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 1 / 12

The Reflection Calculus The Reflection Calculus, RC 0 Λ , in a nutshell: • Closed positive fragment of GLP Λ . 2 / 12

The Reflection Calculus The Reflection Calculus, RC 0 Λ , in a nutshell: • Closed positive fragment of GLP Λ . • No variables. 2 / 12

The Reflection Calculus The Reflection Calculus, RC 0 Λ , in a nutshell: • Closed positive fragment of GLP Λ . • No variables. • Formulas are made up of ⊤ , ∧ and � α � for all α < Λ. 2 / 12

The Reflection Calculus The Reflection Calculus, RC 0 Λ , in a nutshell: • Closed positive fragment of GLP Λ . • No variables. • Formulas are made up of ⊤ , ∧ and � α � for all α < Λ. ⊤ � 1 �� 0 �⊤ ∧ � 5 �⊤ � 7 � ( � 2 �⊤ ∧ � 0 �� 0 �� 0 �⊤ ) 2 / 12

The Reflection Calculus: axioms and rules Let ϕ, ψ, χ be formulas of RC 0 Λ , and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ϕ ϕ ⊢ ψ ϕ ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ⊢ ψ ϕ ⊢ χ ϕ ∧ ψ ⊢ ψ ϕ ⊢ ψ ∧ χ � α �� α � ϕ ⊢ � α � ϕ � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 3 / 12

What are worms? Worms are long cylindrical animals with a tube-like body and no limbs iterated consistency statements: A = � 73 �� ǫ 0 �� 0 �� 42 �� ω �⊤ No one likes to write all of these � s and � s... A = 73 ǫ 0 0 42 ω 4 / 12

Decomposing worms Worms have α -heads and α -remainders. 5 / 12

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 ): h 73 (73 ǫ 0 0 42 ω ) = 73 ǫ 0 5 / 12

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 ): h 73 (73 ǫ 0 0 42 ω ) = 73 ǫ 0 The α -remainder is the rest: r 73 (73 ǫ 0 0 42 ω ) = 0 42 ω 5 / 12

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 ): h 73 (73 ǫ 0 0 42 ω ) = 73 ǫ 0 The α -remainder is the rest: r 73 (73 ǫ 0 0 42 ω ) = 0 42 ω A = h α ( A ) r α ( A ) ≡ RC h α ( A ) ∧ r α ( A ) 5 / 12

Why are worms interesting? For any worms A , B such that min A > α , A ∧ α B ≡ RC A α B 6 / 12

Why are worms interesting? For any worms A , B such that min A > α , A ∧ α B ≡ RC A α B Theorem (Worms are the “core” of RC 0 ) For each formula ϕ of RC 0 Λ there is some worm A such that: ϕ ≡ RC A . 6 / 12

Why are worms interesting? For any worms A , B such that min A > α , A ∧ α B ≡ RC A α B Theorem (Worms are the “core” of RC 0 ) For each formula ϕ of RC 0 Λ there is some worm A such that: ϕ ≡ RC A . Then maybe we could forget about conjunctions? 6 / 12

The Reflection Calculus Let ϕ, ψ, χ be formulas of RC 0 Λ , and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ ϕ ⊢ ϕ ϕ ⊢ ψ ϕ ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ⊢ ψ ϕ ⊢ χ ϕ ∧ ψ ⊢ ψ ϕ ⊢ ψ ∧ χ � α �� α � ϕ ⊢ � α � ϕ � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ ϕ ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ⊢ ψ ϕ ⊢ χ ϕ ∧ ψ ⊢ ψ ϕ ⊢ ψ ∧ χ � α �� α � ϕ ⊢ � α � ϕ � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ A ⊢ ⊤ ϕ ∧ ψ ⊢ ϕ ϕ ⊢ ψ ϕ ⊢ χ ϕ ∧ ψ ⊢ ψ ϕ ⊢ ψ ∧ χ � α �� α � ϕ ⊢ � α � ϕ � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ A ⊢ ⊤ ϕ ⊢ ψ ϕ ⊢ χ ϕ ∧ ψ ⊢ ψ ϕ ⊢ ψ ∧ χ � α �� α � ϕ ⊢ � α � ϕ � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ A ⊢ ⊤ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ � α �� α � ϕ ⊢ � α � ϕ � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ A ⊢ ⊤ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ αα A ⊢ α A � α � ϕ ⊢ � β � ϕ for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ A ⊢ ⊤ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ αα A ⊢ α A α A ⊢ β A for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A be a worm, and α, β < Λ. ϕ ⊢ χ χ ⊢ ψ A ⊢ A ϕ ⊢ ψ A ⊢ ⊤ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ αα A ⊢ α A α A ⊢ β A for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A , B , C be worms, and α, β < Λ. A ⊢ B B ⊢ C A ⊢ A A ⊢ C A ⊢ ⊤ ϕ ⊢ ψ ϕ ⊢ χ ϕ ⊢ ψ ∧ χ αα A ⊢ α A α A ⊢ β A for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let ϕ, ψ, χ be formulas of RC 0 Λ , A , B , C be worms, and α, β < Λ. A ⊢ B B ⊢ C A ⊢ A A ⊢ C A ⊢ ⊤ αα A ⊢ α A α A ⊢ β A for α > β ϕ ⊢ ψ � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β � α � ϕ ⊢ � α � ψ 7 / 12

The Worm Calculus? Let A , B , C be worms, and α, β < Λ. A ⊢ B B ⊢ C A ⊢ A A ⊢ C A ⊢ ⊤ αα A ⊢ α A α A ⊢ β A for α > β A ⊢ B � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β α A ⊢ α B 7 / 12

The Worm Calculus? Let A , B , C be worms, and α, β < Λ. A ⊢ B B ⊢ C A ⊢ A A ⊢ C A ⊢ ⊤ A ⊢ B A ⊢ α C , min B > α A ⊢ B α C αα A ⊢ α A α A ⊢ β A for α > β A ⊢ B � � � α � ϕ ∧ � β � ψ ⊢ � α � ϕ ∧ � β � ψ for α > β α A ⊢ α B 7 / 12

The Worm Calculus? Let A , B , C be worms, and α, β < Λ. A ⊢ B B ⊢ C A ⊢ A A ⊢ C A ⊢ ⊤ A ⊢ B A ⊢ α C , min B > α A ⊢ B α C αα A ⊢ α A α A ⊢ β A for α > β A ⊢ B α A ⊢ α B 7 / 12

The Worm Calculus Let A , B , C be worms, and α, β < Λ. A ⊢ B B ⊢ C A ⊢ C A ⊢ ⊤ A ⊢ B A ⊢ α C , min B > α A ⊢ B α C αα A ⊢ α A α A ⊢ β A for α > β A ⊢ B α A ⊢ α B 7 / 12

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 . ... 8 / 12

Conservativity It’s a bit harder to check that if A ⊢ RC D then A ⊢ WC D : 9 / 12

Conservativity It’s a bit harder to check that if A ⊢ RC D then A ⊢ WC D : • A ⊢ WC ⊤ is an axiom. 9 / 12

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 = α B : We want: if A ⊢ RC α B , then A ⊢ WC α B . 9 / 12

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 = α B : We want: if A ⊢ RC α B , then A ⊢ WC α B . • Induction on the length of A α B . 9 / 12

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 = α B : We want: if A ⊢ RC α B , then A ⊢ WC α B . • Induction on the length of A α B . • Base case: length ( A α B ) = 1. ⊤ ⊢ RC α = ⇒ ⊤ ⊢ WC α 9 / 12