Towards Coherence for Guarded Traces Sergey Goncharov a oder a Paul - PowerPoint PPT Presentation

Towards Coherence for Guarded Traces Sergey Goncharov a oder a Paul Blain Levy b Lutz Schr¨ January, 7, 2019, Birmingham TCS Seminar a Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg b University of Birmingham

Introduction • Recursion / iteration • order-theoretic / unguarded • process-theoretic / guarded • Generic categorical models: • Total: • Axiomatic/synthetic domain theory (Hyland, Fiore, Taylor et al.) • let-ccc’s with fixpoint objects (Crole/Pitts, Simpson) • Traced monoidal categories (Joyal/Street/Verity, Hasegawa) • Elgot monads/theories (Bloom/Esik, Ad´ amek, Milius et al.) • Partial: • Completely iterative monads/theories (Bloom/Esik, Ad´ amek, Milius et al.) • later-modality (Nakano, Appel, Melli` es, Benton, Birkedal et al.) • Partial traced categories (Heghverdi, Scott, Malherbe, Selinger) • Functorial dagger (Milius, Litak) • Here: Unifying framework for guarded and unguarded feedback in monoidal categories

Guarded Fixpoints: Overview guarded traced categories guarded iteration guarded 8 -dimensional recursion Hilbert spaces guarded topos of trees, complete total Conway recursion, Elgot monads complete metric spaces completely complete iterative monads Elgot monads process algebra domain-enriched examples examples

Guarded Fixpoints: Overview FoSSaCS17: G., Schr¨ oder, Rauch, Pir´ og, Unifying Guarded guarded traced and Unguarded Iteration categories guarded iteration guarded 8 -dimensional recursion Hilbert spaces guarded topos of trees, complete total Conway recursion, Elgot monads complete metric spaces : -congruent completely complete iterative monads Elgot monads retraction process algebra domain-enriched examples examples

Guarded Fixpoints: Overview FoSSaCS17: G., Schr¨ oder, Rauch, Pir´ og, Unifying Guarded guarded traced and Unguarded Iteration categories guarded iteration guarded 8 -dimensional recursion Hilbert spaces guarded topos of trees, complete total Conway recursion, Elgot monads complete metric spaces : -congruent completely complete iterative monads Elgot monads retraction FoSSaCS18: G., Schr¨ oder, GTC process algebra domain-enriched examples examples

Motivating Example: Process Algebra In process algebra, we solve tail-recursive process definitions, like x “ a . x ` τ. x ` y More abstractly, we involve a monad T Σ X “ νγ. T p X ` Σ γ q of infinite process trees and axiomatize guardedness of f : X Ñ T Σ Y in a coproduct summand σ : Y 1 Y as follows (in Klesili): f : X Ñ inr Y ` Z g : Y Ñ σ V h : Z Ñ V f : X Ñ Z (vac ` ) (cmp ` ) inl f : X Ñ inr Z ` Y r g , h s ˝ f : X Ñ σ V (par ` ) f : X Ñ σ Z f : Y Ñ σ Z r f , g s : X ` Y Ñ σ Z

Guarded Iteration v.s. Guarded Recursion Guarded iteration is a (partial) operation f : X Ñ Y ` X f : : X Ñ Y with f guarded in X Dualization should yield guarded recursion: f : X ˆ Y Ñ X f : : Y Ñ X Z Can we make sense of this intuition? : a X Y

Guarded Iteration v.s. Guarded Recursion Guarded iteration is a (partial) operation f : X Ñ Y ` X f : : X Ñ Y with f guarded in X Dualization should yield guarded recursion: f : X ˆ Y Ñ X f : : Y Ñ X Z Can we make sense of this intuition? : a X Pivotal Idea: Keep the notion Y of guardedness independent of fixpoint calculations

Going Monoidal

Going Monoidal (We only consider symmetric monoidal categories, think of b “ ` , ˆ ) Identity id: Composition g ˝ f : Tensor g b f : Symmetry:

Going Monoidal: Additional Structure Trace tr p f : U b A Ñ B b U q 1 : A Ñ B Compact closure: unit η : I Ñ A b A ‹ and counit ǫ : A ‹ b A Ñ I - q ‹ is a contravariant involutive endofunctor where p - In compact closed categories, trace is definable and unique, for: * U * U * U U * B U U C 1 The twist of input wires is nonstandard, but bear with me

Iteration and Recursion Iteration and recursion are typically viewed as corner cases: • With b “ ` , we obtain p f : A Ñ B ` A q : “ tr p f ˝ ∇ q : A A B • With b “ ˆ , we obtain p f : A ˆ B Ñ A q : “ tr p ∆ ˝ f q : B A A Corresponding converse definitions can also be produced. So, traces and (Conway) fixpoints are equivalent in the requisite cases!

