Intro Plotkin Algebra to the Rescue Results Don’t try this at home: No-Go Theorems for Distributive Laws Maaike Zwart & Dan Marsden University of Oxford 27 March 2018
Intro Plotkin Algebra to the Rescue Results Overview • Introduction • Monads • Distributive laws • Previously broken distributive laws • Plotkin’s counterexample • General No-Go theorems: the algebraic approach • Generalized Plotkin • And more... • Results
Intro Plotkin Algebra to the Rescue Results Monads A monad is a triple x T , η, µ y , with T an endofunctor and η : 1 ñ T , µ : TT ñ T natural transformations, such that: η T T µ T TT TTT TT Id µ µ T η µ T µ µ TT T TT T
Intro Plotkin Algebra to the Rescue Results Monads A monad is a triple x T , η, µ y , with T an endofunctor and η : 1 ñ T , µ : TT ñ T natural transformations, such that: η T T µ T TT TTT TT Id µ µ T η µ T µ µ TT T TT T • List: L • L p X q set of all finite lists. • η X p x q “ r x s • µ X concatenation.
Intro Plotkin Algebra to the Rescue Results Monads A monad is a triple x T , η, µ y , with T an endofunctor and η : 1 ñ T , µ : TT ñ T natural transformations, such that: η T T µ T TT TTT TT Id µ µ T η µ T µ µ TT T TT T • List: L • L p X q set of all finite lists. • η X p x q “ r x s • µ X concatenation. • Powerset P • P p X q set of all subsets. • η X p x q “ t x u • µ X union.
Intro Plotkin Algebra to the Rescue Results Monads A monad is a triple x T , η, µ y , with T an endofunctor and η : 1 ñ T , µ : TT ñ T natural transformations, such that: η T T µ T TT TTT TT Id µ µ T η µ T µ µ TT T TT T • Distribution D • List: L • D p X q set of all probability • L p X q set of all finite lists. distributions. • η X p x q “ r x s • η X p x q point distribution. • µ X concatenation. • µ X weighted average. • Powerset P • P p X q set of all subsets. • η X p x q “ t x u • µ X union.
Intro Plotkin Algebra to the Rescue Results Monads A monad is a triple x T , η, µ y , with T an endofunctor and η : 1 ñ T , µ : TT ñ T natural transformations, such that: η T T µ T TT TTT TT Id µ µ T η µ T µ µ TT T TT T • Distribution D • List: L • D p X q set of all probability • L p X q set of all finite lists. distributions. • η X p x q “ r x s • η X p x q point distribution. • µ X concatenation. • µ X weighted average. • Powerset P • More examples: Multiset, • P p X q set of all subsets. • η X p x q “ t x u Exception, Reader, Writer, • µ X union. ...
Intro Plotkin Algebra to the Rescue Results Composing Monads with Distributive Laws We can compose monads with the help of a distributive law - Beck 1969 x TS , η T η S , µ T µ S ¨ T λ S y Where λ : ST Ñ TS is a natural transformation satisfying the following axioms. S λ λ S T SST STS TSS η S T T η S µ S T T µ S λ λ ST TS ST TS λ T T λ S STT TST TTS S η T η T S S µ T µ T S λ λ ST TS ST TS
Intro Plotkin Algebra to the Rescue Results Examples There is a distributive law LP ñ PL . It works like the famous ‘times over plus’ distributivity: p a ` b q ˚ c “ a ˚ b ` a ˚ c rt a , b u , t c us ÞÑ tr a , c s , r b , c su Many more work like this: MM ñ MM , LM ñ ML , MP ñ PM , ...
Intro Plotkin Algebra to the Rescue Results Examples There is a distributive law LP ñ PL . It works like the famous ‘times over plus’ distributivity: p a ` b q ˚ c “ a ˚ b ` a ˚ c rt a , b u , t c us ÞÑ tr a , c s , r b , c su Many more work like this: MM ñ MM , LM ñ ML , MP ñ PM , ... Some general results: • If T is a commutative monad, and S a monad defined by linear equations, then there is a distributive law ST ñ TS - Manes and Mulry 2007 . • There are variations on the above theorem for affine and relevant monads - Dahlqvist, Parlant and Silva 2018 .
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus.
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus. However...
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus. However... • We thought we had a distributive law DD ñ DD .
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus. However... • We thought we had a distributive law DD ñ DD . But we made a mistake, which was hard to spot.
