# Given The Following Lambda Expressions And Corresponding Interpretations The Int

Given the following lambda expressions and corresponding interpretations:

- The interpretation of λf.λx.x is natural number 0 (zero). The interpretation of λf λx.(f (f. (… x ))), with n applications of f on x, is the natural number n >0.
- The interpretation of λn.λf.x(f ((n f) x )) is a successor function succ for natural numbers, where n is the formal parameter corresponding to the number whose successor is computed.
- The interpretation of λm.λn.((m succ) n) is the addition function add for two natural numbers, where m and n are the formal parameters corresponding to the numbers whose sum is computed.
- The interpretation of λm.λn.((m add) n)) zero) is the multiplication function mul for two natural numbers, where m and n are the formal parameters corresponding to the numbers whose product is computed.
- The interpretation of λx.λy.x is propositional constant true.
- The interpretation of λx.λy.y is propositional constant false.

Identify the mathematical/logical interpretation for the following expressions. Justify your answer. (In

all these problems, apply the functions on some actual arguments and examine the results; does the

result correspond to some already known interpretation, e.g., true, false, zero, some natural numbers;

does their exist some mapping between interpretation of the result and the corresponding input.)

(a) λx.((x false) true), where x is the formal parameter corresponding to propositional constants.

(b) λn.((n λp.((p false) true)) true), where n is the formal parameter corresponding to natural numbers.

(c) λn.((n λx.λy.λz.z) true), where n is the formal parameter corresponding to natural numbers.

(d) λm.λn.((m (mul n)) (succ zero)), where m and n are formal parameters corresponding to natural numbers.

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