Primitive Recursive Bars Are Inductive Jon Sterling Carnegie - PowerPoint PPT Presentation

Primitive Recursive Bars Are Inductive Jon Sterling Carnegie Mellon University August 17, 2016

⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees. 𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙 “All bars on a spread can be coded as inductive trees.”

⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees. 𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙 “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma)

⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees. 𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙 “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous

In general, implies termination for a class of functional programs on infinite trees. 𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙 “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable)

𝕮𝖘𝖕𝖛𝖝𝖋𝖘'𝖙 𝕮𝖇𝖘 𝖀𝖎𝖋𝖙𝖏𝖙 “All bars on a spread can be coded as inductive trees.” ⋆ ⇒ the Fan Theorem (intuitionistic König’s Lemma) ⋆ ⇒ all functions on the interval 𝕁 ≜ [0, 1] are uniformly continuous ⋆ ⇒ completeness of intuitionistic first-order logic (Bickford, Constable) In general, implies termination for a class of functional programs on infinite trees.

A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws: 1. If 𝑣 ∈ 𝔗 , then there exists an 𝑦 ∈ ℕ such that 𝑣 ⌢ 𝑦 ∈ 𝔗 . 2. If 𝑣 ⌢ 𝑦 ∈ 𝔗 , then also 𝑣 ∈ 𝔗 . 3. Finally, ⟨⟩ ∈ 𝔗 . The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space. ⃗ ⃗ ⃗ ⃗ 𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙 A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree.

1. If 𝑣 ∈ 𝔗 , then there exists an 𝑦 ∈ ℕ such that 𝑣 ⌢ 𝑦 ∈ 𝔗 . 2. If 𝑣 ⌢ 𝑦 ∈ 𝔗 , then also 𝑣 ∈ 𝔗 . 3. Finally, ⟨⟩ ∈ 𝔗 . The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space. ⃗ ⃗ ⃗ ⃗ 𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙 A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws:

2. If 𝑣 ⌢ 𝑦 ∈ 𝔗 , then also 𝑣 ∈ 𝔗 . 3. Finally, ⟨⟩ ∈ 𝔗 . The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space. ⃗ ⃗ ⃗ ⃗ 𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙 A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws: 1. If 𝑣 ∈ 𝔗 , then there exists an 𝑦 ∈ ℕ such that 𝑣 ⌢ 𝑦 ∈ 𝔗 .

3. Finally, ⟨⟩ ∈ 𝔗 . The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space. ⃗ ⃗ ⃗ ⃗ 𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙 A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws: 1. If 𝑣 ∈ 𝔗 , then there exists an 𝑦 ∈ ℕ such that 𝑣 ⌢ 𝑦 ∈ 𝔗 . 2. If 𝑣 ⌢ 𝑦 ∈ 𝔗 , then also 𝑣 ∈ 𝔗 .

The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space. ⃗ ⃗ ⃗ ⃗ 𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙 A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws: 1. If 𝑣 ∈ 𝔗 , then there exists an 𝑦 ∈ ℕ such that 𝑣 ⌢ 𝑦 ∈ 𝔗 . 2. If 𝑣 ⌢ 𝑦 ∈ 𝔗 , then also 𝑣 ∈ 𝔗 . 3. Finally, ⟨⟩ ∈ 𝔗 .

⃗ ⃗ ⃗ ⃗ 𝕿𝖖𝖘𝖋𝖇𝖊𝖙 𝖇𝖙 𝖙𝖖𝖇𝖉𝖋𝖙 𝖇𝖔𝖊 𝖚𝖘𝖋𝖋𝖙 A 𝓽𝓺𝓼𝓯𝓫𝓮 is at the same time a topological space and a non-wellfounded tree. A spread is defined by a predicate 𝔗 on lists of natural numbers subject to some laws: 1. If 𝑣 ∈ 𝔗 , then there exists an 𝑦 ∈ ℕ such that 𝑣 ⌢ 𝑦 ∈ 𝔗 . 2. If 𝑣 ⌢ 𝑦 ∈ 𝔗 , then also 𝑣 ∈ 𝔗 . 3. Finally, ⟨⟩ ∈ 𝔗 . The predicate 𝔗 either defines the finite prefixes of paths down an infinite tree, or it defines the lattice of open sets of a topological space.

