# Introduction to Programming Languages/Ambiguity
### Ambiguity

Compilers and interpreters use grammars to build the data-structures that they will use to process programs. Therefore, ideally a given program should be described by only one derivation tree. However, depending on how the grammar was designed, ambiguities are possible. A grammar is ambiguous if some phrase in the language generated by the grammar has two distinct derivation trees. For instance, the grammar below, which we have been using as our running example, is ambiguous.
In order to see that this grammar is ambiguous we can observe that it is possible to derive two different syntax trees for the string "a * b + c". The figure below shows these two different derivation trees:
Sometimes, the ambiguity in the grammar can compromise the meaning of the sentences that we derive from that grammar. As an example, our English grammar is ambiguous. The sentence "The professor saw the student with the glasses" has two possible derivation trees, as we show in the side figure. In the upper tree, the prepositional phrase "with the glasses" is modifying the verb. In other words, the glasses are the instruments that the professor has used to see the student. On the other hand, in the derivation tree at the bottom the same prepositional expression is modifying "the student". In this case, we can infer that the professor saw a particular student that was possibly wearing glasses at the time he or she was seen.
A particularly famous example of ambiguity in compilers happens in the if-then-else construction. The ambiguity happens because many languages allow the conditional clause without the "else" part. Lets consider a typical set of production rules that we can use to derive conditional statements:
Upon stumbling on a program like the code below, we do not know if the "else" clause is paired with the outermost or with the innermost "then". In C, as well as in the vast majority of languages, compilers solve this ambiguity by pairing an "else" with the closest "then". Therefore, according to this semantics, the program below will print the value 2 whenever a > b and c <= d:
However, the decision to pair the "else" with the closest "then" is arbitrary. Language designers could have chosen to pair the "else" block with the outermost "then" block, for instance. In fact, that grammar we saw above is ambiguous. We demonstrate this ambiguity by producing two derivation trees for the same sentence, as we do in the example figure below:
As we have seen in the three examples above, we can show that a grammar is ambiguous by providing two different parsing trees for the same sentence. However, the problem of determining if a given grammar is ambiguous is in general undecidable. The main challenge, in this case, is that some grammars can produce an infinite number of different sentences. To show that the grammar is ambiguous, we would have to choose, among all these sentences (and there are an infinite number of them), one that could be generated by two different derivation trees. Because the number of potential candidates might be infinite, we cannot simply go over all of them trying to decide if it has two derivation trees or not.
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