In the function\u00a0addx<\/strong>\u00a0the variable y<\/strong>\u00a0<\/strong>is local and\u00a0x<\/strong> is global.<\/p>\n Let’s try applying the function to the value\u00a04.\u00a0<\/strong>As\u00a0y<\/strong> is the parameter, it gets the value 4, which is added to the value of\u00a0x<\/strong>, which is 2.<\/p>\n Here’s what it looks like using syntax trees. Remember, a \\(\\beta\\) conversion is when we bring the value in for a parameter.<\/p>\n Applying the addx function to an argument. Notice that x is a global variable so we don’t explicitly pass a value to it (because it has already been defined).<\/p><\/div>\n This is fairly straightforward because there’s just one function, and it is a fairly simple one, but this is about to get very complex, so let’s build ourselves some machinery to analyse these functions. We’ll start off with some shorthand:<\/p>\n \u03bbx. E<\/strong><\/p>\n is a function that takes\u00a0exactly\u00a0<\/strong>one argument. Note that we don’t care what the function body actually does, as all we’re interested in is keeping track of variables<\/p>\n Similarly:<\/p>\n \u03bbx. E F<\/strong><\/p>\n is also a function that takes\u00a0exactly\u00a0<\/strong>one argument, but this time there are two distinct parts to the function body, for example, if we had something like this\u00a0(\u03bbx. + 1 2) <\/strong>we could say that<\/p>\n\\(\\lambda x. + x\\; y \\equiv \\lambda x. E \\)\n because\u00a0E = + x y<\/strong><\/p>\n We can also say that<\/p>\n\\(\\lambda x. + x\\; y \\equiv \\lambda x. E\\; F \\)\n because\u00a0E = + x\u00a0<\/strong>and\u00a0F = y<\/strong>.\u00a0<\/strong>However, we could equally say that<\/p>\n\\(\\lambda x. + x\\; y \\equiv \\lambda x. E\\; F \\)\n because\u00a0E = +\u00a0\u00a0<\/strong>and\u00a0F = x<\/strong> y<\/strong>, because there are still two parts. What we could\u00a0not<\/strong> say is that\u00a0F = + x y<\/strong>, because then there would be nothing for E.<\/p>\n Here’s one last example:<\/p>\n\\(\\lambda x. x \\equiv \\lambda x. E \\)\n because\u00a0E =\u00a0 x<\/strong>, but<\/p>\n\\(\\lambda x. x \\not\\equiv \\lambda x. E\\; F \\)\n because there’s only one item in the function body.<\/p>\n In\u00a0\u03bb calculus, a local variable is referred to as a\u00a0bound\u00a0<\/strong>variable, so for example,\u00a0x\u00a0<\/strong>is bound in\u00a0(\u03bbx. + x y)<\/strong> because it is in the body of the function\u00a0and<\/em> in the parameter list. What about\u00a0y<\/strong>? It’s clearly not bound because it’s value is coming from somewhere else, however, for reasons that we’ll see shortly it isn’t necessarily<\/em>\u00a0a global variable, so we refer to it as a free<\/strong> variable.<\/p>\n The next section will look how formal steps to identify free and bound variables which will make our lives easier when dealing with complex functions and expressions.<\/p>\n","protected":false},"excerpt":{"rendered":" We’ve already seen local and global variables in Racket. > (define x 2) > (define addx (lambda (y) (+ y x))) In the function\u00a0addx\u00a0the variable y\u00a0is local and\u00a0x is global. Let’s try applying the function to the value\u00a04.\u00a0As\u00a0y is the Continue Reading<\/a><\/span><\/p>\n","protected":false},"author":4,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-435","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/pages\/435","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/comments?post=435"}],"version-history":[{"count":10,"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/pages\/435\/revisions"}],"predecessor-version":[{"id":447,"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/pages\/435\/revisions\/447"}],"wp:attachment":[{"href":"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-json\/wp\/v2\/media?parent=435"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}> (addx 4) \r\n6\r\n<\/pre>\n
![\"\"](\"https:\/\/books.compclassnotes.com\/elementarycomputing\/wp-content\/uploads\/sites\/9\/2018\/11\/Free-01.png\")