How to define arithmetic operations using church numerals

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Reference no: EM132325337

Assignment

1. Different notions of normal form have been discussed in the literature, including the full normal form (or simply normal form) and weak head normal form.

a. What is the difference between a term having a normal form and being a normal form? Write down some example terms.

A term has a normal form when there are still redexes possbile through reduction. A term such as λabc.(( λx.a(λy.xy))b c ) for example can still be normalized.

When a term is in normal form, that means thare are no redexes left for it. Going with the example I mentioned above, λabc.(a(λy.by)c) for example, would be in normal form since there are no further redexes left for the term.

b. If a closed term is a weak head normal form, it has to be an abstraction "λx.M". Why?

c. Indicate whether the following λ-terms have a normal form:

-- (λx.(λy.yx)z)v

-- (λx.xxy)(λx.xxy)

d. Show that the term Ω = (λx.xx)(λx.xx) does not have a normal form. Find a term different from Ω that is not normalizing (i.e., a term such that every reduction sequence starting from it is infinite).

2. It has been shown how to define arithmetic operations using Church numerals.

a. Check that the term ADD = λxyab.(xa)(yab) behaves like the addition function; that is, show that when we apply ADD to two Church numerals, the Church numeral representing their sum is obtained. Hint: Reduce the term (λx.λy.λa.λb.(xa)(yab)nm.

b. Show that the λ-term MULT = λx.λy.λz.x(yz) applied to two Church numerals m and n computes their product m x n.

c. Which arithmetic operation does the term λn.λm.m (MULT n)1 compute?

3. Compute the normal forms of the following terms (Where I = λx.x and K = λxy.x):

a. λy.(λx.x)y

b. λy.y(λx.x)

c. II

d. KI

e. KKK

Reference no: EM132325337

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