# Functional Programming. Functional Programming Languages. Chapter 14. Introduction

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1 Functional Programming Languages Chapter 14 Introduction Functional programming paradigm History Features and concepts Examples: Lisp ML 1 2 Functional Programming Functional Programming Languages The Functional Programming Paradigm is one of the major programming paradigms. FP is a type of declarative programming paradigm Also known as applicative programming and value-oriented programming Idea: everything is a function Based on sound theoretical frameworks (e.g., the lambda calculus) Examples of FP languages First (and most popular) FP language: Lisp Other important FPs: ML, Haskell, Miranda, Scheme, Logo 3 The design of the imperative languages is based directly on the von Neumann architecture Efficiency is the primary concern, rather than the suitability of the language for software development The design of the functional languages is based on mathematical functions A solid theoretical basis that is also closer to the user, but relatively unconcerned with the architecture of the machines on which programs will run 4

2 Mathematical Functions Characteristics of Pure FPLs Def: A mathematical function is a mapping of members of one set, called the domain set, to another set, called the range set. The lambda calculus is a formal mathematical system devised by Alonzo Church to investigate functions, function application and recursion. A lambda expression specifies the parameter(s) and the mapping of a function in the following form??x. x * x * x for the function cube (x) = x * x * x Lambda expressions describe nameless functions Lambda expressions are applied to parameter(s) by placing the parameter(s) after the expression (??x. x * x * x) 3 => 3*3*3 => 27 (??x,y. (x-y)*(y-x)) (3,5) => (3-5)*(5-3) => -4 Pure FP languages tend to Have no side-effects Have no assignment statements Often have no variables! Be built on a small, concise framework Have a simple, uniform syntax Be implemented via interpreters rather than compilers Be mathematically easier to handle 5 6 Importance of FP In their pure form FPLs dispense with notion of assignment claim is: it's easier to program in them also: easier to reason about programs written in them FPLs encourage thinking at higher levels of abstraction support modifying and combining existing programs thus, FPL's encourage programmers to work in units larger than statements of conventional languages: "programming in the large" FPLs provide a paradigm for parallel computing absence of assignment (or single assignment) } provide basis independence of evaluation order } for parallel ability to operate on entire data structures } functional programming Importance of FP FPLs are valuable in developing executable specifications and prototype implementations Simple underlying semantics» rigorous mathematical foundations» ability to operate on entire data structures ideal vehicle for capturing specifications FPLs are very useful for AI and other applications which require extensive symbol manipulation. Functional Programming is tied to CS theory provides framework for viewing decidability questions» (both programming and computers) Good introduction to Denotational Semantics» functional in form 7 8

3 Expressions Key purpose of functional programming: to extend the advantages of expressions (over statements) to an entire programming language Backus* has said that expressions and statements come from two different worlds. expressions: (a + b) * c arithmetic (a + b) = 0 relational (a? b) boolean statements: the usual assortment with assignment singled out assignments alter the state of a computation (ordering is important) e.g. a:= a * i; i:= i + 1 In contrast, ordering of expressions is not side-effecting and therefore not order dependent (Church-Rosser property /Church Diamond) Church-Rosser Theorem We can formally model the process of evaluating an expression as the application of one or more reduction rules. E.g., lambda-calculus includes the beta-reduction rule to evaluate the application of a lambda abstraction to an argument expression. A copy of the body of the lambda abstraction is made and occurrences of the bound variable replaced by the argument. E.g. (? x. x+1) 4 => 4+1 The CR theorem states that if an expression can be reduced by zero or more reduction steps to either expression M or expression N then there exists some other expression to which both M and N can be reduced. This implies that there is a unique normal form for any expression since M and N cannot be different normal forms because the theorem says they can be reduced to some other expression and normal forms are irreducible by definition. It does not imply that a normal form is reachable, only that if reduction terminates it will reach a unique normal form. * "Can programming be liberated from the von Neumann style?", John Backus, Communications of the ACM, 21, 8, pp , 1978.) 9 10 More Expressions With Church-Rosser reasoning about expressions is easier order independence supports fine-grained parallelism Diamond property is quite useful Referential transparency In a fixed context, the replacement of a subexpression by its value is completely independent of the surrounding expression» having once evaluated an expression in a given context, shouldn t have to do it again. Alternative: referential transparency is the universal ability to substitute equals for equals (useful in common subexpression optimizations and mathematical reasoning) 11 FPLs address C.A.R. Hoare's Principles of Structuring 1) Transparency of meaning Meaning of whole expression can be understood in terms of meanings of its subexpressions. 2) Transparency of Purpose Purpose of each part consists solely of its contribution to the purpose of the whole. => No side effects. 3) Independence of Parts Meaning of independent parts can be understood completely independently. In E + F, E can be understood independently of F. 4) Recursive Application Both construction and analysis of structure (e.g. expressions) can be accomplished through recursive application of uniform rules. 5) Narrow Interfaces Interface between parts is clear, narrow (minimal number of inputs and outputs) and well controlled. 6) Manifestness of Structure Structural relationships among parts are obvious. e.g. one expression is subexpression of another if the first is textually embedded in the second. Expressions are unrelated if they are not structurally related. Hoare, Charles Antony Richard. Hints on programming language design., In SIGACT/SIGPLAN Symposium on principles of programming languages, October

