# lec06 - CS 3110 Lecture 6 The Substitution Model of...

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CS 3110 Lecture 6 The Substitution Model of Evaluation In this lecture, we examine how OCaml programs evaluate more closely, building a more formal and precise description of the evaluation process. This is a model of evaluation based on the basic notion of substitution , in which variable names are replaced by values that they are bound to. This corresponds to the mathematical notion that two equal things are interchangeable. A tricky example What is the value of the following expression? (Note that this is not just a definition of a function, it binds two names evil and dummy to functions and then applies evil to three arguments, returning the value of that expression. The names evil and dummy are bound only in the body and not at top level.) let rec evil(f1, f2, n) = let f(x) = 10 + n in if n = 1 then f(0) + f1(0) + f2(0) else evil(f, f1, n-1) and dummy(x) = 1000 in evil(dummy, dummy, 3) We can see that the function evil calls itself recursively, and the result of the function is the result when it is called with n=1 . But what are the values returned by the applications of functions f , f1 and f2 ? To understand what those values are, we need to better understand the OCaml evaluation model, and how variable names like n are bound. Evaluation The OCaml prompt lets you type either a term or a declaration that binds a variable to a term. It evaluates the term to produce a value : a term that does not need any further evaluation. We can define values v as a syntactic class too. Values include not only constants, but tuples of values, variant constructors applied to values, and functions. Running an ML program is just evaluating a term. What happens when we evaluate a term? In an imperative (non-functional) language like Java, we sometimes imagine that there is an idea of a "current statement" that is executing. This isn't a very good model for ML; it is better to think of ML programs as being evaluated in the same way that you would evaluate a mathematical expression. For example, if you see an expression like (1+2)*4, you know that you first evaluate the subexpression 1+2, getting a new expression 3*4. Then you evaluate 3*4. ML evaluation works the same way. As each point in time, the ML evaluator rewrites the program expression to another expression. Assuming that evaluation eventually terminates, eventually the whole expression is a value and then evaluation stops: the program is done. Or maybe the expression never reduces to a value, in which case you have an infinite loop. Rewriting works by performing simple steps called reductions . In the arithmetic example above, the rewrite is performed by doing the reduction 1+2 3 within the larger expression, replacing the occurrence of the subexpression 1+2 with the right-hand side of the reduction, 3 , therefore rewriting (1+2)*4 to 3*4 .

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The next question is which reduction OCaml does. Fortunately, there is a simple rule. Evaluation works by always performing the leftmost reduction that is allowed. So we can describe evaluation precisely by simply
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lec06 - CS 3110 Lecture 6 The Substitution Model of...

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