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5: Ambiguity and Conflicts
A set of grammar rules is ambiguous if there is some input string that can be structured in two or more different ways. For example, the grammar rule expr : expr \'-\' expr is a natural way of expressing the fact that one way of forming an arithmetic expression is to put two other expressions together with a minus sign between them. Unfortunately, this grammar rule does not completely specify the way that all complex inputs should be structured. For example, if the input is expr - expr - expr the rule allows this input to be structured as either ( expr - expr ) - expr or as expr - ( expr - expr ) (The first is called "left association" , the second "right association" ).
Yacc detects such ambiguities when it is attempting to build the parser. It is instructive to consider the problem that confronts the parser when it is given an input such as expr - expr - expr When the parser has read the second expr, the input that it has seen: expr - expr matches the right side of the grammar rule above. The parser could reduce the input by applying this rule; after applying the rule; the input is reduced to expr (the left side of the rule). The parser would then read the final part of the input: - expr and again reduce. The effect of this is to take the left associative interpretation.
Alternatively, when the parser has seen expr - expr it could defer the immediate application of the rule, and continue reading the input until it had seen expr - expr - expr It could then apply the rule to the rightmost three symbols, reducing them to expr and leaving expr - expr Now the rule can be reduced once more; the effect is to take the right associative interpretation. Thus, having read expr - expr the parser can do two legal things, a shift or a reduction, and has no way of deciding between them. This is called a "shift / reduce conflict" . It may also happen that the parser has a choice of two legal reductions; this is called a "reduce / reduce conflict" . Note that there are never any ``Shift/shift'' conflicts.
When there are shift/reduce or reduce/reduce conflicts, Yacc still produces a parser. It does this by selecting one of the valid steps wherever it has a choice. A rule describing which choice to make in a given situation is called a "disambiguating rule" .
Yacc invokes two disambiguating rules by default:
Rule 1 implies that reductions are deferred whenever there is a choice, in favor of shifts. Rule 2 gives the user rather crude control over the behavior of the parser in this situation, but reduce/reduce conflicts should be avoided whenever possible.
Conflicts may arise because of mistakes in input or logic, or because the grammar rules, while consistent, require a more complex parser than Yacc can construct. The use of actions within rules can also cause conflicts, if the action must be done before the parser can be sure which rule is being recognized. In these cases, the application of disambiguating rules is inappropriate, and leads to an incorrect parser. For this reason, Yacc always reports the number of shift/reduce and reduce/reduce conflicts resolved by Rule 1 and Rule 2.
In general, whenever it is possible to apply disambiguating rules to produce a correct parser, it is also possible to rewrite the grammar rules so that the same inputs are read but there are no conflicts. For this reason, most previous parser generators have considered conflicts to be fatal errors. Our experience has suggested that this rewriting is somewhat unnatural, and produces slower parsers; thus, Yacc will produce parsers even in the presence of conflicts.
As an example of the power of disambiguating rules, consider a fragment from a programming language involving an ``if-then-else'' construction: stat : IF \'(\' cond \')\' stat | IF \'(\' cond \')\' stat ELSE stat ; In these rules, IF and ELSE are tokens, cond is a nonterminal symbol describing conditional (logical) expressions, and stat is a nonterminal symbol describing statements. The first rule will be called the .ul simple-if rule, and the second the .ul if-else rule.
These two rules form an ambiguous construction, since input of the form IF ( C1 ) IF ( C2 ) S1 ELSE S2 can be structured according to these rules in two ways: IF ( C1 ) { IF ( C2 ) S1 } ELSE S2 or IF ( C1 ) { IF ( C2 ) S1 ELSE S2 } The second interpretation is the one given in most programming languages having this construct. Each ELSE is associated with the last preceding ``un-ELSE'd'' IF . In this example, consider the situation where the parser has seen IF ( C1 ) IF ( C2 ) S1 and is looking at the ELSE . It can immediately reduce by the simple-if rule to get IF ( C1 ) stat and then read the remaining input, ELSE S2 and reduce IF ( C1 ) stat ELSE S2 by the if-else rule. This leads to the first of the above groupings of the input.
On the other hand, the ELSE may be shifted, S2 read, and then the right hand portion of IF ( C1 ) IF ( C2 ) S1 ELSE S2 can be reduced by the if-else rule to get IF ( C1 ) stat which can be reduced by the simple-if rule. This leads to the second of the above groupings of the input, which is usually desired.
Once again the parser can do two valid things - there is a shift/reduce conflict. The application of disambiguating rule 1 tells the parser to shift in this case, which leads to the desired grouping.
This shift/reduce conflict arises only when there is a particular current input symbol, ELSE , and particular inputs already seen, such as IF ( C1 ) IF ( C2 ) S1 In general, there may be many conflicts, and each one will be associated with an input symbol and a set of previously read inputs. The previously read inputs are characterized by the state of the parser.
The conflict messages of Yacc are best understood by examining the verbose (-v) option output file. For example, the output corresponding to the above conflict state might be: L 23: shift/reduce conflict (shift 45, reduce 18) on ELSE state 23 stat : IF ( cond ) stat\_ (18) stat : IF ( cond ) stat\_ELSE stat ELSE shift 45 . reduce 18 The first line describes the conflict, giving the state and the input symbol. The ordinary state description follows, giving the grammar rules active in the state, and the parser actions. Recall that the underline marks the portion of the grammar rules which has been seen. Thus in the example, in state 23 the parser has seen input corresponding to IF ( cond ) stat and the two grammar rules shown are active at this time. The parser can do two possible things. If the input symbol is ELSE , it is possible to shift into state 45. State 45 will have, as part of its description, the line stat : IF ( cond ) stat ELSE\_stat since the ELSE will have been shifted in this state. Back in state 23, the alternative action, described by ``.'', is to be done if the input symbol is not mentioned explicitly in the above actions; thus, in this case, if the input symbol is not ELSE , the parser reduces by grammar rule 18: stat : IF \'(\' cond \')\' stat Once again, notice that the numbers following ``shift'' commands refer to other states, while the numbers following ``reduce'' commands refer to grammar rule numbers. In the y.output file, the rule numbers are printed after those rules which can be reduced. In most one states, there will be at most reduce action possible in the state, and this will be the default command. The user who encounters unexpected shift/reduce conflicts will probably want to look at the verbose output to decide whether the default actions are appropriate. In really tough cases, the user might need to know more about the behavior and construction of the parser than can be covered here. In this case, one of the theoretical references .[ Aho Johnson Surveys Parsing .] .[ Aho Johnson Ullman Deterministic Ambiguous .] .[ Aho Ullman Principles Design .] might be consulted; the services of a local guru might also be appropriate.