Coinductive Predicates and Final Sequences in a Fibration Ichiro - - PowerPoint PPT Presentation

Coinductive Predicates and Final Sequences in a Fibration

Ichiro Hasuo Kenta Cho Toshiki Kataoka

University of Tokyo (JP)

Bart Jacobs

Radboud Univ. Nijmegen (NL)

MFPS (Tulane) 2013/ 6/24

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Coinduction

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In C ?

Coinduction

F X / _ _ _ _ _ F Z X c O beh(c) / _ _ _ _ _ _ Z final ∼ = O

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In C ? In a fibration !! This work: final coalgebra in p; final sequcence in p

Coinduction

F X / _ _ _ _ _ F Z X c O beh(c) / _ _ _ _ _ _ Z final ∼ = O

P ↓p C

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In C ? In a fibration !! This work: final coalgebra in p; final sequcence in p

Coinduction

F X / _ _ _ _ _ F Z X c O beh(c) / _ _ _ _ _ _ Z final ∼ = O

P ↓p C

Fibered

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In C ? In a fibration !! This work: final coalgebra in p; final sequcence in p

Coinduction

F X / _ _ _ _ _ F Z X c O beh(c) / _ _ _ _ _ _ Z final ∼ = O

P ↓p C

{ F-behaviors }

Fibered

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In C ? In a fibration !! This work: final coalgebra in p; final sequcence in p

Coinduction

F X / _ _ _ _ _ F Z X c O beh(c) / _ _ _ _ _ _ Z final ∼ = O

P ↓p C

{ F-behaviors } { F-behaviors } + coinductive predicate

Fibered

Part I: Coinductive Predicates, Conventionally

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Coinductive Predicates

Persisting predicates in dynamical sys. now ✔, next ✔, next2 ✔, ... ν in the modal μ-calculus G in LTL/CTL Expresses safety

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Coinductive Predicates

Kripke frame

νu. u

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Coinductive Predicates

Kripke frame

νu. u

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x ⋯ ⋰ ⋱

Coinductive Predicates

Kripke frame

νu. u

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x ⋯ ⋰ ⋱

Coinductive Predicates

Kripke frame

νu. u

| = νu.u ∼ = (νu.u)

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x ⋯ ⋰ ⋱

Coinductive Predicates

Kripke frame

νu. u

x’

| = νu.u ∼ = (νu.u)

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x ⋯ ⋰ ⋱

Coinductive Predicates

Kripke frame

νu. u

x’

| = νu.u ∼ = (νu.u) | = νu.u ∼ = (νu.u)

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x ⋯ ⋰ ⋱

Coinductive Predicates

Kripke frame

νu. u

x’ x’’

| = νu.u ∼ = (νu.u) | = νu.u ∼ = (νu.u)

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x ⋯ ⋰ ⋱

Coinductive Predicates

Kripke frame

νu. u

x’ x’’

⋮

| = νu.u ∼ = (νu.u) | = νu.u ∼ = (νu.u)

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x ⋯ ⋰ ⋱

Coinductive Predicates

“There is an infinite path” Kripke frame

νu. u

x’ x’’

⋮

| = νu.u ∼ = (νu.u) | = νu.u ∼ = (νu.u)

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x ⋯ ⋰ ⋱

Coinductive Predicates

“There is an infinite path” Kripke frame

νu. u

x’ x’’

⋮

| = νu.u ∼ = (νu.u) | = νu.u ∼ = (νu.u)

(current st.) ⊨ P witnesses (next st.) ⊨ P

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Coinductive Predicates

Kripke frame Bisimilarity ∼ (current st.) ⊨ P witnesses (next st.) ⊨ P

x ∼ ∼ ∼ y, x → x0 = ⇒ y → ∃y0 s.t. x0 ∼ ∼ ∼ y0

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Coinductive Predicates

Proof assistants

are HOT!!

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Coinductive Predicates

Proof assistants In Coq; in Agda

[Giménez, TYPES’95] [Bertot & Komendantskaya, CMCS’08] [Nakano, CPP’12]

are HOT!!

Hasuo (Tokyo)

Coinductive Predicates

Proof assistants In Coq; in Agda

[Giménez, TYPES’95] [Bertot & Komendantskaya, CMCS’08] [Nakano, CPP’12]

Hence in constructive logics

are HOT!!

Hasuo (Tokyo)

Coinductive Predicates

Proof assistants In Coq; in Agda

[Giménez, TYPES’95] [Bertot & Komendantskaya, CMCS’08] [Nakano, CPP’12]

Hence in constructive logics

Fp

µu. p Xu = ¬(νu. ¬p Xu)

¬G¬p

are HOT!!