⃗ ⃗ 𝕺𝖋𝖏𝖍𝖎𝖈𝖕𝖘𝖎𝖕𝖕𝖊𝖙 𝖇𝖔𝖊 𝕼𝖕𝖏𝖔𝖚𝖙 A list of naturals 𝑣 ∈ ℕ ∗ can be thought of as a neighborhood around a point, or as a prefix of a path through an infinite tree. A stream of naturals 𝛽 ∈ ℕ ℕ can be thought of as an ideal point in the spread (space), or as a path through the spread’s infinite tree. 𝑣 ≺ 𝛽 ( ⃗ 𝑣 approximates 𝛽 ) 𝛽 ∈ ⃗ 𝑣 ( ⃗ 𝑣 is a neighborhood around 𝛽 )

∀ 𝛽 ∈ ⃗ 𝑣 . ∃ 𝑜 ∈ ℕ . 𝛽 [ 𝑜 ] ∈ 𝔆 𝑣 ◁ 𝔆 ⃗ ⃗ ⃗ 𝖀𝖕 𝕮𝖇𝖘 𝕭 𝕺𝖕𝖊𝖋 A bar 𝔆 is a predicate on neighborhoods such that every point “hits it”. More generally, 𝔆 bars a neighborhood 𝑣 when every path through 𝑣 ends up in 𝔆 .

⃗ ⃗ ⃗ 𝖀𝖕 𝕮𝖇𝖘 𝕭 𝕺𝖕𝖊𝖋 A bar 𝔆 is a predicate on neighborhoods such that every point “hits it”. More generally, 𝔆 bars a neighborhood 𝑣 when every path through 𝑣 ends up in 𝔆 . ∀ 𝛽 ∈ ⃗ 𝑣 . ∃ 𝑜 ∈ ℕ . 𝛽 [ 𝑜 ] ∈ 𝔆 𝑣 ◁ 𝔆

Let 𝔗 ♮ ( ⃗ 𝑣 ) ≜ { 𝑦 ∈ ℕ ∣ 𝑣 ⌢ 𝑦 ∈ 𝔗 } . Presupposing 𝑣 ∈ 𝔗 and 𝔆 monotone : ∀ 𝑦 ∈ 𝔗 ♮ ( ⃗ 𝑣 ). 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ∈ 𝔆 𝜃 ϝ 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ◀ 𝑗𝑜𝑒 𝔆 Admissible (by monotonicity): 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝜂 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ 𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊 “All demonstrations of barhood can be analyzed into inductive mental constructions .”

∀ 𝑦 ∈ 𝔗 ♮ ( ⃗ 𝑣 ). 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ∈ 𝔆 𝜃 ϝ 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ◀ 𝑗𝑜𝑒 𝔆 Admissible (by monotonicity): 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝜂 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ 𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊 “All demonstrations of barhood can be analyzed into inductive mental constructions .” Let 𝔗 ♮ ( ⃗ 𝑣 ) ≜ { 𝑦 ∈ ℕ ∣ 𝑣 ⌢ 𝑦 ∈ 𝔗 } . Presupposing 𝑣 ∈ 𝔗 and 𝔆 monotone :

Admissible (by monotonicity): 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝜂 ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ 𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊 “All demonstrations of barhood can be analyzed into inductive mental constructions .” Let 𝔗 ♮ ( ⃗ 𝑣 ) ≜ { 𝑦 ∈ ℕ ∣ 𝑣 ⌢ 𝑦 ∈ 𝔗 } . Presupposing 𝑣 ∈ 𝔗 and 𝔆 monotone : ∀ 𝑦 ∈ 𝔗 ♮ ( ⃗ 𝑣 ). 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ∈ 𝔆 𝜃 ϝ 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ◀ 𝑗𝑜𝑒 𝔆

⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ 𝕵𝖔𝖊𝖛𝖉𝖚𝖏𝖜𝖋 𝕮𝖇𝖘𝖎𝖕𝖕𝖊 “All demonstrations of barhood can be analyzed into inductive mental constructions .” Let 𝔗 ♮ ( ⃗ 𝑣 ) ≜ { 𝑦 ∈ ℕ ∣ 𝑣 ⌢ 𝑦 ∈ 𝔗 } . Presupposing 𝑣 ∈ 𝔗 and 𝔆 monotone : ∀ 𝑦 ∈ 𝔗 ♮ ( ⃗ 𝑣 ). 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ∈ 𝔆 𝜃 ϝ 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ◀ 𝑗𝑜𝑒 𝔆 Admissible (by monotonicity): 𝑣 ◀ 𝑗𝑜𝑒 𝔆 𝑣 ⌢ 𝑦 ◀ 𝑗𝑜𝑒 𝔆 𝜂