GATE CSE 2005 | Question: 83b

Question

Consider the following expression grammar. The semantic rules for expression evaluation are stated next to each grammar production.$$\begin{array}{l|l} E\rightarrow number & E.val = {number.val} \\\qquad \mid \ E \ \ ‘+\text{'} \ E & E^{(1)}.val = E^{(2)}.val + E^{(3)}.val     \\\qquad \mid \ E \ \ ‘\times\text{'} \  E & E^{(1)}.val = E^{(2)}.val \times E^{(3)}.val  \end{array}$$

Assume the conflicts of this question are resolved using yacc tool and an LALR(1) parser is generated for parsing arithmetic expressions as per the given grammar. Consider an expression $3 \times 2 + 1$. What precedence and associativity properties does the generated parser realize?

• A. Equal precedence and left associativity; expression is evaluated to $7$

• B. Equal precedence and right associativity; expression is evaluated to $9$

• C. Precedence of ‘$\times$’ is higher than that of ‘$+$’, and both operators are left associative; expression is evaluated to $7$

• D. Precedence of ‘$+$’ is higher than that of ‘$\times$’, and both operators are left associative; expression is evaluated to $9$

Comments

@sushmita here yaac prefer shift over reduce . so if we push E*E in stack then next we have + and yaac prefer shift on + over reduction of E*E to E so according to you + have more priority.

now consider another example E+E*E. here when we push E+E in stack then againg parser have two options either shift over * or reduce over E+E to E(you can draw state diagram to understand) and yaac resolve shift over reduce and then * get more priority so 

here precedence is same for + and * . expression is evaluated from right to left as shift always get done over reduce so right associative

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If anyone is wondering how is it Right associative then take this string num + num + num , at some point of time during parsing the stack content will be E+E | +num $ so YACC will prefer shift instead of reduce (E ->E+E) and thus at the end the stack content will be E+E+E|\$ and now we will start reducing it.

Is this intuition correct @Shaik Masthan sir? or I can straight away say that LALR(1) is Bottom up so it will follow RMD in reverse , hence it will be right associative

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YACC tool - this has significance. It's behavior define the output.

Answer 1

LALR Parser is type of Bottom up Parser which uses Right most Derivation

For $3×2+1$

$E \rightarrow E * E$ (Both shift and reduce possible but yacc prefers shift)

     $ \rightarrow E * E + E$ 

     $ \rightarrow E * E + 1$

     $ \rightarrow E * 2 + 1$

     $ \rightarrow E * 3$

     $ \rightarrow 3 * 3$

     $ \rightarrow 9$

All the productions are in same level therefore all have same precedence

Therefore Ans is B. Equal precedence and right associativity; expression is evaluated to 9.

Comments

All the productions are at the same level, therefore, all have the same precedence, still + is evaluated first before * so this grammar is Right Associative we can also say this because bottom-up parsers follow rightmost derivation so right most operators will be evaluated first.

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How do we know that there’s no SR conflict for E → 3 reduction?

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@savvysiddharth

Look at part A of the question https://gateoverflow.in/1405/gate2005-83a.

There will be no conflict when you’ll reduce E → 3 this.

Answer 2

None of the Options is correct here ... The answer has to be "precedence of + is higher than * " and "both * and + are right associative"

Here we will never come across an RR conflict because we dont have 2 productions with the same RHS but different LHS ... 

EX : In the grammar, 

S->A/a, 

A->a 

we have 2 productions with the same RHS (which is a) but different LHS (S and A) ... Now while parsing a string I might come across a single state with productions as A->a. and S->a. Now this state will create a conflict on whether should I reduce string "a" to S or A ... So clearly there is an RR conflict here .... 

But in the given grammar it is not the case ... 

While parsing a string say "num+num*num" from the above grammar,I will come across an SR conflict ... When ?? after scanning num+num , I have a choice on whether should I shift on * (as good as giving higher precedence to * over +) or reduce "num+num" to E (as good as giving higher precedence to + over *) ... So here there is an SR conflict ... 

YACC tool always goes in-favour of SHIFT incase of SR conflict (and first reduce incase of RR conflict) ...

So,since we are using YACC to resolve conflicts, here + will be given higher precedence over * but incase if we come across a string like 2+3+5 , it will be right associative ... 

None of the Options is correct here ...

Comments

although your answer explains that in certain cases when the grammar poses an RR conflict, the grammar rule which comes before will get a priority, it still is misleading as far as this grammar and question is concerned.

 

Since the case that you are mentioning here (RR conflict priority decision) doesn't concern the grammar given in question, I think the correct option, according to you, that " precedence of + is higher than * " is simply wrong.

 

Also after reading your answer, I feel that you don't clearly understand the precedence and associativity concepts. Always, remember, precedence is established first. Once the precedence rules are established, and there comes a case where two operators come one after another and have equal precedence (the operators may be same or different), only then, we apply the associativity rule.

 

In the last part of your answer, you are using the associativity rule to derive the precedence of + and x which I find to be a reason in concluding the possibility of confusion that you might have regarding associativity and precedence.

 

Nice catch though regarding RR conflict.

Answer 3

LALR(1) is a bottom-up parser as it uses RMD in reverse
so answer is 9 , not 7
option B