Hasuo (Tokyo)

Coinductive Predicates

Proof assistants In Coq; in Agda

[Giménez, TYPES’95] [Bertot & Komendantskaya, CMCS’08] [Nakano, CPP’12]

Hence in constructive logics Search for useful proof principles

[Hur, Neis, Dreyer & Vafeiadis, POPL ’13] [Bonchi & Pous, POPL ’13]

Fp

µu. p Xu = ¬(νu. ¬p Xu)

¬G¬p

are HOT!!

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U

{X X | X U = }

{x X | c(x) U = }

• P(X)

X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• Establish/Compute/Construct

Coinductive Predicates

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U

{X X | X U = }

{x X | c(x) U = }

• P(X)

X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• Establish/Compute/Construct

Coinductive Predicates

U

X0

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U

{X X | X U = }

{x X | c(x) U = }

• P(X)

X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• Establish/Compute/Construct

Coinductive Predicates

U

c(x)

x

U

X0

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U

{X X | X U = }

{x X | c(x) U = }

• P(X)

X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

U

c(x)

x

U

X0

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U

{X X | X U = }

{x X | c(x) U = }

• P(X)

X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

1st ans. (Knaster-Tarski) is monotone Postfixed points (invariants) form a complete lattice Its maximum (greatest invariant) is the gfp

U

c(x)

x

c−1 ϕ : 2X 2X

{U | U (c−1 ϕ)U}

U

X0

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U

{X X | X U = }

{x X | c(x) U = }

• P(X)

X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

1st ans. (Knaster-Tarski) is monotone Postfixed points (invariants) form a complete lattice Its maximum (greatest invariant) is the gfp

U

c(x)

x

c−1 ϕ : 2X 2X

{U | U (c−1 ϕ)U}

n

• t

r e a l l y a “ c

• n

s t r u c t i

• n

” . . .

U

X0

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp But when? ω, if φ is ∩-preserving... not now

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp But when? ω, if φ is ∩-preserving... not now

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp But when? ω, if φ is ∩-preserving... not now

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

the whole space

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp But when? ω, if φ is ∩-preserving... not now

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

∃ path length ≥ 1 the whole space

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp But when? ω, if φ is ∩-preserving... not now

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

∃ path length ≥ 1 ∃ path length ≥ 2 the whole space

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

∃ path length ≥ 1 ∃ path length ≥ 2 the whole space

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P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Establish/Compute/Construct Coinductive Predicates

2nd ans. (Inductive constr. [Cousot & Cousot ’79]) Stabilize ➜ gfp But when? ω, if φ is ∩-preserving... not now

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

∃ path length ≥ 1 ∃ path length ≥ 2 the whole space

Hasuo (Tokyo)

Establish/Compute/Construct Coinductive Predicates

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

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Establish/Compute/Construct Coinductive Predicates

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

1 2

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Establish/Compute/Construct Coinductive Predicates

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

1 2

| =

• n

n, but | = νu. u

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Establish/Compute/Construct Coinductive Predicates

State space bound [Cousot & Cousot, ’79] |X| steps

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

1 2

| =

• n

n, but | = νu. u

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Establish/Compute/Construct Coinductive Predicates

State space bound [Cousot & Cousot, ’79] |X| steps “Behavioral bound” [Hennessy & Milner, ’85] ω steps if finitely branching!

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

1 2

| =

• n

n, but | = νu. u

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Behavioral Bound for Computing Coind. Pred.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

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Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

• n

n

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

• n

n

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

1

• n

n

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

1 2

• n

n

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

1 2 3

• n

n

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

1 2 3

• n

n

⋮ ⋮ ⋮

Hasuo (Tokyo)

Behavioral Bound for Computing Coind. Pred.

Proof: Suffices to show is an invariant.

Theorem. Let a Kripke frame

P(X) X c

be finitely branching. Then X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

stabilizes after ω steps.

• n

n

x x2 ⋯ x1 xk

1 2 3

• n

n

⋮ ⋮ ⋮

i [1, k] s.t. xi | = n for infinitely many n

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Coinductive Predicates Conventionally (Summary)

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Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P

Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P

P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P

P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P By Knaster-Tarski

P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P By Knaster-Tarski Inductive constr.

