Crafting A Compiler

Charles N. Fischer, Ron K. Cytron, Richard J. LeBlanc

Chapter 6 Bottom-Up Parsing - all with Video Answers

Educators

Chapter Questions

Problem 1

Build the CFSM and the parse table for the grammar shown in Figure 6.2.

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02:25

Using the knitting analogy of Section 6.2.2, show the sequence of LR shift and reduce actions for the grammar of Figure 6.2 on the following strings:
(a) plus plus num num num $$$
(b) $plus num plus num num $$$

Alice .

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09:53

Problem 3

Figures 6.6 and 6.7 trace a bottom-up parse of an input string using the table shown in Figure 6.5. Trace the parse of the following strings.
(a) $q $$
(b) $c $$
(c) a d c $$$

Susan Hallstrom

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06:18

Problem 4

Chris Trentman

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00:42

Problem 5

Show the LR parse table for the CFSM constructed in Exercise 4.

James Chok

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View

Problem 6

Which of following grammars are LR(0)? Explain why.
$$
\text { (a) } \begin{array}{rlrl}
1 & \mathrm{~S} & \rightarrow \text { StmtList } \$ \\
2 & \text { StmtList } & \rightarrow \text { StmtList semi Stmt } \\
3 & & \mid \mathrm{Stmt} \\
4 & \text { Stmt } \quad & \rightarrow \mathrm{s}
\end{array}
$$
$$
\text { (b) } \begin{array}{rlr}
1 & \mathrm{~S} & \rightarrow \text { StmtList } \$ \\
2 & \text { StmtList } & \rightarrow \text { Stmt semi StmtList } \\
3 & \mid \mathrm{Stmt} \\
4 & \text { Stmt } \quad \rightarrow \mathrm{s}
\end{array}
$$
$$
\text { (c) } \begin{array}{rlr}
1 & \mathrm{~S} & \rightarrow \text { StmtList } \$ \\
2 & \text { StmtList } & \rightarrow \text { StmtList semi StmtList } \\
3 & 1 \mathrm{Stmt} \\
4 & \text { Stmt } \quad \rightarrow \mathrm{s}
\end{array}
$$
$$
\text { (d) } \begin{array}{lll}
1 & \mathrm{~S} & \rightarrow \text { StmtList } \$ \\
2 & \text { StmtList } & \rightarrow \mathrm{s} \text { StTail } \\
3 & \text { StTail } & \rightarrow \text { semi StTail } \\
4 & \| \lambda
\end{array}
$$

James Kiss

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04:39

Problem 7

Show that the CFSM corresponding to a LL(1) grammar has the following property. Each state has exactly one kernel item if the grammar is $\lambda$-free.

Chris Trentman

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Problem 8

Prove or disprove that all $\lambda$-free $\mathrm{LL}(1)$ grammars are $\mathrm{LR}(0)$.

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00:35

Problem 9

Explain why the following grammar is unambiguous:
Start $\rightarrow$ Single a
| Double b
Single $\rightarrow 0$ Single 1
101
Double $\rightarrow 0$ Double 11
1011

Amy Jiang

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Problem 10

Show the LR(0) construction for the following grammars:
Data can't copy.

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01:46

Problem 11

Explain why the language defined by the grammar in Exercise 9 is inherently nondeterministic-there is no LALR $(k)$ grammar for such languages.

Celine Ibrahim

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00:47

Problem 12

Given the claim of Exercise 11, explain why the following statement is true or false:
There is no LR $(k)$ grammar for the language
$$
\left\{0^n 1^n a\right\} \cup\left\{0^n 1^{2 n} b\right\}
$$

Audrey Fong

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01:11

Problem 13

Discuss why is it not possible during $\operatorname{LR}(0)$ construction to obtain a shift/reduce conflict on a nonterminal.

Manisha Sarker

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Problem 14

Discuss why there cannot be an unambiguous CFGs for the language
$$
\left\{a^i b^j c^k \mid i=j \text { or } j=k ; i, j, k \geq 1\right\}
$$

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Problem 15

Complete the $\mathrm{LR}(0)$ construction for the grammar in Figure 6.19.

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01:20

Problem 16

16. Show that $\mathrm{LL}(1)$ construction fails for an unambiguous grammar that is $\operatorname{not} \operatorname{LR}(1)$.

Adriano Chikande

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03:27

Problem 17

Show that the grammar in Figure 6.20 is $\mathrm{LR}(0)$.

Raushan Kumar

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Problem 18

Complete the $\operatorname{LR}(0)$ construction for the grammar shown in Figure 6.25. Your state numbers should agree with those shown in the partial LR(0) construction.

