A Calculus for Worms Ana Borges with Joost Joosten Universitat de - - PowerPoint PPT Presentation

A Calculus for Worms

Ana Borges with Joost Joosten

Universitat de Barcelona

Wormshop 2017 December 19

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The Reflection Calculus

The Reflection Calculus, RC0

Λ, in a nutshell:

• Closed positive fragment of GLPΛ.

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The Reflection Calculus

The Reflection Calculus, RC0

Λ, in a nutshell:

• Closed positive fragment of GLPΛ.
• No variables.

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The Reflection Calculus

The Reflection Calculus, RC0

Λ, in a nutshell:

• Closed positive fragment of GLPΛ.
• No variables.
• Formulas are made up of ⊤, ∧ and α for all α < Λ.

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The Reflection Calculus

The Reflection Calculus, RC0

Λ, in a nutshell:

• Closed positive fragment of GLPΛ.
• No variables.
• Formulas are made up of ⊤, ∧ and α for all α < Λ.

⊤ 10⊤ ∧ 5⊤ 7(2⊤ ∧ 000⊤)

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The Reflection Calculus: axioms and rules

Let ϕ, ψ, χ be formulas of RC0

Λ, and α, β < Λ.

ϕ ⊢ ϕ ϕ ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ∧ ψ ⊢ ψ ααϕ ⊢ αϕ αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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What are worms?

Worms are long cylindrical animals with a tube-like body and no limbs iterated consistency statements: A = 73ǫ0042ω⊤ No one likes to write all of these s and s... A = 73 ǫ0 0 42 ω

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Decomposing worms

Worms have α-heads and α-remainders.

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Decomposing worms

Worms have α-heads and α-remainders. The α-head is the largest initial segment with only “big” modalities (at least as big as α): h73(73 ǫ0 0 42 ω) = 73 ǫ0

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Decomposing worms

Worms have α-heads and α-remainders. The α-head is the largest initial segment with only “big” modalities (at least as big as α): h73(73 ǫ0 0 42 ω) = 73 ǫ0 The α-remainder is the rest: r73(73 ǫ0 0 42 ω) = 0 42 ω

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Decomposing worms

Worms have α-heads and α-remainders. The α-head is the largest initial segment with only “big” modalities (at least as big as α): h73(73 ǫ0 0 42 ω) = 73 ǫ0 The α-remainder is the rest: r73(73 ǫ0 0 42 ω) = 0 42 ω A = hα(A)rα(A) ≡RC hα(A) ∧ rα(A)

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Why are worms interesting?

For any worms A, B such that min A > α, A ∧ αB ≡RC AαB

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Why are worms interesting?

For any worms A, B such that min A > α, A ∧ αB ≡RC AαB

Theorem (Worms are the “core” of RC0)

For each formula ϕ of RC0

Λ there is some worm A such that:

ϕ ≡RC A.

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Why are worms interesting?

For any worms A, B such that min A > α, A ∧ αB ≡RC AαB

Theorem (Worms are the “core” of RC0)

For each formula ϕ of RC0

Λ there is some worm A such that:

ϕ ≡RC A. Then maybe we could forget about conjunctions?

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The Reflection Calculus

Let ϕ, ψ, χ be formulas of RC0

Λ, and α, β < Λ.

ϕ ⊢ ϕ ϕ ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ∧ ψ ⊢ ψ ααϕ ⊢ αϕ αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A ϕ ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ∧ ψ ⊢ ψ ααϕ ⊢ αϕ αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ∧ ψ ⊢ ψ ααϕ ⊢ αϕ αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ϕ ∧ ψ ⊢ ψ ααϕ ⊢ αϕ αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ααϕ ⊢ αϕ αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ααA ⊢ αA αϕ ⊢ βϕ for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A be a worm, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A, B, C be worms, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

A ⊢ B B ⊢ C A ⊢ C ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let ϕ, ψ, χ be formulas of RC0

Λ, A, B, C be worms, and α, β < Λ.

A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

A ⊢ B B ⊢ C A ⊢ C ϕ ⊢ ψ αϕ ⊢ αψ

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The Worm Calculus?

Let A, B, C be worms, and α, β < Λ. A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

A ⊢ B B ⊢ C A ⊢ C A ⊢ B αA ⊢ αB

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The Worm Calculus?

