Towards a Flowchart Diagrammatic Language for Monad-based Semantics

Transcription

1 Towards a Flowchart Diagrammatic Language or Monad-based Semantics Julian Jakob Friedrich-Alexander-Universität Erlangen-Nürnberg julian.jakob@au.de

2 Introductory Examples while do What is the most general ormalization or such diagrams?

3 Overview 1 Basic Deinitions 2 Traces, Fixpoints and Iteration 3 Algebraic Operations and Generic Eects

4 Diagrammatic Language edges: nodes: crossing: eedback node: strict node:

5 Deinition Deinition as in algebraic type theory: edges = base edge base edge parallel to edge nodes = base node crossing eedback node node sequential to base node strict nodes

6 Categories In short, a category C is given by: Collection o objects Ob(C) Collection o morphisms Hom(C) Two operations assigning to each morphism its source src( ) Ob(C) and its target tar( ) Ob(C), written : src( ) tar( ). There is an operation assigning to each pair o composable morphisms and g their composition which is a morphism denoted by g or just g and such that src(g ) = src( ) and tar(g ) = tar(g). There is also an operation assigning to each object A Ob(C) an identity morphism id A : A A. These operations are required to be unitary and associative.

7 Diagrams or Categories Object A: A Morphism : A B: Identity id A : A A: A A B Composition g : A C or : A B and g : B C: A B C g

8 Formal Descriptions Deinition (Selinger (2011), Signature) A simple categorical signature Σ consists o a set Σ 0 o object variables, a set Σ 1 o morphism variables, and a pair o unctions dom, cod : Σ 1 Σ 0. Object variables are usually written A,B,C,..., morphism variables are usually written,g,h,..., and we write : A B i dom( ) = A and cod( ) = B. Deinition (Selinger (2011), Interpretation) An interpretation i o a signature Σ a category C consists o a unction i 0 : Σ 0 Ob(C), and or any Σ 1 a morphism i 1 ( ) : i 0 (cod( )) i 0 (dom( )).

9 Coherence Deinition (Selinger (2011), Coherence) A graphical language is coherent, i it is sound and complete. It is sound, i all the axioms o the category are automatically satisied in the graphical language up to isomorphism o diagrams. It is complete i the ollowing condition is true: every equation that is true in the graphical language is a consequence o the axioms.

10 Soundness o the Diagrammatic Language or Categories Let Identity: A B A = B Right Identity: A B A = B Associativity: A B C D g h A B C D = g h

11 Monoidal Categories A monoidal category M has additional structure: A unctor : M M M called the tensor product. An object 1 Ob(M) called the unit object. A natural isomorphism a x,y,z : (x y) z x (y z) called the associator. A natural isomorphism λ x : 1 x x called the let unitor. A natural isomorphism ρ x : x 1 x called the right unitor.

12 Diagrammatic Language or Monoidal Categories Tensor product A B: A Tensor product g : A C B D: Unit object I : A C B g B D

13 Coherence or Monoidal Categories Theorem (Joyal/Street (1991), Coherence or Monoidal Categories) A well-ormed equation between morphism terms in the language o monoidal categories ollows rom the axioms o monoidal categories i and only i it holds, up to planar isotopy, in the graphical language.

14 Symmetry A symmetric monoidal category additionally is equipped with a natural isomorphism, called symmetry B x,y : x y y x Symmetry B A,B : A B B A

15 Coherence or Symmetric Monoidal Categories Theorem (Selinger (2011), Coherence or Symmetric Monoidal Categories) A well-ormed equation between morphisms in the language o symmetric monoidal categories ollows rom the axioms o symmetric monoidal categories i and only i it holds, up to isomorphism o diagrams, in the graphical language.

