Functional Programming

Transcription

1 A System Practical March 1, 2011

2 Outline Preliminaries A System Practical 1 Preliminaries John Backus Issues With von Neumann Languages 2 A System 3 Practical Haskell Ruby

3 John Backus Preliminaries A System Practical John Backus Issues With von Neumann Languages Born Died Directed development of FORTRAN, released in Backus-Naur Form. Received Turing Award in 1977 for his work on programming languages.

4 A System Practical The von Neumann Architecture John Backus Issues With von Neumann Languages

5 A System Practical Excessive "Housekeeping" Code John Backus Issues With von Neumann Languages / Computes dot product of two vectors / public i n t dotproduct ( i n t [ ] a, i n t [ ] b ) { i n t res = 0; for ( i n t i = 0; i < a. l e n g th ; i ++) res += a [ i ] b [ i ] ; return res ; }

6 A System Practical Lack of Useful Algebraic Properties John Backus Issues With von Neumann Languages Example: f (x) + 2f (x) + 3f (x) (1) = ( )f (x) (2) = 6f (x) (3) Compare to: i n t y = 1 f ( x ) + 2 f ( x ) + 3 f ( x ) ; Is y == 6*f(x)?

7 A System Practical Potential Problems John Backus Issues With von Neumann Languages Not if f references global variables. i n t f ( i n t x ) { return someglobalvariable ++; } Or if f interacts with the user. i n t f ( i n t x ) { i n t r e t ; scanf ( "%d ", & r e t ) ; return r e t ; }

8 A System Practical Charasteristics of an FP System The system is not history-sensitive. Functions have no side-effects. Functions are formed by combining other functions. Programs built in an FP system can be transformed by useful algebraic properties.

9 A System Practical Structure of an FP System A set O of objects. A set F of primitive functions. The application operation. A set F of functional forms. A set D of defined functions.

10 Objects Preliminaries A System Practical Objects are the data on which our programs operate. Atoms are numbers and combinations of characters: 12, T, F, FOO Lists are any combination of atoms: < 3, 4, 5 >, < A, B, < C, D >>, φ represents bottom, or undefined Any list containing is equivalent to, and any function that operates on returns.

11 A System Practical Primitive Functions A function is a mapping from the set of objects to the set of objects. Functions accept a single argument and return a single result. The primitive functions are those built-in to the system. Functions are applied with the application operator :. For any function f and any object x, f : x the result of applying f to x.

12 A System Practical Function Application

13 A System Practical Examples of Function Application id : A = A + :< 3, 4 >= 7 2 :< 5, 6, 7 >= 6 : 1 = tl : = (Remember that all functions are preserving). trans :<< 1, 2 >, < 5, 6 >>=<< 1, 5 >, < 2, 6 >>

14 Functional Forms Preliminaries A System Practical Functional forms are expressions that represent functions. Functional forms are operators with functions as their operands. We can use functional forms to construct more useful functions from the supplied primitives.

15 A System Practical Examples of Functional Forms Composition: f g : x f : (g : x) Apply to All: αf :< x 1, x 2,...x n > < f : x 1, f : x 2,...f : x n > Insert: /f :< x 1, x 2,...x n > f :< x 1, /f < x 2,...x n >> Construction: [f 1, f 2,...f n ] : x < f 1 : x, f 2 : x,...f n : x > Condition: (p f ; g) (p : x) = T f : x; g : x Constant: x : y x

16 Composition Preliminaries A System Practical f g

17 Construction Preliminaries A System Practical [f, g]

18 Conditional Preliminaries A System Practical p f ; g

19 Definitions Preliminaries A System Practical A definition, or defined function, associates a name with a functional form. Definitions are analogous to function definitions in von Neumann languages. A Definition consists of a name on the left side and an expression consisting only of functional forms involving primitive and other defined functions on the right side. Takes the form Def name expression

20 A System Practical Examples of Functional Programs Def sub1 [id, 1] Def eq0 eq [id, 0] Def fact eq0 1; [id, fact sub1] Collatz Sequence: Def eq1 eq [id, 1] Def div2 [id, 2] Def mult3add1 + [1, [id, 3]] Def collatzstep even div2; mult3add1 Def collatz eq1 < 1 >; apndl [id, collatz collatzstep]

21 A System Practical Algebraic Properties of FP Systems Unlike von Neumann languages, FP systems exhibit useful algebraic properties. Using these properties, we can manipulate programs algebraically the way we manipulate mathematical expressions in high school algebra. We can prove properties of programs using the programming language itself.

22 A System Practical Examples of Algebraic Laws of the FP System [f 1,...f n ] g [f 1 g,...f n g] αf [g 1,...g n ] [f g 1,...f g 2 ] (p f ; g) h p h f h; g h [...,,...] f f id f

23 A System Practical Properties of Haskell Haskell Ruby Pure functional language: no concept of state, functions have no side-effects (except for I/O). Unlike Backus FP system, handles multiple argument functions by currying. User writes program by specifying type signatures and function definitions. Functions are first class objects and can be passed to other functions.

24 A System Practical Example Haskell Program Haskell Ruby c o l l a t z S t e p : : I n t > I n t c o l l a t z S t e p x = i f x mod 2 == 0 then x div 2 else 3 x + 1 c o l l a t z : : I n t > [ I n t ] c o l l a t z x = i f x == 1 then [ 1 ] else x : ( c o l l a t z $ c o l l a t z S t e p x ) main = do x < getline putstrln $ show $ c o l l a t z $ read x

25 Execution Preliminaries A System Practical Haskell Ruby

26 Pointfree Style Preliminaries A System Practical Haskell Ruby Avoid use of variable names in function definitions. Favor function composition and partial application. Very similar to Backus style.

27 Examples Preliminaries A System Practical Haskell Ruby A function, sumx2, which multiplies every element of a list by 2 and sums the resultant list. Normal: sumx2 xs = sum (map ( 2) xs) Pointfree: sumx2 = sum. map ( 2)

28 A System Practical Properties of Ruby Haskell Ruby Dynamic object-oriented scripting lanugage. Allows methods to accept "blocks" as their final argument: basically anonymous functions. Blocks minimize "housekeeping code" that Backus cited.

29 File I/O Example Preliminaries A System Practical Haskell Ruby C Example: FILE f i n = fopen ( " t e s t. t x t ", " r " ) ; while (! f e o f ( f i n ) ) { f s c a n f ( f i n, "%s \ n ", t e x t ) ; p r i n t f ( "%s \ n ", t e x t ) ; } f c l o s e ( f i n ) ; Ruby Example: IO. foreach ( t e s t. t x t ) do l i n e puts l i n e end

30 Preliminaries A System Practical Imperative programming languages are inflexible, and don t exhibit useful algebraic properties. FP systems allow the user to create programs by combining simpler functions, and these programs can be manipulated algebraically. In modern languages we have both purely functional languages and imperative/object-oriented languages which incorporate functional features.

31 Appendix For Further Reading For Further Reading I J. Backus. Can Programming Be Liberated from the von Neumann Style? A Functional Style and Its Algebra of Programs. Communications of the ACM, 21(8):, Haskell Wiki. Pointfree Ruby Documentation Guides and API Documentation