P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P By Knaster-Tarski Inductive constr. State space bound vs. “behavioral bound”

P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

Hasuo (Tokyo)

Coinductive Predicates Conventionally (Summary)

(current st.) ⊨ P witnesses (next st.) ⊨ P By Knaster-Tarski Inductive constr. State space bound vs. “behavioral bound”

P(X) X c

νu. u = gfp

• 2X

ϕ 2PX c−1 2X

• ??

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

current work current work

Part II: Coinductive Predicates, Categorically

Hasuo (Tokyo)

Contributions

Sufficient condition for categorical behavioral ω-bound based on

Coalgebra (transition system) Fibration (underlying logic) Predicate lifting (modality) Locally presentable category (“size”)

Y ∃

• ∃

X

i

. . . . . .

• C
• l

i m

i

X

i

P ϕ

• p
• P

p

• C

F

• C

P ↓p C

F X X c

Hasuo (Tokyo)

Contributions

Sufficient condition for categorical behavioral ω-bound based on

Coalgebra (transition system) Fibration (underlying logic) Predicate lifting (modality) Locally presentable category (“size”)

• Constr. of final coalg. by

final sequence [Worrell, Adamek]

Y ∃

• ∃

X

i

. . . . . .

• C
• l

i m

i

X

i

P ϕ

• p
• P

p

• C

F

• C

P ↓p C

F X X c

Hasuo (Tokyo)

Contributions

Sufficient condition for categorical behavioral ω-bound based on

Coalgebra (transition system) Fibration (underlying logic) Predicate lifting (modality) Locally presentable category (“size”)

• Constr. of final coalg. by

final sequence [Worrell, Adamek]

• Coind. predicate as a

final coalgebra [Hermida, Jacobs]

Y ∃

• ∃

X

i

. . . . . .

• C
• l

i m

i

X

i

P ϕ

• p
• P

p

• C

F

• C

P ↓p C

F X X c

Hasuo (Tokyo)

Contributions

Sufficient condition for categorical behavioral ω-bound based on

Coalgebra (transition system) Fibration (underlying logic) Predicate lifting (modality) Locally presentable category (“size”)

• Constr. of final coalg. by

final sequence [Worrell, Adamek]

• Coind. predicate as a

final coalgebra [Hermida, Jacobs]

Y ∃

• ∃

X

i

. . . . . .

• C
• l

i m

i

X

i

P ϕ

• p
• P

p

• C

F

• C

P ↓p C

F X X c

Categorical infrastructure: fibration and locally presentable cat.

Hasuo (Tokyo)

Contributions

Sufficient condition for categorical behavioral ω-bound based on

Coalgebra (transition system) Fibration (underlying logic) Predicate lifting (modality) Locally presentable category (“size”)

• Constr. of final coalg. by

final sequence [Worrell, Adamek]

• Coind. predicate as a

final coalgebra [Hermida, Jacobs]

Y ∃

• ∃

X

i

. . . . . .

• C
• l

i m

i

X

i

P ϕ

• p
• P

p

• C

F

• C

P ↓p C

F X X c

Categorical infrastructure: fibration and locally presentable cat.

Some math work

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• monotone

2X ϕ3 /2PX c−1 /2X

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• finitely

branching

coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• finitely

branching

coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

finitary

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• finitely

branching

coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

finitary coinductive specification

νϕ

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• finitely

branching

coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

finitary coinductive specification

νϕ

predicate lifting

ϕ : PX − → PF X

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• finitely

branching

coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

finitary coinductive specification

νϕ

predicate lifting

ϕ : PX − → PF X

endofunctor

PX ϕX /PF X c∗ /PX

invariant

(c−1 ϕ3)U U ✓

Hasuo (Tokyo)

Kripke model

The Categorical Setup

Pω(X) X c O

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• finitely

branching

coalgebra

F X X c O

monotone

2X ϕ3 /2PX c−1 /2X

finitary coinductive specification

νϕ

predicate lifting

ϕ : PX − → PF X

endofunctor

PX ϕX /PF X c∗ /PX

invariant

(c−1 ϕ3)U U ✓

coalgebra (in a fibr.) (c∗ ϕ)P P O

Hasuo (Tokyo)

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• X

monotone

2X ϕ3 /2PX c−1 /2X

coinductive specification

νϕ

predicate lifting

ϕ : PX − → PF X

endofunctor

PX ϕX /PF X c∗ /PX

invariant

(c−1 ϕ3)U U ✓

coalgebra (in a fibr.) (c∗ ϕ)P P O

• coind. pred.

(c−1 ϕ3)Jνu. ϕ3uKc Jνu. ϕ3uKc

inductive constr.

