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We'll be covering recursive algebraic data types, curried functions, purity, generic programming, and category theory.

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- 1. Functional Programming Algebraic Data Types Functional Programming (in C++) What? Why? How (in C++)? unit, primitive type, one value, denoted () - Algebraic Data Types : (), :=, : =, , ⊕⊗ - Functions : a (b c) →→ con 20 - Generic Programming ost : (c◄C) → c int → bo David Sankel 10 - Category Theory : monoid, monad, etc. Sankel Software 1 2 3 Algebraic Data Types unit, primitive type, one value, denoted () + product, binary type operation, denoted a b, “a and b”, one of each ⊗ 4
- 2. Algebraic Data Types Examples: Examples: unit, (), one value true : = () Z : = () product, a b, “a and b” ⊗ false : = () N := Z ⊕ N sum, a b, “a or b” ⊕ bool := true ⊕ false is the same as, a := b Z is implemented with unit. is implemented with, a : = b true is implemented with unit. A N value is the same as a Z value or an N false is implemented with unit. value. A bool value is the same as a true value or a false value. 7 8 9 Examples: Type Functions Type Functions Z : = () Add parameter on left side of : = or := [] : = () N := Z ⊕ N symbol that can be used on the right. L a := [] ⊗ ⊕ (a L a) [] : = () Say ai is a value of type a. and e is the z Z ∈ L a := [] ⊗ ⊕(a L a) N0 = (0,z) value of type []. A value of “L of a” is either a N1 = (1,N0) = (1,(0,z)) (0,e) (1,(ai,(0,e)) N2 = (1,N1) = (1,(1,(0,z))) value of [] or (a value of a and a value of 10 “L of a”). 11 (1,(aj,(1,(ai,(0,e)) 12
- 3. Type Functions Algebraic Data Types Algebraic Data Types (in C++!) T a : = (T a ⊗ T a) ⊕ a - Abstract Our critera for functional concepts in C++ - No (minimal) syntax sugar, it scares Binary tree with values of type 'a' at the - Simple ((), :=, : =, , ⊕⊗ ) away the new bees. leaves. - Powerful - Mixes well with typical C++. - ... - No copycatting other languages and their limitations.. 13 14 15 Algebraic Data Types (in C++!) Algebraic Data Types (in C++!) Algebraic Data Types (in C++!) Easy ones Product Types Sum Types - Can use enum when underlying types - Unit (): use boost::mpl: void_ - z := a ⊗ ⊗ b c. struct. Named - are units, and accessors - aren't used elsewhere. - New unit types: T : = () struct T {}; -a ⊗ b. boost.fusion.vector. Access by - Can use a product type with an index. index - common and error prone - “is the same as”: a := b - Can use polymorphic base class. typedef b a; - boost.fusion.map. Both accessor methods. - not really nice syntax/error prone 16 17 18
- 4. Algebraic Data Types (in C++!) Algebraic Data Types (in C++!) Algebraic Data Types (in C++!) “is implemented with , : =, wrap it in a struct ” Type Functions (:= style) Use “type function trait” from boost.mpl. Sum Types R01 : = double - boost.variant (best option!) struct R01 - arbitrary underlying types { explicit R01( const double impl_ ) none : = (), Op a := none⊕ a - small syntax overhead : impl( impl_ ) {} struct none{}; - access by index or type double impl; template< typename a > }; struct Op { - accessor functions (invariant typedef boost::variant<none,a> a b c ⊕ ⊕ guaranteed) boost::variant<a,b,c> - internal impl access function (invariant type; 19 requirement) 20 }; 21 Algebraic Data Types (in C++!) Algebraic Data Types (in C++!) What the heck is a recursive Type Functions (::= style) product type? Use a wrapper template struct Recursive Types S a := a ⊗ (S a) none : = (), Op a : = none ⊕ a - sum types: use make_recursive_variant struct none{}; - Seems like nonsense!? template< typename a > - product types: use struct Op { make_recursive_variant (see explicit Op( boost::variant<none,a>); paper for details) //... boost::variant<none,a> impl; }; 22 23 24
