Multicategories of multiary lenses Bob Rosebrugh (with Mike Johnson) Department of Mathematics and Computer Science Mount Allison University FMCS 2019-06-02
Backgound Bidirectional Transformations (BX) are n = 2 case, well studied commonly modelled using symmetric and asymmetric lenses ◮ Assume synchronization data specifying when an X state and a Y state are consistent ◮ A lens L : X ← → Y models a (BX) and it includes: ◮ synchronizing data (encoded in various ways) and has ◮ consistency restoration or re-synchronization operator(s) for state changes ◮ Symmetric case: Consistent models with two-way re-synchronization, for example for database interoperation ◮ Asymmetric case: One-way synchronization/restoration with global/local states, for example database view updating 2
Outline ◮ Multidirectional transformations as n -ary lenses ◮ first generalize (binary) symmetric lenses – more propagations ◮ also generalize spans of asymmetric lenses – to wide spans ◮ introduce “amendments” in both cases ◮ Resulting multiary lenses compose via wide spans ◮ A multicategory of multiary lenses ensues 3
Multidirectional Transformations An ( n -ary) multimodel space M = (( A i ) i ≤ n , R ) has ◮ A i a category ◮ objects model states A 1 , A 2 , . . . ; arrows: state updates ◮ R is synchronization data specifying consistent model states A Multidirectional Transformation (MX) has: ◮ a multimodel space M ◮ consistency restoration or re-synchronization operators after a model state changes (update) Generalizing lenses provides models of two types (below): ◮ n -ary symmetric lens ◮ (wide) span of asymmetric lenses 4
n-ary Symmetric Lenses Let M a multimodel space, A i categories, R = ( R , δ ) is a function � | A 1 | × . . . × | A n | (call R elements corrs ) δ = ( δ i ) : R An n -ary symmetric lens (Diskin-Konig-Lawford) M = ( M , p) has: ◮ Multimodel space M ◮ Consistency restoration operators p = (p i ) i ≤ n ◮ Given corr R and update u i from A i = δ i ( R ) in A i ... ◮ p i gives updates of A j = δ j ( R ) target tuple consistent For an example, think of database interoperation 5
� � n -ary Symmetric Lenses Suppose states ( A j ) j ≤ n synchronized by corr R , A i � ❥❥❥❥❥❥❥❥❥❥❥ R R R ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A 1 · · · A j · · · 6
� � n -ary Symmetric Lenses Suppose states ( A j ) j synchronized by corr R , then given an update u i from state A i to target A ′ i A i � ❥❥❥❥❥❥❥❥❥❥❥ R R R ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A i A 1 · · · A j · · · u i � A ′ i 7
✤ ✤ � ✤ ✤ � ✤ ✤ � ✤ ✤ � n -ary Symmetric Lenses Suppose states ( A j ) j synchronized by corr R , then given an update from state A i to target A ′ i the lens propagates updates u ′ j of A j ’s (and A ′ i ) targets A ′′ j A i � ❥❥❥❥❥❥❥❥❥❥❥ R R R ❨ ❨ ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A i A 1 A 1 · · · A j A j · · · u i � p i � j =p R u ′ u ′ ij ( u i ) A ′ A ′ 1 i i u ′ i � A ′′ A ′′ A ′′ i 1 j 8
� ✤ � ✤ ✤ ✤ ✤ ✤ � ✤ � � ✤ � n -ary Symmetric Lenses Suppose states ( A j ) j synchronized by corr R , then given an update from state A i to target A ′ i the lens propagates updates of A j ’s (and A ′ i ) to targets A ′′ j and , re-synchronization data R ′ on ( A ′′ j ) j . A i � ❥❥❥❥❥❥❥❥❥❥❥ R R R ❨ ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A i A 1 A 1 · · · A j A j · · · u i � p i � j =p R u ′ u ′ ij ( u i ) A ′ A ′ 1 i i u ′ i � A ′′ A ′′ A ′′ A ′′ A ′′ A ′′ · · · · · · i � ❚ ❚ ❚ ❚ ❚ ❚ R ′ ❡ i 1 1 j j ❡ ❡ t ❡ ❡ ❡ ❡ t ❡ R ′ R ′ ❡ ❡ 9
n -ary Symmetric Lenses A small example with A 1 = NA = set 2 , A 2 = AC = set 2 ; R = { ( X , X ′ ) | X 1 = X ′ 0 } Formal notation for the propagation operation p i R ′ = p R u ′ j = p R u ′ i = p R i ( u i ) ij ( u i ) j � = i ii ( u i ) Call p R ii ( u i ) the amendment (update) of u i Amendment may arise (be necessary) if, for example, no corr synchronizes A ′ i with other model states. Update u i is R-closed when p R ii ( u i ) = id A i (trivial amendment) Lens L is closed if it is R -closed for all R and u i 10
