Can every lambda expression be reduced to a normal form?
Can every lambda expression be reduced to a normal form?
The Church-Rosser Theorems A corollary of this theorem is that no lambda expression can be reduced to two distinct normal forms. This corollary says that all reduction sequences that terminate will always yield the same result and that result must be a normal form.
What is normal order reduction?
Normal-order reduction is the strategy in which one continually applies the rule for beta reduction in head position until no more such reductions are possible. At that point, the resulting term is in head normal form. One then continues applying head reduction in the subterms. , from left to right.
How do you reduce a lambda expression?
You can follow the following steps to reduce lambda expressions:
- Fully parenthesize the expression to avoid mistakes and make it more obvious where function application takes place.
- Find a function application, i.e. find an occurrence of the pattern (λX.
- Apply the function by replacing (λx.
What is applicative order reduction?
Applicative order means that for applications (F E) (F applied to E) E must be first reduced before it is substituted in the body of F. In our example that means that the inner (twice (square)) must be first be reduced before it is substituted in the body of the outer twice.
What is normal form in lambda calculus?
Normal Form: A lambda expression that cannot be reduced further (by beta-reduction) is called a normal form. If a lambda expression E can be reduced to a normal form, we then say that E has a normal form. In general, a lambda expression may not have a normal form.
What is the main reduction rule of the semantic of the lambda calculus?
Semantics of Lambda Expressions Evaluating a lambda expression is called reduction . The basic reduction rule involves substituting expressions for free variables in a manner similar to the way that the parameters in a function definition are passed as arguments in a function call.
What is normal order evaluation?
A non-strict evaluation order is an evaluation order that is not strict, that is, a function may return a result before all of its arguments are fully evaluated. Normal order evaluation has the property that it terminates without error whenever any other evaluation order terminates without error.
How do you evaluate lambda in calculus?
Evaluating a Lambda Expression A lambda calculus expression can be thought of as a program which can be executed by evaluating it. Evaluation is done by repeatedly finding a reducible expression (called a redex) and reducing it by a function evaluation until there are no more redexes.
What is difference between normal order and applicative order evaluation?
While normal-order evaluation may result in doing extra work by requiring function arguments to be evaluated more than once, applicative-order evaluation may result in programs that do not terminate where their normal-order equivalents do.
What is the difference between normal order and applicative order?
In a purely applicative-order language, all procedures are strict in each argument. In a purely normal-order language, all compound procedures are non-strict in each argument, and primitive procedures may be either strict or non-strict.
What is beta reduction in lambda calculus?
2) Beta Reduction – Basically just substitution. This is the process of calling the lambda expression with input, and getting the output. A lambda expression is like a function, you call the function by substituting the input throughout the expression. Take (λx.
What is normal order reduction in calculus?
Normal order reduction starts by reducing the top-level redex (because the function $K_y$ is already in normal form). Since $K_y$ ignores its argument, $(K_y \\Omega) o y$ in one step. More generally, $K_y N o y$ for any term $N$, and this is the reduction chosen by the normal order strategy.
What is normal order evaluation of lambda expressions?
Normal-order evaluation of a lambda expression is the repeated application of the leftmost reducible function application. In other words, normal order evaluation is the strategy that always reduces the left-most outer-most redex first.
How do you reduce the Order of a function?
More generally, for any term $M$, $M \\Omega o M \\Omega$, and this is the reduction chosen by the applicative order strategy. Normal order reduction starts by reducing the top-level redex (because the function $K_y$ is already in normal form).
How many times can applicative order reduction loop a function?
(Make sure to work it out on paper at least once in your life.) Applicative order reduction must reduce the argument of the function to a normal form, before it can evaluate the top-level redex. Since the argument $\\Omega$ has no normal form, applicative order reduction loops infinitely.
