Start Lecture #8
If we are given an SDD and a parse tree for a given sentence, we would like to evaluate the annotations at every node. Since, for synthesized annotations parents can depend on children, and for inherited annotations children can depend on parents, there is no guarantee that one can in fact find an order of evaluation. The simplest counterexample is the single production A→B with synthesized attribute A.syn, inherited attribute B.inh, and rules A.syn=B.inh and B.inh=A.syn+1. This means to evaluate A.syn at the parent node we need B.inh at the child and vice versa. Even worse it is very hard to tell, in general, if every sentence has a successful evaluation order.
All this not withstanding we will not have great difficulty because we will not be considering the general case.
Recall that a parse tree has leaves that are terminals and internal nodes that are nonterminals.
Definition: A parse tree decorated with attributes, is called an annotated parse tree. It is constructed as follows.
Each internal node corresponds to a production. The node is labeled with the LHS of the production. If there are no attributes for the LHS in this production, we leave the node as it was (I don't believe this is a common occurrence). If there are k attributes for the LHS, we replace the LHS in the parse tree by k equations. The LHS of the equation is the attribute and the right hand side is its value.
Note that the annotated parse tree contains all the information of the original parse tree since we (either leave the node alone—if the LHS had no attributes or) replace the nonterminal A labeling the LHS with a series of equations A.attr=value.
We computed the values to put in this tree for 7+6/3 and on the right is (7-6).
Inherited attributed definitely make the situation more complicated. For a simple example, recall the circular dependency above involving A.syn and B.inh. But we do need them.
Consider the following left-recursive grammar for multiplication of numbers, and the parse tree on the right for 3*5*4.
T → T * F T → F F → num
It is easy to see how the values can be propagated up the tree and the expression evaluated.
When doing top-down parsing, however, we need to avoid left recursion. Consider the grammar below, which is the result of removing the left recursion. Again its parse tree is shown on the right. Try not to look at the semantic rules for the moment.
|T → F T'||T'.lval = F.val||Inherited|
|T.val = T'.tval||Synthesized|
|T' → * F T1'||T'1.lval = T'.lval * F.val||Inherited|
|T'.tval = T'1.tval||Synthesized|
|T' → ε||T'.tval = T'.lval||Synthesized|
|F → num||F.val = num.lexval||Synthesized|
Where on the tree should we do the multiplication 3*5 since there is no node that has 3 and * and 5 as children?
The second production is the one with the *, so that is the natural candidate for the multiplication site. Make sure you see that this production (for the * in 3*5) is associated with the blue-highlighted node in the parse tree. The right operand (5) can be obtained from the F that is the middle child of this T'. F gets the value from its child, the number itself; this is an example of the simple synthesized case we have already seen, F.val=num.lexval (see the last semantic rule in the table).
But where is the left operand?
It is located at the sibling of T' in the parse
tree, i.e., at the F immediately to T's left (we shall see the
significance that the sibling is to the left; it is not significant
that it is immediately to the left).
This F is not mentioned in the production associated with the
T' node we are examining.
So, how does T' get F.val from its sibling?
Answer: The common parent, in this case T, can get the value from F and then our node can inherit the value from its parent.
Bingo! ... an inherited attribute. This can be accomplished by having the following rule at the node T.
T'.lval = F.val (inherited)
This observation yields the first rule in the table.
Now lets look at the second multiplication (3*5)*4, where the parent of T' is another T'. (This is the normal case. When there are n multiplies, n-1 have T' as parent and only one has T).
The pink-highlighted T' is the site for the multiplication. However, it needs as left operand, the product 3*5 that its parent can calculate. So we have the parent (another T' node, the blue one in this case) calculate the product and store it as an attribute of its right child namely the pink T'. That is the first rule for T' in the table.
We have now explained the first, third, and last semantic rules. These are enough to calculate the answer. Indeed, if we trace it through, 60 does get evaluated and stored in the bottom right T', the one associated with the ε-production. Our remaining goal is to get the value up to the root where it represents the evaluation of this term T and can be combined with other terms to get the value of a larger expression.
