The theory behind Functional Programming Functional: based on lambda calculus / combinators, recursion theory, category theory. By contrast with other paradigms: Functions ONLY! Defining functions Defining data types

Category Theory Abstract view of functions as “arrows” with domains and codomains as their “source” and “target” objects Things to watch: How we compose arrows Which arrows can be composed How objects can be seen as identity arrows How we connect categories with functors

Resources on Category Theory +Category+Theory+&ie=utf-8&oe=utf- 8&aq=t&rls=org.mozilla:en- US:official&client=firefox-a

Definition of a Category Tw classes: Objects A,B,C,… and Arrows f,g,h,… For each arrow f : A->B, two objects A=dom(f) B=cod(f) If f:A->B and g:B->C, their composition, denoted g. f : A->C exists For each object A the arrow 1_A : A->A exists For arrows f:A->B,g:B->C,h:C->D composition is associative: h. (g. f) = (h. g).f For f:A->B, 1_A is a unit arrow: f. 1_A=f=1_B. f

Examples of Categories Finite sets with functions Groups with group homomorphisms Vector spaces with linear mappings Graphs with graph homomorphisms Topological spaces with continuous functions Posets and monotone functions

Functional programming seen as a category Objects: monomorphic data types: Integer, Boolean, records of employees Arrows: functions between data types Id:: a -> a provides 1_a for each type a “.” provides composition – and it is defined if types match Function composition is associative

Functors a (covariant) functor F : C → D: – for each object x in C, an object F(x) in D – for each arrow f : x → y in C, an arrow F(f) : F(x) → F(y), such that the following two properties hold: – For every object x in C, F(1 x ) = 1 F(x) ; – For all morphisms f : x → y and g : y → z, F(g ∘ f) = F(g) ∘ F(f). Of special interest: endofunctors F:C->C

The Initial Algebra Semantics for polymorphic data types (1) F-algebra: can be used to represent data structures used in programming, such as lists and trees. Defined such that the following diagram commutes:data structuresprogramminglists

The Initial Algebra Semantics for polymorphic data types (2) the functor F: Set -> Set that sends a set X to 1+X + denotes the usual coproduct (also called “sum”) given by disjoint union, and 1 is a terminal object (i.e. any singleton set) the set N of natural numbers together with the function succ : 1+N->N, is the + of the functions zero : 1->N (whose image is 0) and succ : 1+N->N (which sends an integer n to n+1) is an initial F- algebra

The Initial Algebra Semantics for polymorphic data types (3) Types defined by using least fixed point construct with functor F can be regarded as an initial F- algebra, provided that parametricity holds for the typeleast fixed pointparametricity Examples: data N = Zero | Succ N data List a = Nil | Cons a (List a) map provides the functor definition for lists: it preserves the structure while changing its payload

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