CS 541 Lecture -*- Outline -*-

* Untyped lambda calculus
	advert: close match to Haskell, Scheme, LISP

** The language

*** concrete syntax
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  UNTYPED LAMBDA CALCULUS CONCRETE SYNTAX

  e in Expr
  x in Identifier

e ::= x | \x . e | e e | (e)


  ABSTRACT SYNTAX

  e in Expr
  x in Identifier

e ::= x | (\x . e) | (e e)
---------------------------
*** parsing rules
	above grammar is ambiguous.
	This is the way it is disambiguated
-------------
   PARSING RULES

\x. e e'  ==   (\x. (e e'))
x y z       ==   ((x y) z)
-------------

*** Microsyntax
  (microsyntax not standard)
  x ::= non-blank non-blank*
  non-blank ::= any character except blank, ".", ":", "(", and ")"
	same parsing conventions as typed language

** substitution
	examples: (\x . (+ x x)) 2  ==> (+ 2 2)
		  (\+ . (+ 2 2)) * ==> (* 2 2)
		  (\f. \y . f y) (\y.y) ==> ?
				-no problem-
		  (\x . \y . x y) y ==> ?
				-be careful-
*** occurrences
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     FREE AND BOUND VARIABLES

def: x occurs free in
(i) x,
(ii) (U V) if x occurs free in U or V,
(iii) (\y.U) if x occurs free in U and x =/= y.
-------------

*** free variables, Fv
----------
FV: Expr -> (Set of Identifier)

FV(x) = {x}
FV(\x.e) = FV(e) - {x}
FV(e e') = FV(e) union FV(e')
----------

*** bound variables
-------------
def: x is occurs bound in
(i) \x.U
(ii) (U V) if x occurs bound in U or in V,
(iii) (\y.U) if x occurs bound in U
      or if x == y
-------------
	note: x can be free and bound in the same term:
		(x (\x.x))

***	syntactic substitution
------------
    SYNTACTIC SUBSTITUTION
   FOR FREE OCCURRENCES OF A VARIABLE

[e/x]x         == e
[e/x]y         == y
		    if not(y == x)
[e/x](e1 e2)   == ([e/x]e1)([e/x]e2)
[e/x](\y.e') == \y . [e/x]e'
      where not(y == x), not(y in Fv(e))
------------
	last restriction necessary to preserve lexical scoping
		e.g. [y/x](\y . (x y))  is not (\y . (y y))
			rather (\z . (y z))

**  Church's thesis
	self-application is allowed, for example
		(x x) is a term
	this permits all computations, including the Y combinator
------------------
      CHURCH'S THESIS

All computable functions are definable
 in the (untyped) lambda calculus.
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