Coalgebraic bar recursion and bar induction Tarmo Uustalu, Inst. of - PowerPoint PPT Presentation

Coalgebraic bar recursion and bar induction Tarmo Uustalu, Inst. of Cybernetics, Tallinn joint work with Venanzio Capretta, Nottingham Theory Days at K¨ ao, 29-31 Jan. 2016

This talk Brouwer’s bar induction and Spector’s bar recursion and bar induction are well-known semi-classical principles. We give a “modernized” coalgebraic account that both rationalizes the formulations and avoids unncessary rigidity. We argue that these are useful tools of an (open-minded) applied type theorist.

Bar recursion: traditional formulation Bar recursion: Given a predicate Q on N ∗ that is decidable and suffix-closed. Assume that Q is a bar: every σ : N ω has a prefix ℓ : N ∗ satisfying Q . Suppose we have a set C , an element c : C and a function g : N → C → C . Then there is a unique function f : N ∗ → C such that f ℓ = c , if Q ℓ , and f ℓ = g ( λ n . f ( ℓ + +[ n ])), otherwise.

Bar induction: traditional formulation Bar induction: Given a predicate Q on N ∗ that Q is decidable and suffix-closed. Assume that Q is a bar: every σ : N ω has a prefix ℓ : N ∗ satisfying Q . Given also a predicate R on N ∗ . Suppose that R is inductive: If ℓ satisfies Q , it also satisfies R , else if ℓ + +[ n ] satisfies R for any n : N , then ℓ satisfies R . Then any ℓ : N ∗ satisfies R . N can in principle be replaced with any set A , but one can also argue that only for countable A the principles are acceptable from the constructivist point of view.

Coalgebraic versions: A preview These deal directly with arbitrary (countable) branching. They also remove the need to have the tree continue behind the bar. Coalgebraic bar recursion: If a coalgebra is barred, it is recursive. ( ≈ If all paths in a tree are finite, you can recurse on the tree.) Coalgebraic bar induction: If a coalgebra is barred, it is wellfounded. ( ≈ If all paths in a tree are finite, you can induct on the tree.)

Leading idea We can represent branching types, incl base case detection, as set functors. E.g., F X = 1 + X + X × X . Then we can represent potentially non-wellfounded trees as pointed coalgebras, i.e., as a pair of a coalgebra and an element of the carrier. E.g., ( A , α, a ) = ( N , f : N → 1 + N + N × N , n : N ) where f 0 = inl ∗ f 1 = inr (inl 0)) f ( i + 2) = inr (inr ( i , i + 1)) and n is some fixed number.

� � � � Recursive coalgebras A coalgebra ( A , α ) of an endofunctor F on a category C is said to be recursive , if, for every algebra ( C , γ ), there exists a unique map f : A → C (a coalgebra-to-algebra morphism) satisfying α A FA f Ff C FC γ

� � � � � Wellfounded coalgebras Given a coalgebra ( A , α ) of a pullbacks along monos preserving endofunctor F on a category C with a pullbacks along monos. For any mono j : U ֒ → A , another mono nt α j : nt α U ֒ → A is defined by the pullback α | j nt α U F U nt α j Fj � F A α A The coalgebra ( A , α ) is called wellfounded if, for any mono j : U ֒ → A , if nt α U factors through U , then j is an isomorphism. I.e., nt α U = ⇒ j is iso. U � ��������� � � � � � � nt α j j � � A

Containers, container functors A container is given by a set S of shapes and an S -indexed family of positions for every shape. It defines a set functor � S , P � by � S , P � X = Σ s : S . P s → X .

Finite paths of a coalgebra For any a : A we define a set Path α a of (maximal) paths starting from it coinductively by α a = ( s , h ) e : ¬ P s α a = ( s , h ) p : P s π : Path α ( h p ) end e : Path α a p ≺ π : Path α a We define finiteness of a path inductively by α a = ( s , h ) e : ¬ P s α a = ( s , h ) p : P s π : Path α ( h p ) π ↓ end e ↓ p ≺ π ↓

Barred coalgebras (of container functors) A barred coalgebra is a coalgebra ( A , α ) such that all its paths are finite: ∀ a : A . ∀ π : Path α a . π ↓

Coalgebraic bar recursion and bar induction Coalgebraic bar recursion: If a coalgebra is barred, it is recursive. ( ≈ If all paths in a tree are finite, you can recurse on the tree.) Coalgebraic bar induction: If a coalgebra is barred, it is wellfounded. ( ≈ If all paths in a tree are finite, you can induct on the tree.)

Example: Tabulating stably continuous stream functions Functions f : A ∞ → B can sometimes be tabulated in a tree. What is required is often formulated as stable continuity. But you need bar recursion to actually exploit it.

Stably continuous stream functions A modulus of continuity for a function f : A ∞ → B is a function m : A ∞ → N such that ∀ σ, σ ′ : A ∞ . σ ′ = m σ σ → f σ ′ = f σ The modulus is stable if ∀ σ, σ ′ : A ∞ . σ ′ = m σ σ → m σ ′ = m σ (i.e., it is its own modulus of continuity).

Trees for tabulations We want to represent stream functions f : A ∞ → B as some form of trees. The suitable tree type SF A , B is defined inductively by g : A → SF A , B b : B tell b : SF A , B ask g : SF A , B

A recursive coalgebra from a stably continuous stream function Given a stream function f : A ∞ → B with a stable modulus m : A ∞ → N . We construct a coalgebra on the functor F X = B + ( A → X ) with A ∗ as the carrier: α : A ∗ → B + ( A → A ∗ ) � inl ( f ( ℓ + + σ 0 )) if m ( ℓ + + σ 0 ) ≤ len ℓ α ℓ = inr ( λ a . ℓ + +[ a ]) otherwise This coalgebra is barred! So by bar recursion we can define a function to h : A ∗ → SF A , B . The tabulation of f is h [],