Fibrational View on Breadth-First Search Thorsten Wimann Informatik - PowerPoint PPT Presentation

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 1 / 8 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 E X is countably cocomplete (writing ⊔ and ⊥ ). ¯ f ∗ F Y 3 For every f : X → FY , the functor E Y − − → E FY − → E X has a left-adjoint − � f . 4 Fixed ¯ I ∈ E , s.t. every : p (¯ I ) → C has a cocartesian lifting. i c Definition: Liftings Fib( C , i , c ) of I − → C − → FC I ( ¯ Fib( C , i , c ) ⊆ Coalg ¯ F ) is the fibre above ( C , i , c ). Theorem i c The initial lifting of I − → C − → FC , i.e. initial object of Fib( C , i , c ), is carried by ¯ c ( i ∗ (¯ � � k C := − I )) in E C . k ≥ 0

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 3 / 8 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

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 4 / 8 Alternative description of Sub if C is a topos f X Y ≤ x y 2 In Set: Sub = (Set / (2 , ≤ )) lax

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 4 / 8 Alternative description of Sub if C is a topos f X Y ≤ x y 2 In Set: Sub = (Set / (2 , ≤ )) lax Liftings of F : Set → Set to (Set / 2) = (Set / (2 , =)) lax = (Unary) predicate liftings for F .

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 5 / 8 R ∞ 2 ≥ 0 1 0 1 . . . ∞ 0 f f X Y X Y ֒ → ≥ ≤ x y x y R ∞ 2 ≥ 0 E := (Set / ( R ∞ ≥ 0 , ≥ )) lax FX = P ≤ ω ( R ≥ 0 × X )

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 6 / 8 E := (Set / ( R ∞ FX = P ≤ ω ( R ∞ ≥ 0 , ≥ )) lax ≥ 0 × X ) Lifting of F FX = P ( R ∞ ≥ 0 × X ) lifts to (Set / R ∞ ≥ 0 ) lax by F ( X , d ) = ( FX , ¯ ¯ ¯ d : P ( R ∞ ≥ 0 × X ) → R ∞ d ) with ≥ 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 ) r iff d ( x ) + r ≥ d ( y ) for all x − → y . in (Set / R ∞ ≥ 0 ) lax i c → P ≤ ω ( R ∞ Proposition: Initial lifting of I − → C − ≥ 0 × C ): r 1 r 2 r n d ( x ) = inf { r 1 + . . . + r n | i − → x 1 − → . . . x n − 1 − → x n = x }

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 7 / 8 Example graph with R ≥ 0 -labelled edges Vertices V = N + N = { x 0 , y 0 , x 1 , y 1 , . . . } , i = x 0 . Edges: 2 − 2 n 2 − 2 n − 1 2 − 2 n − 3 − − − → y n − − − − → x n +1 − − − − → y n for all n ∈ N . x n x n y n +1 Visually: 2 − 2 n − 1 x n x n +1 2 − 2 n 2 − 2 n − 2 for all n ≥ 0 y n y n +1 2 − 2 n − 3 d ( y 0 ) = 5 6

Fibrational Construction Reachability Distances References | Thorsten Wißmann | 8 / 8 For every f : X → FY : ( f ∗ · ¯ r F Y ( d Y : Y → R ∞ ≥ 0 ))( x ) = sup { d Y ( y ) − r | x − → y } and its left-adjoint is: r � f ( d X : X → R ∞ ( − ≥ 0 ))( y ) = inf { d X ( x ) + r | x → y } . − ¯ E FY f ∗ F Y E Y E X ⊢ � f − Corollary For ( C , i , c ) the initial lifting ( C , d ) is constructed by c ( i ∗ (¯ � � k − I )) in E C . k ≥ 0

Fibrational Construction Reachability Distances References | Thorsten Wißmann | ∞ / 8 Simone Barlocco, Clemens Kupke, Jurriaan Rot. “Coalgebra Learning via Duality”. In: Foundations of 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 .