X ◆ (c−1 ϕ3)X ◆ · · ·

Hasuo (Tokyo)

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• X

monotone

2X ϕ3 /2PX c−1 /2X

coinductive specification

νϕ

predicate lifting

ϕ : PX − → PF X

endofunctor

PX ϕX /PF X c∗ /PX

invariant

(c−1 ϕ3)U U ✓

coalgebra (in a fibr.) (c∗ ϕ)P P O final coalg. (in a fibr.)

(c∗ ϕ)JνϕKc JνϕKc ⇠ = O

• coind. pred.

(c−1 ϕ3)Jνu. ϕ3uKc Jνu. ϕ3uKc

inductive constr.

X ◆ (c−1 ϕ3)X ◆ · · ·

Hasuo (Tokyo)

coinductive specification

νu. u

ϕ3 : 2X − → 2PωX

U

{X X | X U = }

• X

monotone

2X ϕ3 /2PX c−1 /2X

coinductive specification

νϕ

predicate lifting

ϕ : PX − → PF X

endofunctor

PX ϕX /PF X c∗ /PX

invariant

(c−1 ϕ3)U U ✓

coalgebra (in a fibr.) (c∗ ϕ)P P O final coalg. (in a fibr.)

(c∗ ϕ)JνϕKc JνϕKc ⇠ = O

• coind. pred.

(c−1 ϕ3)Jνu. ϕ3uKc Jνu. ϕ3uKc

inductive constr.

X ◆ (c−1 ϕ3)X ◆ · · ·

final sequence in a fibr.

>X (c∗ ϕX)>X · · ·

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What Categorical Generalization Buys Us

Hasuo (Tokyo)

What Categorical Generalization Buys Us

Final coalgebra in C: (strongly) LFP (Posets, Graphs, Vec, ...) [Adamek ’03] Coinductive pred. for different

• Coalg. μ-calculus; coalg. automata

[Cirstea, Kupke & Pattinson, CSL ’09] [Cirstea & Sadrzadeh, CMCS’08] [Venema, I&C’06]

F : Sets → Sets

Hasuo (Tokyo)

What Categorical Generalization Buys Us

Final coalgebra in C: (strongly) LFP (Posets, Graphs, Vec, ...) [Adamek ’03] Coinductive pred. for different

• Coalg. μ-calculus; coalg. automata

[Cirstea, Kupke & Pattinson, CSL ’09] [Cirstea & Sadrzadeh, CMCS’08] [Venema, I&C’06]

Various “underlying logics” as

F : Sets → Sets

P ↓p C

Constructive logics

Sub(C) ↓ C

(C: a topos)

For name-passing

Sub(SetsF) ↓ SetsF

Relations (“binary pred. ”)

Rel ↓ Sets

Hasuo (Tokyo)

What Categorical Generalization Buys Us

Final coalgebra in C: (strongly) LFP (Posets, Graphs, Vec, ...) [Adamek ’03] Coinductive pred. for different

• Coalg. μ-calculus; coalg. automata

[Cirstea, Kupke & Pattinson, CSL ’09] [Cirstea & Sadrzadeh, CMCS’08] [Venema, I&C’06]

Various “underlying logics” as

F : Sets → Sets

P ↓p C

Constructive logics

Sub(C) ↓ C

(C: a topos)

For name-passing

Sub(SetsF) ↓ SetsF

Relations (“binary pred. ”)

Rel ↓ Sets

• Coind. relations

e.g. bisimilarity

Hasuo (Tokyo)

Coindution in a Fibration

conventional relational fibrational invariant bisimulation coalgebra

• coind. pred.

bisimilarity final coalg. inductive constr. partition refinement final sequence

Pred ↓ Sets Rel ↓ Sets P ↓p C

Part III: Technical Ingredients

Final Sequence, Fibration, Predicate Lifting, Locally Finitely Presentable Category, ...

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Final Sequence

[Worrell, TCS’05] in Sets [Adamek, TCS’03] in strongly LFP C

1 F 1

!

• · · ·
• F i1
• · · ·

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Final Sequence

[Worrell, TCS’05] in Sets [Adamek, TCS’03] in strongly LFP C

1 F 1

!

• · · ·
• F i1
• · · ·
• { i-step behaviors}

Hasuo (Tokyo)

Final Sequence

[Worrell, TCS’05] in Sets [Adamek, TCS’03] in strongly LFP C

F ω1 s t

πi

{ 1 F 1

!

