X, d) = (FX, ¯ d) with ¯ d : P(R∞

≥0 × X) → R∞ ≥0

¯ d(M) := sup{d(x) − r | (r, x) ∈ M}.

Lemma: for every c : C → FC and d : C → R∞

≥0

c : (C, d) → ¯ F(C, d) in (Set/R∞

≥0)lax

iff d(x) + r ≥ d(y) for all x

r

− → y.

Proposition: Initial lifting of I

i

− → C

c

− → P≤ω(R∞

≥0 × C):

d(x) = inf{r1 + . . . + rn | i

r1

− → x1

r2

− → . . . xn−1

rn

− → xn = x}

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Example graph with R≥0-labelled edges

Vertices V = N + N = {x0, y0, x1, y1, . . .}, i = x0. Edges: xn

2−2n

− − − → yn xn

2−2n−1

− − − − → xn+1 yn+1

2−2n−3

− − − − → yn for all n ∈ N. Visually: xn xn+1 yn yn+1

2−2n−1 2−2n−3 2−2n 2−2n−2

for all n ≥ 0 d(y0) = 5 6

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For every f : X → FY : (f ∗ · ¯ FY (dY : Y → R∞

≥0))(x) = sup{dY (y) − r | x r

− → y} and its left-adjoint is: (− f (dX : X → R∞

≥0))(y) = inf{dX(x) + r | x r

− → y}. EY EFY EX

¯ FY f ∗ − f ⊢

Corollary

For (C, i, c) the initial lifting (C, d) is constructed by

• k≥0

− k

c(i∗(¯

I)) in EC.

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Simone Barlocco, Clemens Kupke, Jurriaan Rot. “Coalgebra Learning via Duality”. In: Foundations

• f Software Science and Computation
• Structures. Ed. by Miko

laj Boja´ nczyk, Alex Simpson. Cham: Springer International Publishing, 2019, pp. 62–79. isbn: 978-3-030-17127-8. Thorsten Wißmann, Stefan Milius, Shin-ya Katsumata, J´ er´ emy Dubut. A Coalgebraic View on Reachability. 2019. eprint: arXiv:1901.10717.