Identify the redexes in the following expressions, and...

Identify the redexes in the following expressions, and determine whether each redex is innermost, outermost, neither, or both:
$$1+(2 * 3)$$$$
(1+2) *(2+3)$$
fst $(1+2,2+3)$$$
(\backslash x->1+x)(2 * 3)$$

Key Concepts

Redex

A redex (short for reducible expression) is a part of an expression that matches a pattern for which a reduction rule applies, such as applying a function to an argument or evaluating an arithmetic operation. Identifying redexes is a fundamental task in rewriting systems and lambda calculus since they represent the points at which computation can take a step.

Innermost Redex

An innermost redex is one that does not contain any other redex within it. This means no further reducible expressions exist inside the innermost redex, so it is often the natural candidate for evaluation under call-by-value or applicative-order strategies.

Outermost Redex

An outermost redex is one that is not contained within any other redex, representing the largest reducible segment in a given expression. Under normal-order evaluation, the outermost redex is typically reduced first, ensuring that the overall structure of the expression is simplified from the top down.

Beta Reduction

Beta reduction is the specific process in lambda calculus where a function application is simplified by substituting its argument for the bound variable in the function body. This operation makes the entire application a redex and is one of the central computation steps in functional programming languages and formal systems.

Evaluation Strategy

Evaluation strategy refers to the order in which redexes are chosen and reduced during the computation. Common strategies include reducing innermost redexes first (applicative order) or outermost redexes first (normal order). The choice of strategy affects the efficiency and sometimes even the termination of the computation.

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