CHA2545 Last Lecture
LECTURE:
LAMBDA CALCULUS SEMANTICS
TUTORIAL:
Revision
School of Computing
and Mathematics,
University of
LAMBDA CALCULUS


it is a notation for describing the behaviour of ALL
COMPUTABLE FUNCTIONS.
LAMBDA CALCULUS functions are the
denotations in our semantic definition. We will
study briefly its semantics.
School of Computing
and Mathematics,
University of
The Problem with recursive
definitions - Russell’s Paradox for
sets:
We can define SETS of things which are sets
e.g. X = { {1,2}, {2,4} } is a set of sets.
 We can define sets implicitly via properties

e.gX ={x : x is a set containing 3 elements}
 We can define sets of things that contain themselves:
e.g. X = {x : x is a set of sets}
X contains itself…!
 Define Z = { x : x does not contain itself}
does Z contain itself!!
School of Computing
and Mathematics,
University of
Russell’s Paradox is related to recursive
Lambda calculus functions:
At the heart of the paradox is self-reference
- recursive functions are self-referential
- basic lambda calculus was shown to be inconsistent by a
similar argument, which led to “typed” Lambda calculus ie
without “type theory” to limit expression we can introduce
paradoxes into recursive definitions.
School of Computing
and Mathematics,
University of
Fixed point Semantics
A recursive function can be recast as the solution to fixed
point equation H f = f
 One way of finding fixed points is to use a “Fixed Point
Combinator”. This is a function which computes fixed
points of other functions.
Most commonly used is Y = lf ( lx f (x x) ) ( lx f (x x) )
Then for all F, F (YF) = YF. PROOF:
YF = lf ( lx f (x x) ) ( lx f (x x) ) F
= (lx F (x x) ) ( lx F (x x) )
= F ( (lx F (x x))(lx F (x x)) )
= F ( lg (lx g (x x))(lx g (x x))F )
= F (YF)

School of Computing
and Mathematics,
University of
LAMBDA CALCULUS - Fixed Point Semantics
But consider the following function:
f = x. y.( x=y => y+1, f x (f (x-1) (y-1)) )
In this case H has many fixed points !!
E.g.
u. v. u+1
u. v. (u=v => u+1, ) are fixed points of H.
School of Computing
and Mathematics,
University of
Fixed point Semantics via
Approximations

An approximation f’ of a function f is defined as
If f’ s is defined then f’ s = f s
Eg f’ 0 = 0,
f’ 1 = 1,
f’ n = undefined for n>1
Is an approximation of
f n = n*n
School of Computing
and Mathematics,
University of
Fixed point theorem
A static solution of the fixed point equation can be given by
approximations:
f = Hn(, as n tends to infinity
H H H H H H are improving approximations of f
Example: For the factorial function:
H = g.n.(n=0 => 1, n*g(n-1))
H n.(n=0 => 1, 
Graph(H  = (0,1), (1, ), (2, ), (3, ), (4, )…
H H g.n.(n=0 => 1, n*g(n-1)) (n.(n=0 => 1, 
Graph(H
H (0,1), (1, 1), (2, ), (3, ), (4, )…
School of Computing
and Mathematics,
University of
LAMBDA CALCULUS - Fixed Point Semantics
Assume H (f ) = f.
Then Hn(as n tends to infinity, gives us the
“LEAST DEFINED” fixed point f of H, which is
defined as the “non-recursive” meaning of
function f
School of Computing
and Mathematics,
University of
CALCULUS: operational semantics
An operational semantics gives us an abstract
but precise way to execute functions
(programs).
A expression is in NORMAL FORM if it is a
abstraction - i.e. It cannot be reduced.
Operational Semantics: Repeatedly apply the
conversion rules to an “application” until it is
in normal form.
School of Computing
and Mathematics,
University of
CALCULUS: order of application
Two main ones:
 Left-most innermost (call by value)
..basically reduce arguments of a function before
reducing the function
Left-most outermost (call by name)
(also called normal order reduction)
..basically reduce the outer-most function
without reducing its arguments

School of Computing
and Mathematics,
University of
CALCULUS: operational semantics
Problem:
the ORDER of application sometimes makes a
difference!
E.g. Try
(x. y.y) ( (v.vv)(z.zz) )
School of Computing
and Mathematics,
University of
CALCULUS: Church-Rosser Theorem
(Paraphrase)
If a Calculus Application can be reduced to a
normal form then
-- that normal form is UNIQUE up to naming
-- the normal form can be reached using normalorder reduction
Corollary: We now have a nice operational
semantics for calculus and hence pure
functional programming
School of Computing
and Mathematics,
University of
LAMBDA CALCULUS - Fixed Point Semantics
= Operational Semantics
BIG THEOREM: The least defined fixed point of
f IS OPERATIONALLY THE SAME AS f
NB All above is paraphrased in that I have
extracted all the maths/domain theory out to
give you the gist.
School of Computing
and Mathematics,
University of