16 Traced Monoidal Categories Deinition (Traced Monoidal Categories) A symmetric monoidal category (C,, 1, b) (where b is the symmetry) is said to be traced i it is equipped with a natural amily o unctions TrA,B : C(A, B ) C(A, B) satisying ive axioms: it is natural in A and B (let/right tightening), and dinatural in (sliding) vanishing: Tr Y A,B ( ) = Tr A,B (Tr A Y,B ( ))( : A Y B Y ) superposing: TrC A,C B (id C ) = id C TrA,B ( )( : A B ) yanking: Tr, (b, ) = id

17 Axioms or Traced Categories Trace: Naturality (Let Tightening): h = h Naturality (Right Tightening): h = h

18 Axioms or Traced Categories Dinaturality (Sliding): h = h Vanishing(I): = I Vanishing ( ): =

19 Axioms or Traced Categories Superposing: = Yanking: =

20 Coherence or Traced Categories Theorem (Joyal/Street (1991), Coherence or Traced Monoidal Categories) A well-ormed equation between morphisms in the language o symmetric traced categories ollows rom the axioms o symmetric traced categories i and only i it holds in the graphical language up to isomorphism o diagrams.

21 Strictness Deinition (Hasegawa (2003), Strictness) Consider a traced monoidal category C with trace Tr. We say h : Y is strict in C (with respect to the trace Tr) i the ollowing condition holds: For any : A B and g : A Y B Y, (id B h) = g (id A g) : A B Y A B A Y id A h g B Y id B h implies TrA,B ( ) = Tr A,B Y (g) : A B.

22 Strictness Strictness: A B A Y h = Y h g B Y A B A B = g

23 Uniormity Deinition (Hasegawa (2003), Uniormity) Let C be a traced monoidal category with trace Tr, and S be a class o arrows o C. We say Tr is uniorm (or: Tr satisies the uniormity principle) or S i, or any h : Y o S, the condition o strictness holds.

24 Trace-Fixpoint Correspondence Theorem (Hasegawa (1997), Conway Operator Correspondence to Traces) For any category with inite products, to give a Conway operator is to give a trace (where inite products are taken as the monoidal structure). : A = Tr A, ( ) : A g : A B Tr A,B (g) = π B, (g (id A π B, )) : A B

25 Diagrammatic Language or Product Categories A inite product category can be described as a symmetric monoidal category, together with natural amilies o copy and erase maps, with a number o axioms (omitted here, see Selinger (2011)). A : A A A (copy) A A A! A : A 1 (erase) A

26 Some Axioms or the Graphical Language = = = =

27 Coherence or Traced Product Categories Theorem (Selinger (2011), Coherence or Traced Product Categories) A well-ormed equation between morphism terms in the language o traced product categories ollows rom the respective axioms i and only i it holds in the graphical language, up to isomorphism o diagrams, and the diagrammatic manipulations corresponding to the axioms.

28 Conway Operators Deinition (Simpson/Plotkin (2000), Conway Operator) A parameterized ixpoint operator on C is a amily o unctions ( ) : C(A, ) C(A, ) which is natural in A and satisies the ixpoint equation = id A, : A or : A. A Conway operator on C is a parameterized ixpoint operator ( ) which satisies dinaturality: ( st A,, g ) = id A, (g st A,Y, ) : A or : A Y and g : A Y diagonal property: ( (id A )) = ( ) : A or : A.

29 Uniormity or Conway Operators Deinition (Hasegawa (2003), Plotkin s Principle) Let ( ) be a parameterized ixpoint operator on a category with inite products. We say h : Y is strict (with respect to ( ) ) i the ollowing condition holds: For any : A and g : A Y Y,h = g (id A h) A A Y id A h implies g = h : A Y. g Y h

30 Dualities Products are dual to coproducts. Traced coproduct categories can be constructed in the same way as product categories, i.e. a symmetric monoidal category plus maps A : A + A A and! A : 0 A (given a coproduct structure). The axioms are dual to the axioms or traced product categories. The distinction does not matter or deinitions o Conway operators or similar notions.

31 Monads Deinition (Moggi (1991), Kleisli Triple) A Kleisli triple over a category C is a triple (T ; η; ), where T : Ob(C) Ob(C), η A : A TA or A Ob(C), : TA TB or : A TB and the ollowing equations hold: ηa = id TA (1) η A = or : A TB (2) g = (g ) or : A TB and g : B TC (3)

32 Elgot Monads Deinition (Complete Elgot Monad) A monad T over C is a complete Elgot monad i it possesses an operator, called iteration, sending any : T (Y + ) to : TY satisying the ollowing conditions: ixpoint: [η, ] = ; naturality: g = ([T (inl) g, η inr] ) or any g : Y TZ; codiagonal: (T ([id, inr]) g) = g or any g : T ((Y + ) + ); uniormity: h = T (id h) g h = g or any g : Z T (Y + Z) and h : Z