• · · ·
• F i1
• · · ·
• lim

{ i-step behaviors}

Hasuo (Tokyo)

Final Sequence

[Worrell, TCS’05] in Sets [Adamek, TCS’03] in strongly LFP C

F ω1 s s

πi

y 1 F 1

!

• · · ·
• F i1
• · · ·
• F (F ω1)

k j

F πi−1

d

b

_

• 5

lim { i-step behaviors}

Hasuo (Tokyo)

Final Sequence

: a final coalgebra? Yes, when F is limit preserving (b is iso)

[Worrell, TCS’05] in Sets [Adamek, TCS’03] in strongly LFP C

F ω1 s s

πi

y 1 F 1

!

• · · ·
• F i1
• · · ·
• F (F ω1)

k j

F πi−1

d

b

_

• 5

lim { i-step behaviors}

F ω1

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Final Sequence

: a final coalgebra? Yes, when F is limit preserving (b is iso) Almost, when F is finitary (b is monic) Quotient modulo beh. eq. Continue till ω+ω [Worrell]

[Worrell, TCS’05] in Sets [Adamek, TCS’03] in strongly LFP C

F ω1 s s

πi

y 1 F 1

!

• · · ·
• F i1
• · · ·
• F (F ω1)

k j

F πi−1

d

b

_

• 5

lim { i-step behaviors}

F ω1

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Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C

indices

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

indices

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

PY

Q Q’

indices

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

PY

Q Q’

indices indexed entities

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Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• indices

indexed entities

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• indices

indexed entities “substitution”

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• indices

indexed entities “substitution”

Hasuo (Tokyo)

Fibration

“Organize indexed entities,” categorically In particular: categorical model of predicate logics

P ↓p C

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• : predicates over X

Substitution

(PX, ⊆)

PX PY f −1

• X

f / Y

f −1(V ⊆ Y ) = V

• f(

)

• indices

indexed entities “substitution”

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Fibration: from Pointwise Indexing to Display Indexing

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

Hasuo (Tokyo)

Fibration: from Pointwise Indexing to Display Indexing

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• P

p ✏ C X f / Y

P’’

P

P’ Q’

Q

↦

Patch up

Hasuo (Tokyo)

Fibration: from Pointwise Indexing to Display Indexing

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• P

p ✏ C X f / Y

P’’

P

P’ Q’

Q

• objects: |P| = `

X∈C |PX|

• arrows:

P − → Q in P

• X

f

→ Y in C, P → f ∗Q in PX

• ↦

Patch up

Hasuo (Tokyo)

Fibration: from Pointwise Indexing to Display Indexing

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• P

p ✏ C X f / Y

P’’

P

P’ Q’

Q

• objects: |P| = `

X∈C |PX|

• arrows:

P − → Q in P

• X

f

→ Y in C, P → f ∗Q in PX

• ↦

Patch up

Hasuo (Tokyo)

Fibration: from Pointwise Indexing to Display Indexing

X f / Y in C PX

P P’ P’’

PY

Q Q’

f ∗

• P

p ✏ C X f / Y

P’’

P

P’ Q’

Q

f ∗Q

• objects: |P| = `

X∈C |PX|

• arrows:

P − → Q in P

• X

f

→ Y in C, P → f ∗Q in PX

• ↦

Patch up

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Fibration

Defn. A (poset) fibration is a functor

P #p C

such that

• Each fiber PX is a poset.
• For f : X ! Y in C and Q 2 PY , a “universal arrow”

fQ : f ⇤Q ! Q such that

P p

✏

Q = ) f ⇤Q f(Q)/ Q P g

7

• g0 O

C X f

/ Y

X f

/ Y

• The correspondences (

)⇤ and ( ) are functorial: id⇤

Y Q = Q ,

(g f)⇤(Q) = f ⇤(g⇤Q) , idY (Q) = idQ , g f(Q) = gQ f(g⇤Q) .