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01:34

Problem 19

Which of the grammars in Exercise 10 are LR(0)? Justify your answers.

James Kiss

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01:13

Problem 20

Complete the SLR(1) construction for the grammar shown in Figure 6.26. Show the resulting parse table.

James Kiss

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Problem 21

Extend the grammar given in Figure 6.21 to accommodate standard expressions involving addition, subtraction, multiplication, and division. Model the syntax and semantics for these operators according to Java or $\mathrm{C}$

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13:01

Problem 22

Extend the grammar as directed in Exercise 21, but introduce an exponentiation operator that is right-associating. Let the operator (denoted by " $\star$ ") have the highest priority, so that the value of $3+4 \times 5 \star 2$ is 103 .

Susan Hallstrom

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00:49

Problem 23

Repeat Exercise 22, but add the ability to enclose expressions with parentheses to control how expressions are grouped together. Thus, the value of $((3+4) \times 5) \star 2$ is 1225 .

Ernest Castorena

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01:34

Problem 24

Which of the grammars in Exercise 10 are SLR(1)? Justify your answers.

James Kiss

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01:25

Problem 25

Generalize the algorithm given in Section 6.5.1 from SLR(1) to SLR $(k)$.

Nick Johnson

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01:10

Problem 26

Show that the following holds for any LALR(1) construction:
(a) For any state $s$ containing the item $\mathrm{A} \rightarrow \alpha \bullet \beta$,
$$
\text { ItemFollow }((s, \mathrm{~A} \rightarrow \alpha \bullet \beta)) \subseteq \text { Follow }(\mathrm{A})
$$
(b)

Adriano Chikande

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Problem 27

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Problem 28

Recall the EvalItemPropGraph algorithm given in Figure 6.28. Using the grammars in Figure 6.31 and Exercise 27 as a guide, show how to generate a LALR(1) grammar that requires $n$ iterations for ItemFollow sets to converge in EvalitemPropGraph.

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Problem 29

For the grammar shown in Figure 6.35, complete the LALR(1) construction from Figure 6.37.

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01:34

Problem 30

Which of the grammars in Exercise 10 are LALR(1)? Justify your answers.

James Kiss

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Problem 31

Show the LR(1) construction for the grammar in Exercise 4.

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03:26

Problem 32

Define the quasi-identical states of an $L R(1)$ parsing machine to be those states whose kernel productions are identical. Such states are distinguished only by the lookahead symbols associated with their productions. Given the LR(1) machine built for Exercise 31, complete the following:
(a) List the quasi-identical states of the LR(1) machine.
(b) Merge each set of quasi-identical states to obtain an LALR(1) machine.

Chris Trentman

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00:57

Problem 33

Starting with the CFSM built in Exercise 4, compute the LALR(1) lookahead information. Compare the resulting LALR(1) machine with the machine obtained in Exercise 32 .

Kian Manafi

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01:18

Problem 34

34. Which of the grammars in Exercise 10 are LR(1)? Justify your answers.

Adriano Chikande

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12:36

Problem 35

Consider a grammar $G$ and its LALR(1) construction. Suppose that a shift/reduce conflict occurs in G's LALR(1) construction. Prove that G's LR(1) construction also contains a shift/reduce conflict.

Chris Trentman

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03:53

Problem 36

Describe an algorithm that computes LALR(1) and then splits states as needed in an attempt to address conflicts. Take note of the issue raised in Exercise 35 .

Chris Trentman

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Problem 37

Using a grammar for the C programming language, try to extend the syntax to allow nested function definitions. For example, you might allow function definitions to occur inside any compound statement.
Report on any difficulties you encounter and discuss possible solutions. Justify the particular solution you adopt.

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Problem 38

Using a grammar for the $C$ programming language, try to extend the syntax so that a compound statement can appear to compute a value. In other words, allow a compound statement to appear wherever a simple constant or identifier could appear. Semantically, the value of a compound statement could be the value associated with its last statement.
Report on any difficulties you encounter and discuss possible solutions. Justify the particular solution you adopt.

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Problem 39

In Figure 6.3, Marker(2) pushes a state on the parse stack. In the bottomup parse shown in Figures 6.6 and 6.7, stack cells show both the state and the input symbol causing the state's shift onto the stack. Explain why the input symbol's presence in the stack cell is superfluous.

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Problem 40

Recall the dangling else problem introduced in Chapter 5. A grammar for a simplified language that allows conditional statements follows:
Explain why the grammar is or is not LALR(1).