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus. However... • We thought we had a distributive law DD ñ DD . But we made a mistake, which was hard to spot. • We are not alone. • Several examples of mistakes in the literature. • According to Bonsangue, Hansen, Kurz, and Rot: “It can be rather difficult to prove the defining axioms of a distributive law.”
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus. However... • We thought we had a distributive law DD ñ DD . But we made a mistake, which was hard to spot. • We are not alone. • Several examples of mistakes in the literature. • According to Bonsangue, Hansen, Kurz, and Rot: “It can be rather difficult to prove the defining axioms of a distributive law.” • What to do now? Is there a distributive law at all?
Intro Plotkin Algebra to the Rescue Results All that glisters is not gold Distributive laws are often quite intuitive, like times over plus. However... • We thought we had a distributive law DD ñ DD . But we made a mistake, which was hard to spot. • We are not alone. • Several examples of mistakes in the literature. • According to Bonsangue, Hansen, Kurz, and Rot: “It can be rather difficult to prove the defining axioms of a distributive law.” • What to do now? Is there a distributive law at all? • Our goal: to find general principles that tell us when no distributive law exists.
Intro Plotkin Algebra to the Rescue Results Previous Results • No distributive law DP ñ PD - Plotkin / Varacca and Winskel 2005 • No distributive law PD ñ DP - Varacca 2003, without proof • No monad structure on PD - Dahlqvist and Neves 2018 • No monad structure on PP - Klin and Salamanca 2018 • No distributive law TP ñ PT , with T satisfying some technical conditions. - Klin and Salamanca 2018
Intro Plotkin Algebra to the Rescue Results Previous Results • No distributive law DP ñ PD - Plotkin / Varacca and Winskel 2005 • No distributive law PD ñ DP - Varacca 2003, without proof • No monad structure on PD - Dahlqvist and Neves 2018 • No monad structure on PP - Klin and Salamanca 2018 • No distributive law TP ñ PT , with T satisfying some technical conditions. - Klin and Salamanca 2018 What is so special about powerset?
Intro Plotkin Algebra to the Rescue Results Plotkin’s Proof Main idea is to chase a specially chosen element: 1 2 t c , d u P DP p X q Ξ “ t a , b u ` round the naturality diagram: λ X Ξ ? DP p f q PD p f q ? ? λ X
Intro Plotkin Algebra to the Rescue Results Plotkin’s Proof Main idea is to chase a specially chosen element: • Cleverly choose functions so that on the bottom row, the 1 2 t c , d u P DP p X q Ξ “ t a , b u ` unit laws can be applied: round the naturality diagram: f p a q “ a f p c q “ a f p b q “ b f p d q “ b λ X Ξ ? DP p f q PD p f q ? ? λ X
Intro Plotkin Algebra to the Rescue Results Plotkin’s Proof Main idea is to chase a specially chosen element: • Cleverly choose functions so that on the bottom row, the 1 2 t c , d u P DP p X q Ξ “ t a , b u ` unit laws can be applied: round the naturality diagram: f p a q “ a f p c q “ a f p b q “ b f p d q “ b λ X Ξ ? DP p f q PD p f q t a , b u ? λ X
Intro Plotkin Algebra to the Rescue Results Plotkin’s Proof Main idea is to chase a specially chosen element: • Cleverly choose functions so that on the bottom row, the 1 2 t c , d u P DP p X q Ξ “ t a , b u ` unit laws can be applied: round the naturality diagram: f p a q “ a f p c q “ a f p b q “ b f p d q “ b λ X Ξ PD p f q DP p f q t a , b u t a , b u λ X
Intro Plotkin Algebra to the Rescue Results Plotkin’s Proof Main idea is to chase a specially chosen element: • Cleverly choose functions so that on the bottom row, the 1 2 t c , d u P DP p X q Ξ “ t a , b u ` unit laws can be applied: round the naturality diagram: f p a q “ a f p c q “ a f p b q “ b f p d q “ b λ X Ξ • Take an inverse image to PD p f q DP p f q learn fact about λ p Ξ q : t a , b u t a , b u λ X
Intro Plotkin Algebra to the Rescue Results Plotkin’s Proof Main idea is to chase a specially chosen element: • Cleverly choose functions so that on the bottom row, the 1 2 t c , d u P DP p X q Ξ “ t a , b u ` unit laws can be applied: round the naturality diagram: f p a q “ a f p c q “ a f p b q “ b f p d q “ b λ X Ξ • Take an inverse image to PD p f q ´ 1 DP p f q learn fact about λ p Ξ q : t a , b u t a , b u λ X
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