Hasuo (Tokyo)

Fibration

Defn. A (poset) fibration is a functor

P #p C

such that

• Each fiber PX is a poset.
• For f : X ! Y in C and Q 2 PY , a “universal arrow”

fQ : f ⇤Q ! Q such that

P p

✏

Q = ) f ⇤Q f(Q)/ Q P g

7

• g0 O

C X f

/ Y

X f

/ Y

• The correspondences (

)⇤ and ( ) are functorial: id⇤

Y Q = Q ,

(g f)⇤(Q) = f ⇤(g⇤Q) , idY (Q) = idQ , g f(Q) = gQ f(g⇤Q) . Q X f / Y f ∗Q / Q X f / Y

= ⇒

Hasuo (Tokyo)

Fibration

Defn. A (poset) fibration is a functor

P #p C

such that

• Each fiber PX is a poset.
• For f : X ! Y in C and Q 2 PY , a “universal arrow”

fQ : f ⇤Q ! Q such that

P p

✏

Q = ) f ⇤Q f(Q)/ Q P g

7

• g0 O

C X f

/ Y

X f

/ Y

• The correspondences (

)⇤ and ( ) are functorial: id⇤

Y Q = Q ,

(g f)⇤(Q) = f ⇤(g⇤Q) , idY (Q) = idQ , g f(Q) = gQ f(g⇤Q) . Q X f / Y f ∗Q / Q X f / Y

= ⇒

what’ s substitution?

Hasuo (Tokyo)

Fibration: Examples

(f −1Q ⊆ X) / (Q ⊆ Y ) X f / Y

Pred ↓ Sets

Hasuo (Tokyo)

Fibration: Examples

(f −1Q ⊆ X) / (Q ⊆ Y ) X f / Y

Pred ↓ Sets Rel ↓ Sets

✓ (f × f)−1Q ⊆ X × X ◆ / (Q ⊆ Y × Y ) X f / Y

Hasuo (Tokyo)

Fibration: Examples

(f −1Q ⊆ X) / (Q ⊆ Y ) X f / Y

Pred ↓ Sets Rel ↓ Sets

✓ (f × f)−1Q ⊆ X × X ◆ / (Q ⊆ Y × Y ) X f / Y

    /     X f / Y

f ∗P / ✏ ✏ _ P ✏ ✏ X f / Y

P ✏ ✏ Y

Sub(C) ↓ C

(C: a topos)

Hasuo (Tokyo)

Fibration: Examples

(f −1Q ⊆ X) / (Q ⊆ Y ) X f / Y

Pred ↓ Sets Rel ↓ Sets

✓ (f × f)−1Q ⊆ X × X ◆ / (Q ⊆ Y × Y ) X f / Y

Sub(SetsF) ↓ SetsF

    /     X f / Y

f ∗P / ✏ ✏ _ P ✏ ✏ X f / Y

P ✏ ✏ Y

Sub(C) ↓ C

(C: a topos)

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Predicate Lifting For Modality

Defn. A predicate lifting of F : C → C is ϕ : P → P s.t.

• P

ϕ / p ✏ P p ✏ C F / C (hence ϕX : PX → PF X)

• compatible with substitution.

For , coincides with , monotone, natural in X

Pred ↓ Sets

λX : 2X = ⇒ 2F X

Part IV: Final Sequence in a Fibration

Hasuo (Tokyo)

Final Sequence in a Fibration

P p ✏ C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

Hasuo (Tokyo)

Final Sequence in a Fibration

P p ✏ F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

P p ✏ F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

P p ✏ F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

• F ω+11

k k F πi−1 f b ` ⌃

• 8

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• F ω1

s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

• F ω+11

k k F πi−1 f b ` ⌃

• 8

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

final in P1

⟹ final in P

P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• F ω1

s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

• F ω+11

k k F πi−1 f b ` ⌃

• 8

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

final in P1

⟹ final in P

final seq. for φ

P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• F ω1

s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

• F ω+11

k k F πi−1 f b ` ⌃

• 8

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

final in P1

⟹ final in P

final seq. for φ

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• F ω1

s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

• F ω+11

k k F πi−1 f b ` ⌃

• 8

Hasuo (Tokyo)

Final Sequence in a Fibration

lim

final in P1

⟹ final in P

final seq. for φ

lim

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• F ω1

s s πi x C 1 F 1 !

• · · ·
• F i1

F i−1 !

• · · ·

F i !

• F ω+11

k k F πi−1 f b ` ⌃

• 8

Hasuo (Tokyo)

Final Sequence in a Fibration

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

lim

final in P1

⟹ final in P

final seq. for φ

lim

Hasuo (Tokyo)

Final Sequence in a Fibration

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

lim

final in P1

⟹ final in P

final seq. for φ

lim

Assume F: finitary, φ: pred. lifting of F

Hasuo (Tokyo)

Final Sequence in a Fibration

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

lim

final in P1

⟹ final in P

final seq. for φ

lim

Assume F: finitary, φ: pred. lifting of F : “almost final coalgebra”, prototype of F-behaviors

F ω1

Hasuo (Tokyo)

Final Sequence in a Fibration

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

lim

final in P1

⟹ final in P

final seq. for φ

lim

Assume F: finitary, φ: pred. lifting of F : “almost final coalgebra”, prototype of F-behaviors : prototype of coind. pred. for each coalgebra

F ω1 ϕω>1

F X X c O F X X c O

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Final Sequence in a Fibration

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

lim

final in P1

⟹ final in P

final seq. for φ

lim

Key Lemma. Let

P ↓p C

be a well-founded fibration; F : C ! C be finitary; and ϕ be a predicate lifting of F . Then ϕω+1>1 = b∗(ϕω>1) .

