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CHAPTER 4. FORMAL DESCRIPTION OF LANGUAGE use visual studio .net data matrix printing tomake barcode data matrix in .net Office Word Exhibit 4.14. Dissection o barcode data matrix for .

NET f a lambda expression. A lambda expression, with name: Useful interpretation: 2 = x. y.

x(xy). the number two Breakdown of elements 2= D data matrix barcodes for .NET eclares the symbol 2 to be a name for the following expression. x.

The function header names the parameter, x . Everything that follows this . is the expression body.

y.x(xy) The body of the original expression is another expression with a parameter named y . Parameter names are purely arbitrary; this expression would still have the same meaning if it were rewritten with a di erent parameter name, as in: q.

x(xq) x(xy) This is the body of the inner expression. It contains a reference to the parameter y and also references to the parameter x from the enclosing expression..

Reduction. Consider a lamb da expression which represents a function. At the abstract level, the meaning, or semantics, of the expression is the mathematical function that it computes when applied to an argument.

Intuitively, we want to be able to freely replace an expression by a simpler expression that has the same meaning. The rules for beta and eta reduction permit us to do so. The main evaluation rule for lambda calculus is called beta reduction and it corresponds to the action of calling a function on its argument.

A beta reducible expression is an application whose left part is a lambda expression. We also use the term beta redex as a shortening of reducible expression . When a lambda expression is applied to an argument, the argument formula is substituted for the bound variable in the body of the expression.

The result is a new formula. A second reduction rule is called eta reduction. Eta reduction lets us eliminate one level of binding in an expression of the form x.

f (x). In words, this is a special case in which the lambda argument is used only once, at the end of the body of the expression, and the rest of the body is a lambda expression applied to this parameter. If we apply such an expression to an argument, one beta reduction step will result in the simpler form f (x).

Eta reduction lets us make this transformation without supplying an argument. Speci cally, eta reduction permits us to replace any expression of the form x.f (x), where f represents a function, by the single symbol f .

After a reduction step, the new formula may still contain a redex. In that case, a second reduction step may be done. When the result does not contain a beta-redex or eta-redex, the reduction process is complete.

We say such a formula is in normal form. Many lambda expressions contain nested expressions. When such an expression is fully parenthesized it is clear which arguments belong to which function.

When parentheses are omitted, remember that function application associates to the left; that is, the leftmost argument is substituted rst for the parameter in the outermost expression. We now describe in more detail how reduction works. When we reduce a formula (or subformula) of the form H = (( x.

F )G), we replace H by the formula F , where F is obtained from F by. 4.3. SEMANTICS Exhibit 4.15. Lambda calcu lus formulas that represent numbers.

0= 1= 2= x. y.y x.

y.xy x. y.

x(xy). The formula for zero has n VS .NET barcode data matrix o occurrences of its rst parameter in its body. Note that it is the same as the formula for F .

Zero and False are also represented identically in many programming languages. The formula for the integer one has a single x in its body, followed by a y. The formula for two has two x s.

The number n will be represented by a formula in which the rst parameter occurs n times in succession.. substituting G for each re ference to x in F . Note that if F contains another binding x, the references to that binding are not replaced. For example, (( x.

xy)(zw)) reduces to ((zw)y) and (( x.x( x.(xy)))(zz)) reduces to (zz)( x.

(xy)). When an expression containing an unbound symbol is used as an argument to another lambda expression, special care must be taken. Any occurrence of a variable in the argument that was free before the substitution must remain free after the substitution.

It is not permitted for a variable to be captured by an unrelated during substitution. For example, it is not permitted to apply the reduction rule to the formula (( x.( y.

x))(zy)), since y is free in (zy), but after substitution, that occurrence of y would not be free in ( y.(zy)). To avoid this problem, the parameter must be renamed, and all of its bound occurrences must be changed to the new name.

Thus (( x.( y.x))(zy)) could be rewritten as (( x.

( w.x))(zy)), after which the reduction step would be legal. Examples of Formulas and Their Reductions The formulas T and F in Exhibit 4.

13 accomplish the equivalent of branching by manipulating their parameters. They take the place of the conditional statement in a programming language. T (true) returns its rst argument and discards the second.

Thus it corresponds to the IF..THEN statement which evaluates the THEN clause when the condition is true.

Similarly, the formula F (false) corresponds to the IF..ELSE clause.

It returns its second parameter just as an IF statement evaluates the second, or ELSE clause, when the condition is false. The successor function, S, applied to any integer, gives us the next integer. Exhibit 4.

16 shows the lambda formula that computes this function. Given any formula for a number n, it returns the formula for n + 1. The function ZeroP (zero predicate) tests whether its argument is equal to the formula for zero.

If so, the result is T , if not, F . Exhibit 4.17 shows how we would call S and ZeroP .

The process of carrying out these computations will be explained later. Church was able to show that lambda calculus can represent all computation, by representing.
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