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CHA2545 Last Lecture

CHA2545 Last Lecture. LECTURE: LAMBDA CALCULUS SEMANTICS TUTORIAL: Revision. 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.

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CHA2545 Last Lecture

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  1. CHA2545 Last Lecture • LECTURE: • LAMBDA CALCULUS SEMANTICS • TUTORIAL: • Revision

  2. 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.

  3. 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.g X ={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!!

  4. 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.

  5. 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)

  6. 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.

  7. 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

  8. 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, )…

  9. 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

  10. 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.

  11. 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

  12. CALCULUS: operational semantics • Problem: • the ORDER of application sometimes makes a difference! • E.g. Try • (x. y.y) ( (v.vv)(z.zz) )

  13. 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 normal-order reduction • Corollary: We now have a nice operational semantics for calculus and hence pure functional programming

  14. 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.

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