4 Properties of Pure Expressions What are some FPLs? Value is independent of evaluation order Expressions can be evaluated in parallel Referential transparency No side-effects (Church Rosser) Inputs to an expression are obvious from written form Effects of operation are obvious from written form LISP the first FPL, ~1958 Pure FPLs have no side effects Haskell and Miranda are the two most popular examples Some FPLs try to be more practical and do allow some side effects Lisp and it s dialects (e.g. Scheme) ML (Meta Language) and SML (Standard ML) => Meet Hoare's principles well => Good attributes to extend to all programming (?) Lisp Scheme Defined by John McCarthy* ~1958 as a language for AI. Originally, LISP was a typeless language with only two data types: atom and list LISP s lists are stored internally as single-linked lists Lambda notation was used to specify functions. Function definitions, function applications, and data all have the same form If the list (A B C) is interpreted as data it is a simple list of three atoms, A, B, and C but if interpreted as a function application, it means that the function named A is applied to the two parameters, B and C. Example (early Lisp): (defun fact (n) (cond ((lessp n 2) 1)(T (times n (fact (sub1 n)))))) Common Lisp is the ANSI standard Lisp specification In the mid 70 s Sussman and Steele (MIT) defined Scheme as a new LISP-like Language Goal was to move Lisp back toward it s simpler roots and incorporate ideas which had been developed in the PL community since Uses only static scoping More uniform in treating functions as first-class objects which can be the values of expressions and elements of lists, assigned to variables and passed as parameters. Includes the ability to create and manipulate closures and continuations.» A closure is a data structure that holds an expression and an environment of variable bindings in which it s to be evaluated. Closures are used to represent unevaluated expressions when implementing FPLs with lazy evaluation.» A continuation is a data structure which represents the rest of a computation Example: (define (fact n) (if (< n 2) 1 (* n (fact (- n 1))))) Scheme has mostly been used as a language for teaching Computer programming concepts where as Common Lisp is widely used as a practical language. * Recursive Functions of Symbolic Expressions and their Computation by Machine (Part I), John McCarthy, CACM, April

