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CAS810: WEEK 3

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). LAMBDA CALCULUS.

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CAS810: WEEK 3

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  1. 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).

  2. LAMBDA CALCULUS • 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.

  3. LAMBDA CALCULUS -- USE • 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)

  4. LAMBDA CALCULUS -- SYNTAX <lambda-exp> ::= <var> | (<lambda-exp> ) l<var><lambda-exp> | % abstraction <lambda-exp> <lambda-exp> % application NB: • If there are no brackets, expressions are LEFT ASSOCIATIVE - abc = ((a)b)c • We ENRICH syntax with numbers and the function (b->x,y) meaning if b then x else y • Scope rules..

  5. 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

  6. LAMBDA CALCULUS - substitution The calculus “evaluates” using substitution and use of CONVERSION RULES: • 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.

  7. 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

  8. 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))

  9. 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?????

  10. 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

  11. The Fixed Point Equation • Every recursive function f has a corresponding the fixed point equation H f = f where H is constructed as shown in the slide above. So - a non-recursive function which satisfies H f = f can be thought of as the “meaning” f

  12. 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.

  13. 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) } We can also DEFINE functions using graphs, and approximate recursive functions with them. NB g is an FINITE APPROXIMATION of factorial.

  14. 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, ^)…

  15. Example - Conclusion H ^, H H ^, HH 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.

  16. 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?

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