33 Elgot Monads Fixpoint: Y = Y Y Naturality: Y g Z = g Y Z

34 Elgot Monads Uniormity: Z h Y Z = g h Y Z Z h Y Z = g Y Z

35 Elgot Monads Codiagonal: g Y = g Y

36 Elgot Monads g Y = g Y = g Y = g Y

37 Elgot Monads g Y = g Y

38 Algebraic Operations Deinition (Algebraic Operation) Given a monad T on a symmetric monoidal category C, an algebraic operation α on T is a amily o maps (T ) v (T ) w, demanding that the exponentials (T ) v and (T ) w exist in C, such that ( ) id α = α Y ( ) id : T ( Y ) w T ( Y ) w. Further, the ollowing interaction between exponentials and the strength must commute: τ w (id α Y ) = α v τ v : TY T ( Y ) w, where τ z = curry(τ(id ev) assoc) : A TB z T (A B) z or any z, assoc being the obvious associativity o the tensor.

39 Algebraic Operations (T ) v α (T ) w ( ) id ( ) id (TY ) v TY v α Y id α Y (TY ) w TY w τ v τ w T ( Y ) v α v T ( Y ) w

40 Example: Nondeterminism To express nondeterminism we can use the powerset monad T = P. The algebraic operation is v : (T ) 2 T with v(u, v) = u v. This means, or two possible continuations both are possible or put dierently one is chosen nondeterministically.

41 Generic Eect Deinition (Generic Eect) Given an monad T, a generic eect is any Kleisli morphism : n Tm. Equivalently, i C = Sets, and n N, : n Tm is given by a tuple ( 1 Tm,..., n Tm).

42 Generic Eects or the Nondeterminism Monad In nondeterminism the generic eects are: stop : 1 T 0 toss : 1 T 2 Our monad is the powerset monad, so T 0 = P0 corresponds to the set with the single element being the empty set and T 2 = P2 is the powerset o the set {1, 2}, which is {, {1}, {2}, {1, 2}}.

43 Diagrammatic Representation or Generic Eects Example: Nondeterminism. From toss : 1 T 2, where T 2 = T (1 + 1), one can construct Toss : T ( + ) in the ollowing way id toss τ 1 T 2 T ( (1 + 1)) T dist T ( (1 + 1)) T ( + ) Toss: Toss

44 Duality Theorem (Duality) To construct an algebraic operation α x : (T ) v (T ) w rom a generic eect g : w Tv, there is the sequence id,id uncurry(α v curry(η snd)) w w w Tv with curry(η snd) w (Tv) v α v (Tv) w uncurried. To ind a term α x : (T ) v (T ) w given g : w Tv, we need to curry the ollowing: (T ) v id g w (T ) v τ Tv T ((T ) v ev v) T

45 Proo For the proo the ollowing equations have been veriied: α x = curry(ev τ(id g)) g = uncurry(α v curry(η snd)) id, id ( ) id curry(ev τ(id g)) = curry(ev τ(id g)) ( ) id τ w (id curry(ev τ(id g))) = curry(ev τ(id g))τ v

46 Reerences Peter Selinger (2011) A Survey o Graphical Languages or Monoidal Categories New Structures or Physics Lecture Notes in Physics Volume 813, pp Springer Verlag. Andre Joyal, Ross Street (1991) The geometry o tensor calculus I Advances in Mathematics 88(1), pp Masahito Hasegawa (2003) The Uniormity Principle on Traced Monoidal Categories Electronic Notes in Theoretical Computer Science Volume 69, pp Masahito Hasegawa (1997) Recursion rom cyclic sharing: traced monoidal categories and models o cyclic lambda calculi Proc. Typed Lambda Calculi and Applications (TLCA?97) Springer Lecture Notes in Computer Science 1210, pp 196?213.. Eugenio Moggi (1991) Notions o Computation and Monads Inormation and Computation Volume 93, pp Alex Simpson, Gordon Plotkin (2000) Complete axioms or categorical ixed-point operators Proc. 15th Logic in Computer Science LICS 2000, pp 30?41.

47 The End