Partial Recursive Functions and Finality Gordon Plotkin Laboratory - PowerPoint PPT Presentation

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Partial Recursive Functions and Finality Gordon Plotkin Laboratory for the Foundations of Computer Science, School of Informatics, University of Edinburgh SamsonFest, Oxford, May 2013 Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Outline Introduction 1 Natural numbers objects in monoidal categories 2 Weak representability of partial recursive functions 3 Strong representability of partial recursive functions 4 Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Outline Introduction 1 Natural numbers objects in monoidal categories 2 Weak representability of partial recursive functions 3 Strong representability of partial recursive functions 4 Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ = F ( D ) Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ = F ( D ) = D → D Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = F ( D ) = D → D = 1 + N Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = F ( D ) = D → D = 1 + N The solution 1 zero succ 1 + N α − → N − − → N ← − − N equivalently is initial. Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = F ( D ) = D → D = 1 + N The solution N α − 1 − − → 1 + N is final Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = D → D = F ( D ) = 1 + N The solution 1 zero succ 1 + N α − → N − − → N ← − − N equivalently is initial. The solution N α − 1 − − → 1 + N is final Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = F ( D ) = D → D = 1 + N The solution 1 zero succ 1 + N α − → N equivalently − − → N ← − − N is initial. The solution N α − 1 − − → 1 + N is final Initiality ⇒ Primitive Recursion Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = F ( D ) = D → D = 1 + N The solution 1 zero succ 1 + N α − → N equivalently − − → N ← − − N is initial. The solution N α − 1 − − → 1 + N is final Initiality ⇒ Primitive Recursion Finality ⇒ Kleene’s µ -Recursion Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Motivation Domain equations D ∼ D ∼ N ∼ = F ( D ) = D → D = 1 + N The solution 1 zero succ 1 + N α − → N equivalently − − → N ← − − N is initial. The solution N α − 1 − − → 1 + N is final Initiality ⇒ Primitive Recursion Finality ⇒ Kleene’s µ -Recursion So there should be a categorical account of the partial recursive functions Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Sets and partial functions The category pSet of sets and partial functions f : X ⇀ Y has: A “tensor" functor given by cartesian product on objects, and on partial functions f : X ⇀ X ′ , g : Y ⇀ Y ′ by: ( f × g )( x , y ) ≃ � fx , gy � The one-point sets ✶ functions as an identity for cartesian product. Distributive binary sums, where, as before X + Y = ( { 0 } × X ) + ( { 1 } × Y ) (Remark: pSet does have finite binary products: 1 = ∅ X × Y = X + ( X × Y ) + Y but they don’t help.) Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions pSet has a final ( I + − )-coalgebra β ✲ ✶ + Y Y h ✶ + h ❄ ❄ ✲ ✶ + N N α − 1 Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions pSet has a final ( I + − )-coalgebra β ✲ ✶ + Y Y ✶ + h h ❄ ❄ ✶ + N N ✛ α Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions pSet has a final ( I + − )-coalgebra β ✲ ✶ + Y Y h ✶ + h ❄ ❄ ✶ + N N ✛ α Using Kleene equality we can write this out as an equation:  0 ( β ( y ) ≃ inl ( ∗ ))  ≃ h ( y ′ ) + 1 ( β ( y ) ≃ inr ( y ′ )) h ( y ) undefined ( β ( y ) ↑ )  Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions pSet has a final ( I + − )-coalgebra β ✲ ✶ + Y Y h ✶ + h ❄ ❄ ✶ + N N ✛ α Using Kleene equality we can write this out as an equation:  0 ( β ( y ) ≃ inl ( ∗ ))  ≃ h ( y ′ ) + 1 ( β ( y ) ≃ inr ( y ′ )) h ( y ) undefined ( β ( y ) ↑ )  with, setting s = def inr − 1 ◦ β , unique solution µ k ∈ N . β ( s k ( y )) ≃ inl ( ∗ ) ≃ def h ( y ) Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Outline Introduction 1 Natural numbers objects in monoidal categories 2 Weak representability of partial recursive functions 3 Strong representability of partial recursive functions 4 Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Natural numbers algebras Context a monoidal category C Unit: I Tensor Product of objects: A ⊗ B Tensor product of morphisms: g f A − → A ′ B − → B ′ f ⊗ g → A ′ ⊗ B ′ A ⊗ B − − Structural isomorphisms: a A , B , C : A ⊗ ( B ⊗ C ) ∼ l A : I ⊗ A ∼ r A : A ⊗ I ∼ = ( A ⊗ B ) ⊗ C = A = A satisfying standard equations. and a natural numbers algebra zero succ − − − − → N ← − − − − N I Plotkin Partial Recursive Functions and Finality

Introduction Natural numbers objects in monoidal categories Weak representability of partial recursive functions Strong representability of partial recursive functions Representing natural numbers functions Natural numbers algebra zero succ − − − − → N ← − − − − N I The morphism k = def succ k ◦ zero : I → N represents k ∈ N . The morphism f : N n → N m represents f : N n → N m if f ◦� k 1 , . . . , k n � = f ( k 1 , . . . , k n ) ( for all k 1 , . . . , k n ∈ N ) (where we define � c 1 , . . . , c n � : I ⊸ A 1 ⊗ . . . ⊗ A n to be: c 1 ⊗ ... ⊗ c n I − → I ⊗ . . . ⊗ I − − − − − − → A 1 ⊗ . . . ⊗ A n for c i : I ⊸ A i .) Plotkin Partial Recursive Functions and Finality