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Problem 41

Consider the following grammar:
$\begin{array}{lll}1 & \text { Start } & \rightarrow \text { Stmt } \$ \\ 2 & \text { Stmt } & \rightarrow \text { Matched } \\ 3 & \quad \text { I Unmatched } \\ 4 & \text { Matched } \quad & \text { if e then Matched else Matched } \\ 5 & \quad \text { o other } \\ 6 & \text { Unmatched } \rightarrow \text { if e then Matched else Unmatched } \\ 7 & \quad \text { if e then Unmatched }\end{array}$
(a) Explain why the grammar is or is not LALR(1).
(b) Is the language of this grammar the same as the language of the grammar in Exercise 40 ? Why or why not?

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00:41

Problem 42

Repeat Exercise 41, adding the production Unmatched $\rightarrow$ other to the grammar.

Natalie Anderson

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Problem 43

Consider the following grammar:
$\begin{array}{lll}1 & \text { Start } & \rightarrow \text { Stmt } \$ \\ 2 & \text { Stmt } & \rightarrow \text { Matched } \\ 3 & \quad \text { Unmatched } \\ 4 & \text { Matched } & \rightarrow \text { if e then Matched else Matched } \\ 5 & & \text { I other } \\ 6 & \text { Unmatched } & \text { if e then Matched else Unmatched } \\ 7 & \quad \text { if e then Stmt }\end{array}$
(a) Explain why the grammar is or is not LALR(1).
(b) Is the language of this grammar the same as the language of the grammar in Exercise 40 ? Why or why not?

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Problem 44

Based on the material in Exercises 40,41 , and 43 , construct an LALR(1) grammar for the language defined by the following grammar:
$$\begin{array}{ll}1 & \text { Start } \rightarrow \text { Stmt } \$ \\ 2 & \text { Stmt } \rightarrow \text { if e then Stmt else Stmt } \\ 3 & \text { if e then Stmt } \\ 4 & \text { while e Stmt } \\ 5 & \text { repeat Stmt until e } \\ 6 & \text { other }\end{array}$$

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01:23

Problem 45

Show that there exist non-LL(1) grammars that are
(a) $\operatorname{LR}(0)$
(b) SLR(1)
(c) LALR(1)

James Kiss

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Problem 46

Normally, an LR parser traces a rightmost derivation (in reverse).
(a) How could an LR parser be modified to produce a leftmost parse as LL(1) parsers do? Describe your answer in terms of the algorithm in Figure 6.3.
(b) Would it help if we knew that the LR table was constructed for an LL grammar? Explain your reasoning.

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01:08

Problem 47

For each of the following, construct an appropriate grammar:
(a) The grammar is $\operatorname{SLR}(3)$ but not $\operatorname{SLR}(2)$.
(b) The grammar is $\operatorname{LALR}(2)$ but not LALR(1).
(c) The grammar is $\operatorname{LR}(2)$ but not $\operatorname{LR}(1)$.
(d) The grammar is LALR(1) and SLR(2) but not SLR(1).

James Kiss

Numerade Educator

02:23

Problem 48

Construct a single grammar that has all of the following properties:
$\bullet$ It is SLR(3) but not $\operatorname{SLR}(2)$.
$\bullet$ It is LALR(2) but not LALR(1).
$\bullet$ It is LR(1).

James Kiss

Numerade Educator

Problem 49

For every $k>1$ show that there exist grammars that are $\operatorname{SLR}(k+1)$, $\operatorname{LALR}(k+1)$, and $\operatorname{LR}(k+1)$ but not $\operatorname{SLR}(k), \operatorname{LALR}(k)$, or $\operatorname{LR}(k)$.

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02:23

Problem 50

Consider the grammar generated by $1 \leq i, j \leq n, i \neq j$ using the following template:
$$
\begin{aligned}
& \mathrm{S} \rightarrow \mathrm{X}_i \mathrm{z}_i \\
& \mathrm{X}_i \rightarrow \mathrm{y}_j \mathrm{X}_i \\
& \mid \mathrm{y}_j
\end{aligned}
$$
The resulting grammar has $O\left(n^2\right)$ productions.
(a) Show that the CFSM for this grammar has $O\left(2^n\right)$ states.
(b) Explain why the grammar is, or is not, SLR(1).

James Kiss

Numerade Educator

06:22

Problem 51

The bottom-up parsing techniques given in this chapter are more powerful than top-down techniques given in Chapter 5 .
Using the alphabet $\{a, b\}$, devise a language that is not $\operatorname{LL}(k)$ for any $k$ but is $\operatorname{LR}(k)$ for some $k$. What property of $\operatorname{LR}(k)$ parsing allows such a grammar to be constructed?

Trang Hoang

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