This is the archived website of SI 413 from the Fall 2012 semester. Feel free to browse around; you may also find more recent offerings at my teaching page.
The starter code for today's class is a "beefed-up" calculator language that includes things like assignment and comparison. Here's the grammar (bison style):
run: stmt run | stmt stmt: ares STOP ares: VAR ASN bres | bres bres: res | bres BOP res res: exp | res COMP exp exp: exp OPA term | term term: term OPM factor | factor factor: NUM | LP bres RP | VAR
And here's a simple program in this language:
x := (5 + 3) * 2; x > 3 - 7;
Now check out the parse tree for that "simple" program:
That's pretty messy. We saw in lab how the grammar can be simplified somewhat to help shrink down the waste, but that only goes so far. As you know from implementing some simple interpreters in your lab, all this complication in the grammar directly affects the speed of interpreting (or compiling) programs in the language. When programs get really big, and we want to compile them really quickly, this becomes an issue.
An even more critical issue can arise when the syntax of our grammar doesn't match the desired semantics of the language. For example, the natural way to make an operator be left-associative is to write the grammar using left recursion. But, as you know, left recursion doesn't work with top-down parsers. So we rewrite the grammar to be right-recursive (using all those tail rules). This makes the parsing go faster, and is fine if you are writing the parser and interpreter together (like in your lab from last week). But if the program gets more complicated, we will notice quickly that the parse tree that results from this top-down grammar with tail rules is essentially right-associative rather than left. In other words, the parse tree (syntax) doesn't match the semantic meaning of the program. What can we do about this?
Abstract Syntax Trees (ASTs) provide a way of representing the semantic meaning of our program in a simple way that is independent of the actual syntax. This is a little confusing, so I'll emphasize again that the AST has nothing to do with the syntax - it's an abstraction away from the syntax of the language! Here's an example of the AST for the same example from before:
x := (5 + 3) * 2; x > 3 - 7;
The first thing you should notice is that this is a lot simpler than the previous parse tree. This AST represents the actual meaning of our small program - what that program is actually supposed to do.
Notice how the arithmetic operations like times and greater-than are represented. The operation goes up top (or any function call, for that matter), and the arguments go below it, in order.
The top three nodes (assign, print, and null) are a bit different - they are statements rather than expressions. Remember from way back when that an expression is a code fragment that returns or computes some value, whereas a statement doesn't return any value; rather, a statement does something. These statements are ordered, and we represent that ordering by a child relationship: the last child of a statement is the next statement in sequence. Of course the special "null" statement indicates that this sequence ends.
You should also observe that this AST is simpler in most respects than the preceding parse tree, but it also has some new things that the parse tree didn't. For example, in the program above, it's implicit that an expression should be printed out if it's not part of an assignment. The AST makes this explicit.
One more thing to make sure you notice: the AST does not depend on a single programming language. For example, I bet you could write the Scheme program or the C++ program corresponding to the AST above pretty easily. Moreover, a Scheme program that does exactly the same thing could generate exactly the same AST as a program in another language! Of course, the specific components of some language will affect what's in the AST (for example, no
lambdas in our calculator language), but the point is that the AST doesn't depend on any syntactical choices of the language - only on the meaning of the program.
Assume booleans are a "real" type in your language (i.e., they're not just integers like in C), and you have a program like
(7 > 2) + 3. This is an error because
7 > 2 is a boolean, and you can't add a boolean to a number. Specifically, it's a type error, because the problem is that the types of the two operands for the plus operator are "incompatible", meaning basically that they can't be added together.
When we talk about compilation, a lot of things are divided between "static" or "dynamic". This sometimes means different things in different context, but usually it's:
Generally speaking, the more we can do statically (at compile-time), the faster everything is. Here we're specifically talking about catching type errors, and this is always a nice thing to do at compile-time if we can. Why? Well, maybe this type error is buried somewhere deep in the user's code. Maybe the program does some big, long, difficult computation, and then at the very end, just as it's about to display the brilliant result, it gets a type error and crashes. What a tragedy! If we had compile-time (static) type checking, that wouldn't happen, because that error would have been noticed by the compiler before the program ever started running.
The AST is created at compile-time, just after the parsing phase. So any type-checking we do with the AST will be static-time checking. For a simple expression like
(7 > 2) + 3, the process is pretty simple: starting at the leaves, we label every node in the AST with its type. Whenever there's an operation like
+, we check that the types of the arguments are compatible, and then label the node with the type of whatever results. In this case, the
> node gets labeled as a boolean, and of course 3 is a number, so when we go to label the
+ node, we see that these are incompatible and give an error. (By the way, the statement nodes like "assign" and "print" just have type void.)
Hooray! Errors are a good thing - they save the programmer time and help assure that the actual program (with no errors) works as intended. But static type checking gets a lot more complicated when we have variables. Look back up at the AST for this program above:
x := (5 + 3) * 2; x > 3 - 7;
To do the type-checking here, we would need to know the type of
x in the second statement, in order to check that it's actually a number and therefore compatible with
3 - 7 in the
> operation. Of course you can wee that, yes,
x is going to be a number here. But how would the compiler know? In order to do this kind of type checking, we would need static typing, which means that the types of every variable are known at compile-time. For example, Java has static typing; that's why you have to declare the types of all your variables! Scheme, on the other hand, is not statically typed.
But before we can really talk about static typing, we have to answer the more fundamental question of what does a variable refer to, at any given point in the program. This means we need to know about things like scope. Which is what the next unit's all about!