Going up is easy, just synthesize. I named the attribute tval, for term-value. It is generated at the ε-production from the lval attribute and propagated back up. At the T node it is called simply val. At the right we see the annotated parse tree for this input.
Homework: Extend this SDD to handle the left-recursive, more complete expression evaluator given earlier in this section. Don't forget to eliminate the left recursion first.
It clearly requires some care to write the annotations.
Another question is how does the system figure out the evaluation order, assuming one exists? That is the subject of the next section.
Remark: Consider the identifier table. The lexer creates it initially, but as the compiler performs semantic analysis and discovers more information about various identifiers, e.g., type and visibility information, the table is updated. One could think of this as some sort of inherited/synthesized attribute pair that during each phase of analysis is pushed down and back up the tree. However, it is not implemented this way; the table is made a global data structure that is simply updated. The compiler writer must ensure manually that the updates are performed in an order respecting any dependences.
The diagram on the right illustrates a great deal. The black dotted lines comprise the parse tree for the multiplication grammar just studied when applied to a single multiplication, e.g. 3*5.
Each synthesized attribute is shown in green and is written to the right of the grammar symbol at the node where it appears in the annotated parse tree. This is also the node corresponding to the semantic rule calculating the value.
Each inherited attribute is shown in red and is written to the left of its grammar symbol at the node where it appears in the annotated parse tree. Note that this value is calculated by a semantic rule at the parent of this node.
Each green arrow points to the synthesized attribute calculated from the attribute at the tail of the arrow. These arrows either go up the tree one level or stay at a node. That is because a synthesized attribute can depend only on the node where it is defined and that node's children. The computation of the attribute is associated with the production at the node at its arrowhead. In this example, each synthesized attribute depends on only one other attribute, but that is not required.
Each red arrow points to the inherited attribute calculated from the attribute at the tail. Note that, at the lower right T' node, two red arrows point to the same attribute. This indicates that the common attribute at the arrowheads, depends on both attributes at the tails. According to the rules for inherited attributes, these arrows either go down the tree one level, go from a node to a sibling, or stay within a node. The computation of the attribute is associated with the production at the parent of the node at the arrowhead.
The graph just drawn is called the dependency graph. In addition to being generally useful in recording the relations between attributes, it shows the evaluation order(s) that can be used. Since the attribute at the head of an arrow depends on the on the one at the tail, we must evaluate the head attribute after evaluating the tail attribute.
Thus what we need is to find an evaluation order respecting the arrows. This is called a topological sort. The rule is that the needed ordering can be found if and only if there are no (directed) cycles. The algorithm is simple.
If the algorithm succeeds in deleting all the nodes, then the deletion order is a suitable evaluation order and there were no directed cycles.
The topological sort algorithm is non-deterministic (Choose a node) and hence there can be many topological sort orders.
Given an SDD and a parse tree, performing a topological sort constructs a suitable evaluation order or announces that no such order exists.
However, it is very difficult to determine, given just an SDD, whether no parse trees have cycles in their dependency graphs. That is, it is very difficult to determine if there are suitable evaluation orders for all parse trees. Fortunately, there are classes of SDDs for which a suitable evaluation order is guaranteed, and which are adequate for our needs.
As mentioned above an SDD is S-attributed if every attribute is synthesized. For these SDDs all attributes are calculated from attribute values at the children since the other possibility, the tail attribute is at the same node, is impossible since the tail attribute must be inherited for such arrows. Thus no cycles are possible and the attributes can be evaluated by a postorder traversal of the parse tree.
Since postorder corresponds to the actions of an LR parser when reducing the body of a production to its head, it is often convenient to evaluate synthesized attributes during an LR parse.
Unfortunately, it is hard to live without inherited attributes. For example, we need them for top-down parsing of expressions. Fortunately, our needs can be met by a class of SDDs called L-Attributed definitions for which we can easily find an evaluation order.
from the left, and hence the name L-attributed.