Let A, B, C be worms, and α, β < Λ. A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β αϕ ∧ βψ ⊢ α

• ϕ ∧ βψ
• for α > β

A ⊢ B B ⊢ C A ⊢ C A ⊢ B A ⊢ αC A ⊢ BαC , min B > α A ⊢ B αA ⊢ αB

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The Worm Calculus?

Let A, B, C be worms, and α, β < Λ. A ⊢ A A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β A ⊢ B B ⊢ C A ⊢ C A ⊢ B A ⊢ αC A ⊢ BαC , min B > α A ⊢ B αA ⊢ αB

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The Worm Calculus

Let A, B, C be worms, and α, β < Λ. A ⊢ ⊤ ααA ⊢ αA αA ⊢ βA for α > β A ⊢ B B ⊢ C A ⊢ C A ⊢ B A ⊢ αC A ⊢ BαC , min B > α A ⊢ B αA ⊢ αB

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When restricting the language to worms, the calculi are equivalent

• Theorem. For any two worms A and D,

A ⊢RC D if and only if A ⊢WC D.

• Proof. It’s easy to see that if A ⊢WC D then A ⊢RC D.

...

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Conservativity

It’s a bit harder to check that if A ⊢RC D then A ⊢WC D:

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Conservativity

It’s a bit harder to check that if A ⊢RC D then A ⊢WC D:

• A ⊢WC ⊤ is an axiom.

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Conservativity

It’s a bit harder to check that if A ⊢RC D then A ⊢WC D:

• A ⊢WC ⊤ is an axiom.
• Consider just the case where D = αB:

We want: if A ⊢RC αB, then A ⊢WC αB.

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Conservativity

It’s a bit harder to check that if A ⊢RC D then A ⊢WC D:

• A ⊢WC ⊤ is an axiom.
• Consider just the case where D = αB:

We want: if A ⊢RC αB, then A ⊢WC αB.

• Induction on the length of AαB.

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Conservativity

It’s a bit harder to check that if A ⊢RC D then A ⊢WC D:

• A ⊢WC ⊤ is an axiom.
• Consider just the case where D = αB:

We want: if A ⊢RC αB, then A ⊢WC αB.

• Induction on the length of AαB.
• Base case: length(AαB) = 1.

⊤ ⊢RC α = ⇒ ⊤ ⊢WC α

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Conservativity

It’s a bit harder to check that if A ⊢RC D then A ⊢WC D:

• A ⊢WC ⊤ is an axiom.
• Consider just the case where D = αB:

We want: if A ⊢RC αB, then A ⊢WC αB.

• Induction on the length of AαB.
• Base case: length(AαB) = 1.

⊤ ⊢RC α = ⇒ ⊤ ⊢WC α

• Induction hypothesis: For any worms E, F such that

length(EF) < length(AαB), E ⊢RC F = ⇒ E ⊢WC F

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Conservativity: induction step

Goal: break down the statement A ⊢RC αB into one or more smaller provability statements. Then use the induction hypothesis.

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Conservativity: induction step

Goal: break down the statement A ⊢RC αB into one or more smaller provability statements. Then use the induction hypothesis. For example: A ⊢ αB

RC ⇓

⇑ hα(A) ⊢ αhα(B) and A ⊢ rα(B)

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Conservativity: induction step

Goal: break down the statement A ⊢RC αB into one or more smaller provability statements. Then use the induction hypothesis. For example: A ⊢ αB

RC ⇓

⇑

WC

hα(A) ⊢ αhα(B) and A ⊢ rα(B)

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Conservativity: induction step

Goal: break down the statement A ⊢RC αB into one or more smaller provability statements. Then use the induction hypothesis. For example: A ⊢ αB

RC ⇓

⇑

WC

hα(A) ⊢ αhα(B) and A ⊢ rα(B) And so on.

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Conclusions

• It’s possible to eliminate conjunction from RC0

Λ.

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Conclusions

• It’s possible to eliminate conjunction from RC0

Λ.

• Modulo ⊤ ⊢ α, there is a syntactical algorithm to decide WC.

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Conclusions

• It’s possible to eliminate conjunction from RC0

Λ.

• Modulo ⊤ ⊢ α, there is a syntactical algorithm to decide WC.
• Worms are great!

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Thank you

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