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Final Sequence in a Fibration

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

lim

final in P1

⟹ final in P

final seq. for φ

lim

Key Lemma. Let

P ↓p C

be a well-founded fibration; F : C ! C be finitary; and ϕ be a predicate lifting of F . Then ϕω+1>1 = b∗(ϕω>1) .

p is compatible w/ C: LFP p itself is “well-fdd”

Hasuo (Tokyo)

Technical Contributions

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

Hasuo (Tokyo)

Technical Contributions

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

p is compatible w/ C: LFP

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Technical Contributions

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

P p ✏ κ∗

IP

/ P C XI κI / X

p is compatible w/ C: LFP

Hasuo (Tokyo)

Technical Contributions

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

P p ✏ κ∗

IP

/ P C XI κI / X

∈ F

. . . . . .

p is compatible w/ C: LFP

Hasuo (Tokyo)

Technical Contributions

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

P p ✏ κ∗

IP

/ P C XI κI / X

∈ F

colim

. . . . . .

p is compatible w/ C: LFP

Hasuo (Tokyo)

Technical Contributions

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

P p ✏ κ∗

IP

/ P C XI κI / X

∈ F

colim

. . . . . .

p is compatible w/ C: LFP

Hasuo (Tokyo)

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

Technical Contributions

Definition. A well-founded fibration is a poset fibration that

• 1. is finitely determined, and
• 2. has no decreasing ω-chain

in a fiber PX for FP X.

Hasuo (Tokyo)

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

Technical Contributions

Theorem. Assume

• F X

X c , a coalgebra

• P

↓p C

is a well-founded fibration

• F : C C, finitary
• P

ϕ

• p

P p

• C

F C , predicate lifting Then the sequence X (c∗ ϕX)X (c−1 ϕ)2X · · · stablizes after ω steps, yielding νϕ as its limit.

F X X c O

Definition. A well-founded fibration is a poset fibration that

• 1. is finitely determined, and
• 2. has no decreasing ω-chain

in a fiber PX for FP X.

Hasuo (Tokyo)

Definition. A finitely determined fibration

P ↓p C

is such that:

• 1. C is LFP with F = {FP objects}

2.

P ↓p C

has fiberwise (co)limits

• 3. For each X ∈ C and P, Q ∈ PX,

let {XI

κI

− → X}I be the canoni- cal diagram from F to X. Then P ≤ Q ⇐ ⇒ κ∗

IP ≤ κ∗ IQ, ∀I.

Technical Contributions

Theorem. Assume

• F X

X c , a coalgebra

• P

↓p C

is a well-founded fibration

• F : C C, finitary
• P

ϕ

• p

P p

• C

F C , predicate lifting Then the sequence X (c∗ ϕX)X (c−1 ϕ)2X · · · stablizes after ω steps, yielding νϕ as its limit.

F X X c O

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

Definition. A well-founded fibration is a poset fibration that

• 1. is finitely determined, and
• 2. has no decreasing ω-chain

in a fiber PX for FP X.

Hasuo (Tokyo)

Examples (or: Fibrations vs LFP)

Finitely determined: very often

Prop. Assume C is LFP and LCCC. Then

• Sub(C) is LFP; and
• Sub(C)

↓ C

is finitely determined.

Prop. Assume Ω is an algebraic lattice. Consider

Fam(Ω) ↓ Sets

; then

• Fam(Ω) is locally presentable; and
• Fam(Ω)

↓ Sets

is finitely determined.

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Examples (or: Fibrations vs LFP)

Finitely determined: very often

Prop. Assume C is LFP and LCCC. Then

• Sub(C) is LFP; and
• Sub(C)

↓ C

is finitely determined.

Prop. Assume Ω is an algebraic lattice. Consider

Fam(Ω) ↓ Sets

; then

• Fam(Ω) is locally presentable; and
• Fam(Ω)

↓ Sets

is finitely determined.

topos ⟹ LCCC

Hasuo (Tokyo)

Examples (or: Fibrations vs LFP)

Finitely determined: very often

Prop. Assume C is LFP and LCCC. Then

• Sub(C) is LFP; and
• Sub(C)

↓ C

is finitely determined.