- 5. What the heck is a recursive Laziness (in C++!) Laziness (in C++!) product type? How can we represent a computation of a How can we represent a computation of a S a := a ⊗ (S a) value in C++? value in C++? - Seems like nonsense!? - A 0 argument function. - Nope - Think of the recursion as being a computation of type (S a) - ... 25 26 27 Laziness (in C++!) What the heck is a recursive Functions product type? How can we represent a computation of a f:A → B value in C++? S a := a ⊗ (S a) - A 0 argument function. A set S of pairs (a,b) where a A b B. For ∈ ∈ - Streams of type a. See paper for a template< typename a> direct implementation. every a A, there exists exactly one ∈ struct lazy { typedef boost::function< a () > corresponding pair in S. type; }; 28 29 30
- 6. What is a C++ function? What is a C++ function? What is a C++ function? bool f(int); bool f(int); => int bool → bool f(int); => int bool → What about multiple arguments? 31 32 33 What is a C++ function? Lets try another case. bool f2(int, int); 34
- 7. What is a C++ function? What is a C++ function? Currying bool f3(int, int); (int int World) ⊗ ⊗ → Translation a⊗b→c { ++someglobalvar; (World bool) ⊗ to a → (b → c) R f(A1, A2,...,An); return true; } ^^^ (A1 ⊗ A2...An World) (World R) ⊗ → ⊗ - function returns a function (convenient) - (int int) bool doesn't work! → ⊗ - → is right associative - Consider the corresponding set. - Works with any function where - Introduce new parameter the domain is a product. and return value, World... 37 38 39 What is a C++ function? Translation R f(A1, A2,...,An); (A1⊗A2...An⊗World)→(World⊗R) ^^^ A1→A2...An→World→(World⊗R) ^^^ 40
- 8. Functions (in C++) gfp library (netsuperbrain.com/gfp) - gfp::ciof (curried io function) - Converts a c++ function pointer into a function as we formulated. - gfp::cfunc (curried function) - cfunc<a,b,c>::type => function<function<c (b)> a> 43 Generic Programming Formulation: - Emptiable, a type class - A typeclass is a set of pairs (a,p) - a is a : = type or type function - p is a profile that fits certain patterns and laws - Our restrictions - At most one pair per type - Profile is a simple value 46
- 9. Generic Programming (in C++!) Polymorphic Values: - Same type inference trick doesn't work for values. - Introduce resolve: resolve<std::vector<int>>( emptyThing ); - Polymorphic values must follow a Certain trait . . . 51 Generic Programming (in C++!) Generic Programming (in C++!) Generic Programming (in C++!) Polymorphic Values: Polymorphic Values & Functions: Polymorphic Classes: struct EmptyThing - Covers all generics we care about - Extendable collections of polymorphic { template<typename T> - Cannot be extended to support new values and functions. struct result; types without modification of underlying - Use partial template instantiation to //... code. select supported type. result<this_type(vector<int>*)>::type - No relation between related - The polymorphic entities select operator()( vector<int>* dummy ); polymorphic values and functions. appropriate instantiation when //... used. } emptyThing; 52 53 54
- 10. Category Theory Category Theory Category Theory (Monoids) - deals abstractly with mathematical Monoids There are lots of monoids! structures and relations between them. A 0∈A - int, +, 0 : sum monoid + :A → A → A - bool, &&, true: all monoid - Gives some guidance as to what to do - string, concat, “”: string monoid with generics. - a → m: function monoid A, 0, and + form a monoid when + is - m monoid - Very powerful and expressive. associative and 0 is an identity for - forwards monoid operations to results. +. - io m: io monoid. Similar to function monoid. 55 56 57 Category Theory (Monoids) Category Theory Functional Programming (in C++!) Quick example: Much much more: FP Benefits: - header : Message → string - Cleaner design → Cleaner code - contents : Message → string - functor/pointed: functions, containers... - less code/static types → Less bugs - Powerful tools gfp::cfunc<Message,string> - idiom (applicative functors): FRP, streams FP (in C++) Benefits: payload = gfp::mplus( header )( contents ); - No need to switch languages - monad: arbitrary computations - Integrates well - called point free (pointless) programming!. - Easy to use (no special syntax) - foldable: compress collections - Highly Capable 58 59 60

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