159'331 Programming Languages

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159.331 Programming Languages Algorithms

• Lecture 21 - Functional Programming Languages -
  Part 5
• Lambda Calculus Some Theory

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Some Theory

• What was our theory for Imperative and OO
  programs?
• Not much but is in fact grounded out in models
  of machines - memory and automata
• For Functional languages, there are some very
  useful theoretical ideas and underpinning
  concepts
• Look at semantics and Lambda calculus - albeit
  somewhat superficially in this paper.

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What Are Semantics again?

• Any property of a construct can be defined as its
  semantics or meaning
• In an expression checker the semantics of an
  expression like 2 3 can be its value
• Ina type checker the semantics of 2 3 can
  be the type int for integer
• In an infix to postfix translator the semantics
  of 2 3 can be the string 2 3
• There are several methods of defining semantics

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An Example - A Let Expression

• E let x E1 in E2
• We can use this in various context with
  different semantics
• First consideration is for its syntax as a
  production from a grammar for expressions
  (remember EBNF)

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Informal Semantics

• E let x E1 in E2
• Occurrences of x in E2 denote the value E1
• The value of E2 is the value of the whole
  expression E
• Example E2 is 1 x
• E1 might be 2, hence E2 is 1 2 and E is 3
  under some assumptions about arithmetic of
  integers

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Attribute Grammar

• Attribute val of E denotes a value. Attribute
  env for environment binds variables to values.
  The operation bind(x,v,env) creates a new
  environment with x bound to v
• bindings for all other variables are as in env
• E.val E2.val value of E is the value of E2
• E1.env E.env variable bindings in E1 are the
  same as in E
• E2.env bind( x, E1.val, env ) In E2, x is
  bound to the value of E1

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Operational Semantics

• E let x E1 in E2
• The interpreter eval takes two parameters
• An expression to be evaluates
• And an expression with variable bindings
• eval( E, env ) eval( E2, bind( x, eval(E1,env),
  env ) )

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Denotational Semantics

• The meaning of the expression E, written as E
  is a function from environments to values. Thus
  E env, the application of E to
  environment env is a value
• let x E1 in E2 env
• E2 bind(x, E1 env, env )

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Natural Semantics

• Read the logical formula env - E v as
• In environment env, expression E has value v
• The rule for let-expressions is
• env - E1 v1 bind( x, v1, env ) - E2 v2
• env - let x E1 in E2 v2

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An Aside on Logic Formulae

• Suppose we want formal way of defining the
  value of plus E1 E2 as the sum of the values
  E1 and E2
• We can write this as a logic formula
• E1 v1 E2 v2
• plus E1 E2 v1 v2
• In other words if E1 has the value v1 and
  E2 has the value v2 then plus E1 E2 has the
  value v1 v2

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The moral of this story

• There are many ways of saying the same thing
• Different communities have different syntax
  which help in different ways
• When I use a word, it means just what I choose
  it to mean--neither more nor less. - Humpty
  Dumpty, in Through the Looking Glass by Lewis
  Carroll
• Do not be afraid of all this
• We will return to denotational semantics when we
  discuss Logic later

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Lambda Calculus

• Alonzo Churchs notation for studying types
• Supposedly inspired by Whitehead and Russell
  notation for the class of all xs such that f(x)
  to wit xf(x) - moved the carat down
  and it became a lambda ?
• ?x.M is used for a function with parameter x and
  body M
• So ?x.xx is a function that maps 5 to 5 5
• Functions are written next to their arguments so
  f a is the application of function f to argument
  a
• (?x.xx) 5 applies our function to 5 yielding
  25

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• We call formulas like (?x.xx) 5 terms
• Chirchs original formulation of ?-calculus was
  for general properties of functions not tied to
  any particular problem area. The integer 5 and
  the multiplication operator below to the problem
  area arithmetic and are not part of the pure
  calculus
• A grammar for terms in the pure ?-calculus is
• M x (M1 M2 ) (?x.M)
• We use letters f, x, y, z for variables and M,
  N,P, Q for terms
• A term is a variable x, or an application (M N)
  of function M to N or an abstraction (?x.M)

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• We are allowed constants such as c which can
  represent values like integers and operations on
  data structures like lists
• c can stand for basic constants like true and
  nil or constant functions like head and
• Pure ?-calculus is untyped - Functrions can be
  applied freely and it even makes sense to write
  (x,x) where x is applied to itself
• A functional programming language is essentially
  a ?-calculus with appropriate constants

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Going Further

• ?-calculus is a fascinating topic in its own
  right - we could base a whole course around it.
• Generally if you learn how it works and the
  various results that arise from setting up
  certain constants and semantics for them we
  can see how to generate a functional language
• We ascribe meaning to meaning -)

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Summary

• Various Semantics - various ways to set up a
  semantics
• Introduction to ?-Calculus - terms, grammar -
  variables applications and abstractions
• Some of the older books on programming language
  s (eg in Massey Library) give a treatment
  based on ?-calculus up front. - eg Ravi Sethi
  - Programming Languages- Concepts and Cosntructs

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Further Reading

• See Bal Grune Chapter 4
• See Sebesta Chapter 15
• See also Hudaks ACM article on Conception,
  Evolution and Application of Functional
  Programming Languages
• Next - Logic Programming Languages