Case 2c must be handled specially whenever it occurs. We will try to avoid it.
The top picture to the right illustrates the other cases and suggests why there cannot be any cycles.
The picture immediately to the right corresponds to a fictitious R-attributed definition. One reason L-attributed definitions are favored over R, is the left to right ordering in English. See the example below on type declarations and also consider the grammars that result from eliminating left recursion.
We shall see that the key property of L-attributed SDDs is that they can be evaluated with two passes over the tree (an euler-tour order) in which we evaluate the inherited attributes as we go down the tree and the synthesized attributes as we go up. The restrictions L-attributed SDDs place on the inherited attributes are just enough to guarantee that when we go down we have all the values needed for the inherited attributes of the child.
|A → B C||B.inh = A.inh|
C.ihn = A.inh - B.inh + B.syn
A.syn = A.inh * B.inh + B.syn - C.inh / C.syn
|B → X||X.inh = something|
B.syn = B.inh + X.syn
|C → Y||Y.inh = something|
C.syn = C.inh + Y.syn
The table on the right shows a very simple grammar with fairly general, L-attributed semantic rules attached. Compare the dependencies with the general case shown in the (red-green) picture of L-attributed SDDs above.
The picture below the table shows the parse tree for the grammar in
The triangles below B and C represent the parse tree for X and Y.
The dotted and numbered arrow in the picture illustrates the
evaluation order for the attributes; it will be discussed shortly.
The rules for calculating A.syn, B.inh, and C.inh are shown in the table. The attribute A.inh would have been set by the parent of A in the tree; the semantic rule generating A.inh would be given with the production at the parent. The attributes X.syn and Y.syn are calculated at the children of B and C respectively. X.syn can depend of X.inh and on values in the triangle below X; similarly for Y.syn.
The picture to the right
shows that there is an evaluation
order for L-attributed definitions (again assuming no case 2c).
We just need to follow the (Euler-tour) arrow and stop at all the
As in the pictures above, red signifies inherited attributes and
Specifically, the evaluations at the numbered stops are
More formally, do a depth first traversal of the tree and evaluate inherited attributes on the way down and synthetic attributes on the way up. This corresponds to a an Euler-tour traversal. It also corresponds to a call graph of a program where actions are taken at each call and each return
The first time you traverse an edge visit a node (on the way down), evaluate its inherited attributes (in programming terms, the parent evaluates the inherited attributes of the children and passes them as arguments to the call). The second time you traverse an edge and leave node (on the way back up), you evaluate the synthesized attributes (in programming terms the child returns the value to the parent).
The key point is that all attributes needed will have already been evaluated. Consider the rightmost child of the root in the diagram on the right.
|D → T L||L.type = T.type||inherited|
|T → INT||T.type = integer||synthesized|
|T → FLOAT||T.type = float||synthesized|
|L → L1 , ID||L1.type = L.type||inherited|
|addType(ID.entry,L.type)||synthesized, side effect|
|L → ID||addType(ID.entry,L.type)||synthesized, side effect|
When we have side effects such as printing or adding an entry to a table we must ensure that we have not added a constraint to the evaluation order that causes a cycle.
For example, the left-recursive SDD shown in the table on the right propagates type information from a declaration to entries in an identifier table.
The function addType adds the type information in the second argument to the identifier table entry specified in the first argument. Note that the side effect, adding the type info to the table, does not affect the evaluation order.
Draw the dependency graph on the board.
Note that the terminal ID has an attribute called
(given by the lexer) that gives its entry in the identifier table.
The nonterminal L can be considered to have (in addition to the
inherited attribute L.type)
a dummy synthesized attribute, say AddType, that is a place holder
for the addType() routine.
AddType depends on the arguments of addType().
Since the first argument is from a child, and the second is an
inherited attribute of this node, we have legal dependences for a
Thus we have an L-attributed definition.
Homework: For the SDD above, give the annotated parse tree for