Well-behaved lenses An n -ary symmetric lens M = ( M , p) is well-behaved if satisfies: ◮ (Stable): Identities propagate to identities: p R ij ( id A i ) = id A j 1 ≤ j ≤ n and p R i ( id A i ) = R ◮ (Refl2): Amended updates propagate like the unamended: p R ij (p R ii ( u i ) u i ) = p R ij ( u i ) for j � = i , ◮ (Refl3): Amended updates are closed: p R ii ( u i ) u i is R -closed 11
Asymmetric Amendment Lenses ◮ Asymmetric lenses arose as strategy for the database View Update Problem. ◮ First were set based, then algebraic, and categorical ◮ As delta lenses, spans represent (binary) symmetric lenses ◮ Amendments a recent addition (D-K-L), paralleling n -ary symmetric lens 12
Asymmetric Amendment Lenses An aa-lens L = ( A , B , g , p ) has ◮ categories A , B the base and view model spaces � B a functor called the get ◮ g : A ◮ p = { p A | A ∈ A } family of operations delivering � B ′ for any view state update v : g ( A ) � A ′′ the putback of v ◮ p A b ( v ) : A � B ′′ the amendment of v ◮ p A a ( v ) : B ′ also satisfying B ′′ = g ( A ′′ ) � B ′ is closed if its amendment is identity View update v : g ( A ) L a closed aa-lens if all view updates are closed Small examples with A = set 3 ; B = set 2 . . . 13
� � � � Asymmetric Amendment Lenses The aa-lenses compose: For L = ( A , B , g , p ), L ′ = ( B , C , h , q ) the composite is L ′ L = ( A , C , hg , r ) where b ( q g ( A ) � C ′ , define r A b ( w ) = p A for w : hg ( A ) ( w )) b a ( q g ( A ) a ( w ) = h ( v ) · q g ( A ) for v = p A ( w )), define r A ( w ) a b as in: b ( q g ( A ) p A ( w )) b A A A A ” A ” A ” q g ( A ) ( w ) � B ′ v b B ′ B ′ g ( A ) g ( A ) g ( A ) g ( A ) g ( A ) B ” B ” B ” B ” B ” � C ′ C ′ C ′′ C ′′ C ′′ C ′′′ C ′′′ C ′′′ hg ( A ) hg ( A ) hg ( A ) w q g ( A ) h ( v ) ( w ) a 14
Well-Behaved aa-Lenses aa-lens is L = ( A , B , g , p ) is well-behaved if it is: ◮ (Stable) Identities putback to identities ◮ (Refl0) B ′ = g ( X ) for X ∈ A implies p A a ( v ) = id B ′ (if B ′ is a g image, then no amendment) ◮ (Refl2) putback of the amended update same as unamended p A b ( p A a ( v ) v ) = p A b ( v ) ◮ (PutGet) the get of a putback is the amended update Stable, Refl0 and PutGet compose, as does closed, but not Refl2, and it turns out that. . . Convenient category spg-lens has arrows those aa-lenses satisfying Stable and PutGet properties. 15
� spg Lens Spans and n -ary Symmetric Lenses Two Constructions: First, from n -wide span L of spg-lens (domains S ) to n -ary symmetric M L : � ❥❥❥❥❥❥❥❥❥❥❥ S S S ❨ ❨ ( g j , p j ) ( g i , p i ) ❨ ❏ ❨ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ❨ ❏ ❨ ( g 1 , p 1 ) � ❨ ❨ ❨ ❨ ❨ ❨ ❨ A i A 1 · · · A j · · · Let M L = ( M , p) where M = (( A i ) , R ) and ◮ R = | S | (objects of S ), δ i ( S ) = g i ( S ) � A ′ ◮ Define propagation p i for u i : g i ( S ) i by ◮ the j’th put of u i is the j’th get of the i’th put: p S ij ( u i ) = g j ( p S i , b ( u i )) ◮ the M L amendment of u i is its L i amendment Note that M L is stable, closed if the L i are 16
� � � � � spg Lens Spans and n -ary Symmetric Lenses Next, from n -ary symmetric M to n -wide span L M of spg-lens: With M = ( M , p) and M = (( A i ) , R ) Objects of the head S of L M are the corr set R � δ i ( R ′ ) } Arrows S ( R , R ′ ) = { ( u i ) i ≤ n | u i : δ i ( R ) A 1 � ❥❥❥❥❥❥❥❥❥❥❥ R ⑧ R R ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ⑧ ❨ ❨ � ⑧ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A 1 A 2 A 2 · · · A n A n · · · u 1 u 2 u n A ′ A ′ A ′ A ′ A ′ A ′ · · · · · · 1 � ❚❚❚❚❚❚❚❚❚❚❚ R ′ 1 2 � ❄ 2 ❡ ❡ n n ❡ ❡ ❡ ❡ ❡ ❡ ❄ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ R ′ R ′ ❡ ❡ 17
spg Lens Spans and n -ary Symmetric Lenses For the lens ( L M ) i ◮ Define g i ( R ) = δ i ( R ) on objects, similarly on arrows � A ′ ◮ For u i : g i ( R ) i the put is ( p R p R i , b ( u i )) j = ij ( u i ) j � = i ( p R p R i , b ( u i )) i = ii ( u i ) u i and the put amendment is that for p i The two constructions are not mutually inverse, and in any case wide spans of spg lenses and n -ary symmetric lenses with same behaviour should be made equivalent . 18
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