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lambda-calculus代写 programming language代写 function代写

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lambda-calculus代写

CM20256 / CM50262 Functional Programmin

lambda-calculus代写 This coursework is about using the lambda-calculus as a programming language. Your so- lutions should be submitted

Coursework 2 – CM50262 version

Due: 6pm Wednesday May 2nd 2018

 This coursework is about using the lambda-calculus as a programming language. Your so- lutions should be submitted via Moodle as a PDF file. You may write the solutions using appropriate software (LATEX, Word, etc.) or as a clearly legible handwritten document that you scan to PDF (the University printers have a scanning function). Throughout this assign- ment, you could of course write your solutions as pure lambda-terms, but that would be very hard to read. Make use of abbreviations to ensure that your solutions to these exercises are readable!lambda-calculus代写

lambda-calculus代写
lambda-calculus代写

 Part 1 (Ordered triples 10%):lambda-calculus代写

 It is useful in programming to collect multiple terms together as pairs, triples etc. Ordered triples may be encoded in the lambda-calculus as follows:

hM1, M2, M3i , Lp.p M1 MM3

The idea is that the variable p will be bound to a “projection” operator which will select one of M1 , M2 or M3  when the programmer wants to retrieve one of the elements of the  triple.

a)Show that the following term Ø -reduces to M2:

hM1, M2, M3i (Lx.Ly.Lz.y)

You may assume that the terms M1 , M2 and M3 do not contain x , y or z as free variables.lambda-calculus代写

b)Defineterms 1 , 2 and 3 such that for any terms M1 , M2 and M3 ,

i hM1, M2, M3i !Ø Mi

for i = 1, 2, 3 and demonstrate that 2 works as intended.

Part 2 (Working with numbers – 40%):

We will now employ our ordered triples to define some useful functions that operate on the Church numerals.

a)Define a function T that operates on triples of Church numerals suchthat

Tha, b, ci !Ø hb, c, c + 1i

and show that it works. You may make use of the successor function succ without explanation, as well as the i functions defined above.

b)Prove by induction on n that, for all Church numerals n,

lambda-calculus代写
lambda-calculus代写

The.operation is subtraction curtailed at zero, that is:

Part 3 (Fibonacci numbers – 10%):lambda-calculus代写

Below is the definition of a Haskell function that com- putes the Fibonacci numbers.

fib :: Int -> Int fib 0 = 1

fib 1 = 1

fib n = fib (n-1) + fib (n – 2)

Define a lambda-term F which implements this function, making use of the Y combinator to implement recursion and your function T to implement the calculations of  n — 1 and n — 2 . You may make use of the following operations on Church numerals and Booleans

from the lectures: addition, test for zero, if-then-else.lambda-calculus代写

Part 4 (Research question  40%):lambda-calculus代写

 Write a short essay (around 2 pages) on combinatory logic and its relationship with the lambda-calculus and functional programming. You should:

  • give a definition of what the combinators S , K and I are — this can be in terms of lambda-calculus if youlike
  • argue that the combinators are as powerful as the full L-calculus
  • provide some historical background on the discovery of combinatorylogic
  • briefly discuss how combinators are relevant to the implementation of functional pro- gramminglanguages
  • give references to support your claims and fifindings.lambda-calculus代写

This is a research question which deliberately goes beyond what we are covering in lec- tures. Please conduct your own research to find sources and material to help you answer it. Wikipedia is a good place to start, but you should avoid using it as a definitive reference. A really good answer would go beyond the material covered by the obvious Wikipedia pages.

lambda-calculus代写
lambda-calculus代写

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