Transcript Programs without Bugs in the 21st Century? (or

CAS810: WEEK 3
LECTURE:
LAMBDA CALCULUS
PRACTICAL/TUTORIAL:
(i) Do exercises given out
(ii) Go through ARJ’s supporting WEB notes (from main
web page).
School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS
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A Formal System is a formal language together
with a well-defined way of “reasoning” with
expressions in the language. EG predicate
calculus
Lambda Calculus is a formal system designed by
Alonzo Church in 1940, developed by Scott and
Strachey c.1970.
School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS -- USE
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it is a notation for describing the behaviour of
functions - it can be used to represent ALL
COMPUTABLE FUNCTIONS.
It is used (as other formal systems) to theorise
about computation. These theories direct and
support work in compiler design, language design,
interface design etc ..
LAMBDA CALCULUS expressions are the
denotations in our semantic definitions (next week)
School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS -- SYNTAX
<lambda-exp> ::=
<var> | (<lambda-exp> )
l<var><lambda-exp> |
<lambda-exp> <lambda-exp>
% abstraction
% application
NB:
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If there are no brackets, expressions are LEFT ASSOCIATIVE - abc =
((a)b)c
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We ENRICH syntax with numbers and the function (b->x,y) meaning if b
then x else y
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Scope rules..
School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS -- WHY?
It has simple, clear structure / syntax providing a
simple model of function definition and application
 used extensively as a MEANING domain for
programming languages
 allows us to treat functions as values, and to
define functions without giving them a name! EG:
lx.x
- lambda abstraction - identity function
lx.x+1 - lambda abstraction - successor function
(lx.x+1)y - application
(lx.x+1)(ly.y+1) - application
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School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS - substitution
The calculus “evaluates” using substitution and use
of CONVERSION RULES:
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An occurrence of variable x is BOUND in a lambda
expression M if lx.N is a sub-expression of M and
x occurs within N. Otherwise, the occurrence of x is
FREE.
[M/x]N means substitute expression M for all free
occurrences of x in N.
School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS - Conversion Rules
(i)If y does not occur free in N then
lx.N cnva ly[y/x]N
(ii) Reduction (lx.M)N cnvb [N/x]M
(iii) Reduction - if x is NOT FREE in M
(lx.M)N cnvc M
School of Computing and Mathematics, University of Huddersfield
LAMBDA CALCULUS -- EXAMPLES
lx.x
- lambda abstraction - identity function
lx.x+1 - lambda abstraction - successor function
(lx.x+1)y - application
(lx.xx)(ly.y) - application
recursive functions in enriched syntax..
f = ln.(n=0 -> 1, n*f(n-1))
School of Computing and Mathematics, University of Huddersfield
Introduction to the Semantics of
l-calculus
Well, you can evaluate them to normal form using the
conversion rules…
But recursive functions - what do they MEAN?????
School of Computing and Mathematics, University of Huddersfield
Recursive Functions - The Fixed Point
Equation
E.g. Factorial Function
f = ln.(n=0 -> 1, n*f(n-1))
Consider:
H = lg.ln.(n=0 -> 1, n*g(n-1))
H f = (lg.ln.(n=0 -> 1, n*g(n-1))) f
= ln.(n=0 -> 1, n*f(n-1))
= f by the definition above.
Hf = f ... is the fixed point equation
School of Computing and Mathematics, University of Huddersfield
The Fixed Point Equation
Every recursive function f has a corresponding
the fixed point equation
Hf=f
where H is constructed as shown in the slide
above.
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So - a non-recursive function which satisfies H f = f
can be thought of as the “meaning” f
School of Computing and Mathematics, University of Huddersfield
Problem...
But consider
f = lx. ly.( x=y -> y+1, f x (f (x-1) (y-1)) )
In this case H has many fixed points !!
E.g.
lu. lv. u+1 is a fixed point of H. There
are many others.
School of Computing and Mathematics, University of Huddersfield
Solving the Fixed Point Equation “Graphs” of Functions
The Graph of a function f is a list of pairs (n, f n)
where n is an input value to f.
E.G. Part of the graph for factorial is
g = {(0,1), (1,1), (2,2), (3,6), (4,24), (5,120), (6,720) }
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We can also DEFINE functions using graphs, and
approximate recursive functions with them.
NB g is an FINITE APPROXIMATION of factorial.
School of Computing and Mathematics, University of Huddersfield
A static solution to the FPE
The static solution of the FP equation turns out to
be (lots of maths later…):
f = Hn(^)
as n tends to infinity.
eg For the factorial:
H = lg.ln.(n=0 -> 1, n*g(n-1))
H ^ = ln.(n=0 -> 1, ^)
Graph(H ^) = (0,1), (1, ^), (2, ^), (3, ^), (4, ^)…
H H ^ = lg.ln.(n=0 -> 1, n*g(n-1)) (ln.(n=0 -> 1, ^))
Graph(H H ^) = (0,1), (1, 1), (2, ^), (3, ^), (4, ^)…
School of Computing and Mathematics, University of Huddersfield
Example - Conclusion
H ^, H H ^, H H H ^, are improving approximations of
any recursive function f, where H appears in f’s fixed
point equation.
SO, the meaning of recursive functions can be given
via a “fixed point semantics”. The MYSTERY of
recursive functions having infinite fixed points is
cleared up -
Hn(^) = the least defined fixed point.
School of Computing and Mathematics, University of Huddersfield
Exercises
(1) Consider the following Lambda applications:
1.1 (lx.xx)(ly.y)
1.2 (ly.lz.z -> x,y)(lx.ly.xy)
1.3 (lg.ln.(n=0 -> 1, n*g(n-1)) (ly.y+1)) 3
- For each instance of each variable in 1.1-1.3, say whether the
instance of the variable occurs free or bound.
- Use the conversion rules to find the normal forms for 1.1-1.3.
(2) If f is the factorial function, and H f = f, derive H3(^) and H4(^).
Hence derive graph(H3(^) ), graph(H4(^) ) and verify that they are improving
approximations to f.
(3) If H is the general function in the Cmeans of the while loop, derive H1(^)
and H2(^) and their graphs. Are they approximations of the while-loop?
School of Computing and Mathematics, University of Huddersfield