5 ML Haskell ML (Meta Language) is a strict, static-scoped functional language with a Pascal-like syntax that was defined by Robin Milner et. al. in It was the first language to include statically checked polymorphic typing. Uses type declarations, but also does type inferencing to determine the types of undeclared variables Strongly typed (whereas Scheme is essentially typeless) and has no type coercions Includes exception handling and a module facility for implementing abstract data types, garbage collection and a formal semantics. Most common dialect is Standard ML (SML) Example: fun cube (x : int) = x * x * x; Similar to ML (syntax, static scoped, strongly typed, type inferencing) Different from ML and most other FPLs in that it is purely functional -- no variables, no assignment statements, and no side effects of any kind Some key features: - Uses lazy evaluation (evaluate no subexpression until the value is needed) - Has list comprehensions, which allow it to deal with infinite lists Example: fib 0 = 1 fib 1 = 1 fib (n + 2) = fib (n + 1) + fib n Some FP concepts Curried Functions A number of interesting programming language concepts have arisen, including: Curried functions Type inferencing Polymorphism Higher-order functions Functional abstraction Lazy evaluation The logician Frege noted in 1883 that we only need functions of one argument. We can replace a function f(x,y) by a new function f (x) that when called produces a function of another argument to compute f(x,y). That is: (f'(x))(y) = f (x,y) Haskell Curry developed combinatorial logic which used this idea. We call f a curried form of the function f. Two operations: To curry : ((a,b) -> c) -> (a -> b -> c) To uncurry : (a -> b -> c) -> ((a,b) -> c) 19 20

6 Type Inferencing Polymorphism Def: ability of the language to infer types without having programmer provide type signatures. SML ex: fun min(a:real,b) = if a > b then b else a type of a has to be given, but then that s sufficient to figure out the type of b and the type of min What if type of a is not specified? Could be ints or bools or Haskell (as with ML) guarantees type safety (if it compiles, then it s type safe) Haskell ex: eq = (a = b) a polymorphic function that has a return type of bool, assumes only that its two arguments are of the same type and can have the equality operator applied to them. Overuse of type inferencing in both languages is discouraged declarations are a design aid declarations are a documentation aid declarations are a debugging aid ML: fun factorial (0) = 1 = factorial (n) = n * factorial (n - 1); ML infers factorial is an integer function: int -> int Haskell: factorial (0) = 1 factorial (n) = n * factorial (n - 1) Haskell infers factorial is a (numerical) function: Num a => a -> a Polymorphism Higher Order Functions ML: fun mymax(x,y) = if x > y then x else y; SML infers mymax is an integer function: int -> int fun mymax(x: real,y) = if x > y then x else y; SML infers mymax is real Haskell: mymax(x,y) = if x > y then x else y; Haskell infers factorial is an Ord function Definitions: zero-order functions: data in the traditional sense. first-order functions: functions that operate on zero-order functions. second-order functions: operate on first order In general, higher-order functions (HOFs) are those that can operate on functions of any order as long as types match. HOFs are usually polymorphic Higher-order functions can take other functions as arguments and produce functions as values. Applicative programming has often been considered the application of first-order functions. Functional programming has been considered to include higher-order functions: functionals

7 Functional Abstraction Functional programming allows functional abstraction that is not supported in imperative languages, namely the definition and use of functions that take functions as arguments and return functions as values. supports higher level reasoning simplifies correctness proofs Simple examples: Map and filter in Scheme (map square ( )) => ( ) (filter prime (between 1 15)) => ( ) (define (map f l) ; Map applies a function f to elements of list l returning list of the results. (if (null? l) nil (cons (f (first l)) (map f (rest l))))) (define filter (f l) ; (filter f l)returns a list of those elements of l for which f is true (if (null? L ) nil (if (f (first l)) (cons (first l) (filter f (cdr l))) (filter f (cdr l))))))) Lazy evaluation An evaluation strategy in which arguments to a function are evaluated only when needed for the computation. Supported by many FPLs including Scheme, Haskell and Common Lisp. Very useful for dealing with very large or infinite streams of data. Usually implemented using closures data structures containing all the information required to evaluate the expression. Its opposite, eager evaluation, is the usual default in a programming language in which arguments to a function are always evaluated before the function is applied Applications of Functional Languages Summary: ILs vs FPLs Lisp is used for artificial intelligence applications Knowledge representation Machine learning Natural language processing Modeling of speech and vision Embedded Lisp interpreters add programmability to some systems, such as Emacs Scheme is used to teach introductory programming at many universities FPLs are often used where rapid prototyping is desired. Pure FPLs like Haskell are useful in contexts requiring some degree of program verification Imperative Languages: Efficient execution Complex semantics Complex syntax Concurrency is programmer designed Functional Languages: Simple semantics Simple syntax Inefficient execution Programs can automatically be made concurrent 27 28

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