Guarded Categories

Partially Guarded Morphisms A monoidal category is guarded if it is equipped with distinguished families Hom ‚ p A b B , C b D q Ď Hom p A b B , C b D q , drawn as follows B D A C where • A is unguarded input • B is guarded input • C is unguarded output • D is guarded output The idea is to allow feedback only on p A , D q , which we call a guardedness profile of f . Hence we introduce axioms:

The Axioms

Some (Easy) Observations • There is a greatest notion of guardedness, Hom ‚ p A b B , C b D q “ Hom p A b B , C b D q • There is a least (vacuous) notion of guardedness, g h • Axioms are stable under 180 0 -rotations, hence C is guarded iff C op is guarded, i.e. we maintain duality of recursion and iteration

Ideal Guardedness A particularly common case is ideal guardedness A guarded ideal is a family Hom § p X , Y q Ď Hom p X , Y q closed under finite tensors and composition with any morphism on both sides The general form of a partially guarded morphism over a guarded ideal is p 1 n n 1 q n 1

Ideal Guardedness A particularly common case is ideal guardedness A guarded ideal is a family Hom § p X , Y q Ď Hom p X , Y q closed under finite tensors and composition with any morphism on both sides The general form of a partially guarded morphism over a guarded ideal is p 1 n n 1 q n 1 In the (co-)Cartesian case this simplifies greatly, generating standard notions, e.g. f : X Ñ 2 Y ` Z iff r inl , g s h X Ý Ý Ñ Y ` W Ý Ý Ý Ý Ý Ñ Y ` Z with some g P Hom § p W , Y ` Z q and h : X Ñ Y ` W

Guarded Traces

Guarded Traced Categories A guarded category is guarded traced if it is equipped with a trace: satisfying a collection of axioms adapted from the standard case Guarded Conway iteration/recursion operators are obtained analogously to the standard case

Non-Ideal Case: Contractive Maps Consider the category CMS of inhabited complete metric spaces and non-expansive maps Let f : X ˆ Y Ñ Z be guarded in Y if for all x P X , f p x , - - q is contractive This makes CMS into a guarded traced monoidal category (fixpoints calculated via Banach’s fixpoint theorem) but not ideally guarded, because a contraction factor depends on x and may not be chosen uniformly Question: Is there a parametrically contractive map in CMS that is not uniformly contractive?

Unguarded Recursion as Guarded Recursion • A standard way to do recursion with monads is in the category C T ‹ with T -algebras as objects and C -morphisms of carriers as morphisms Example: C = point-free dcpo’s and continuous functions; T = lifting monad X ÞÑ X K • Alternatively, following [Milius and Litak, 2013], we consider guarded recursion operators on C where C is ideally guarded over Hom § p X , Y q “ t f ˝ η | f : TX Ñ Y u Example: with C and T as above, we allow only recursion on X of f : X K ˆ Y Ñ X

Unguarded Recursion as Guarded Recursion We consider the following axioms: • Dinaturality: B B B B g g A C A C C A • Squaring (is not a property of Conway recursion but a property of Conway uniform recursion): B B B A A A A A Theorem: There is a bijective correspondence between guarded squarable dinatural operators on C and unguarded squarable dinatural on C T ‹

More Examples.. • The topos of trees (guardedness by later-operator) • Non-pointed order-enriched monads (e.g. non-empty powerset, probability distributions) 1 • Hybrid iteration semantics : (“guardedness” = “progressiveness”) 0 . 5 1 2 3 4 : Goncharov, Jakob, and Neves 2018, A Semantics for Hybrid Iteration

Guarded Traces in Hilbert Spaces

Finite-Dimensional Hilbert Spaces Recall the multiplicative compact closed category of relations p Rel , ˆ , 1 q - q ‹ Relations can be thought of as Boolean matrices, with transposition p - and (unparamerized) trace being the trace of the square matrices ¨ ˛ b 11 ¨ ¨ ¨ b 1 n ÿ tr ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ‚ “ b ii ˚ ‹ ˝ b n 1 ¨ ¨ ¨ b nn i Analogously, linear operators on finite-dimensional Hilbert spaces can be represented as matrices over a field – we stick to the field of reals Thus, Hilbert spaces are compact closed with tensors p f b g qp x b y q “ f p x q b g p y q , R as tensor unit, X ‹ “ X on objects, f ‹ as the unique adjoint operator x f p x q , y y “ x x , f ‹ p y qy and unit/counit induced by inner products

Infinite-Dimensional Hilbert Spaces More generally, Hilbert spaces are vector spaces with inner products, a complete as a normed spaces under the induced norm | | x | | “ x x , x y Category Hilb : • Objects are Hilbert spaces • Morphisms are bounded linear operators, i.e. | | f p x q| | ď c ¨ | | x | | for a fixed c and every x • Monoidal structure as before • Adjointness for operators still works and p f b g q ‹ “ f ‹ b g ‹ , f ‹‹ “ f , id ‹ “ id, p f ˝ g q ‹ “ g ‹ ˝ f ‹