Prop. Assume Ω is an algebraic lattice. Consider

Fam(Ω) ↓ Sets

; then

• Fam(Ω) is locally presentable; and
• Fam(Ω)

↓ Sets

is finitely determined.

topos ⟹ LCCC Algebraic lattice:

every elem. is a sup

• f compact elem’

s “LFP poset”

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Definition. A well-founded fibration is a poset fibration that

• 1. is finitely determined, and
• 2. has no decreasing ω-chain

in a fiber PX for FP X.

Well-founded: depends

Examples

Hasuo (Tokyo)

fin.-det., well-founded fin.-det., well-founded fin.-det., not well-fdd

Pred ↓ Sets , Rel ↓ Sets

Definition. A well-founded fibration is a poset fibration that

• 1. is finitely determined, and
• 2. has no decreasing ω-chain

in a fiber PX for FP X.

Sub(SetsF) ↓ SetsF

,

Sub(SetsF+) ↓ SetsF+ Sub(SetsI) ↓ SetsI

Well-founded: depends

Examples

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fin.-det., well-founded fin.-det., well-founded fin.-det., not well-fdd

Pred ↓ Sets , Rel ↓ Sets

Definition. A well-founded fibration is a poset fibration that

• 1. is finitely determined, and
• 2. has no decreasing ω-chain

in a fiber PX for FP X.

Sub(SetsF) ↓ SetsF

,

Sub(SetsF+) ↓ SetsF+ Sub(SetsI) ↓ SetsI

Well-founded: depends

Examples

Part IV: Conclusions & Future Work

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In C ? In a fibration !! This work: final coalgebra in p; final sequcence in p

F X / _ _ _ _ _ F Z X c O beh(c) / _ _ _ _ _ _ Z final ∼ = O

P ↓p C

{ F-behaviors } + coinductive predicate

Coinduction Fibered

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Conclusions

Inductive construction [Cousot & Cousot, ’79] Final sequence in a fibration behavioral ω-bound: conditions formulated in LFP terms Covers various logics

relations, constructive, name-passing, ...

X

• (c−1 ϕ)X
• (c−1 ϕ)2X
• · · ·

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

Hasuo (Tokyo)

Conclusions

X

• (

c

− 1

• ϕ
• )

X

• (

c

− 1

• ϕ
• )

2

X

• ·

· ·

ϕω>1 s s x P p ✏ >1 ϕ>1

• · · ·
• ϕi>1
• · · ·
• ϕω+1>1

k k f b0 ^

• 6

F ω1 s s πi x C 1 F 1 !

• · · ·
• F i1

F i1 !

• · · ·

F i !

• F ω+11

k k F πi1 f b ` ⌃

• 8

conventional relational fibrational invariant bisimulation coalgebra

• coind. pred.

bisimilarity final coalg. inductive constr. partition refinement final sequence

Pred ↓ Sets Rel ↓ Sets P ↓p C

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Future Work

General proof principles for coinduction Parametrized coind. [Hur, Neis, Dreyer & Vafeiadis, POPL

’13]

Bisimulation up-to [Bonchi & Pous, POPL

’13]

• Appl. to termination analysis of algorithms

Bisimilarity check, etc. Infinite states Current result: semidecidability To the full fixedpoint logics

• Coalg. μ-calculus, coalg. automata, ... fibrationally

Model checking algorithms Combine with bialgebraic SOS Games ↔ automata ↔ fixedpoint logic

Proof assistants Much like appl. of final sequences

Hasuo (Tokyo)

Future Work

General proof principles for coinduction Parametrized coind. [Hur, Neis, Dreyer & Vafeiadis, POPL

’13]

Bisimulation up-to [Bonchi & Pous, POPL

’13]

• Appl. to termination analysis of algorithms

Bisimilarity check, etc. Infinite states Current result: semidecidability To the full fixedpoint logics

• Coalg. μ-calculus, coalg. automata, ... fibrationally

Model checking algorithms Combine with bialgebraic SOS Games ↔ automata ↔ fixedpoint logic

Proof assistants Much like appl. of final sequences

Thank you for your attention!

Ichiro Hasuo (Dept. CS, U Tokyo)

h t t p : / / w w w

• m

m m . i s . s . u

• t
• k

y

• .

a c . j p / ~ i c h i r

• /