How do you learn Bottom-up Parsing Algorithm.docx

Bottom-up Parsing Algorithm by ChatGPT

Introduction
The purpose of this thread is for learning Bottom-up Parsing including LR(0), SLR, LR(1)
and LALR Parsing.
Method
By asking questions.

LR(1) and LALR Parsing Algorithm
Q1: Please explain
Please explain LALR bottom up parsing.
LALR (Look-Ahead LR) parsing is a type of bottom-up parsing technique used in
compilers to analyze and process context-free grammars. It is a variant of the LR
parsing family and is commonly used in practical parser generators like YACC and
Bison because it offers a good balance between power and efficiency.
Key Concepts in LALR Parsing
1. Bottom-Up Parsing:
o Constructs a parse tree for a given input string starting from the leaves
(tokens) and works upward to the root (start symbol).
o Utilizes shift-reduce operations to manage parsing decisions:
 Shift: Push the next input symbol onto the stack.
 Reduce: Replace symbols on the stack with a non-terminal based
on a production rule.
2. Lookahead:
o Uses a lookahead symbol to decide the correct parsing action (shift or
reduce).
o Lookahead ensures the parser makes decisions only when it is confident,
avoiding ambiguities.
3. LR Parsing:
o LALR parsers are derived from LR(1) parsers, which use:
 L: Left-to-right scanning of the input.
 R: Rightmost derivation in reverse (bottom-up approach).
 1: One-symbol lookahead.
4. Key Optimization:

o LALR parsers combine the states of a canonical LR(1) parser to reduce
the size of the parsing table, making it smaller and more memory-
efficient.
o It does this without significantly compromising the language recognition
power, though it can lead to rare cases of reduced precision (e.g., inability
to distinguish certain conflicts).
How LALR Parsing Works
1. Start with LR(1) Parsing:
o Construct the canonical collection of LR(1) items, which includes:
 States representing the parsing process.
 Each state contains a set of items (augmented production rules
with lookahead).
2. Merge Compatible States:
o LALR parsers identify similar states (states with identical core items but
different lookahead symbols) and merge them to create a more compact
LALR parsing table.
o This reduces the number of states compared to the full LR(1) parser.
3. Create the LALR Table:
o The LALR parsing table has:
 Action Table: Specifies shift, reduce, or accept actions based on
the current state and lookahead symbol.
 Goto Table: Specifies the next state based on the current state
and a non-terminal.
4. Parsing Process:
o Use the LALR table to parse the input string by performing shift and
reduce operations as guided by the table.
Example of LALR Parsing
Grammar:
S Aa
→ ∣ bAc ∣ dc ∣ bdaS to Aa ,|, bAc ,|, dc ,|, bda A dA to d
→
Steps:

1. Construct LR(1) Items:
o Create the canonical LR(1) item sets, including lookahead symbols for
each item.
2. Merge States:
o Identify states with identical cores and merge them, retaining distinct
lookahead symbols.
3. Build the LALR Table:
o Combine the merged states into a more compact parsing table.
4. Parse the Input:
o For an input like bda, use the LALR table to guide shift-reduce actions
until the input is successfully parsed.
Advantages of LALR Parsing
1. Memory Efficiency:
o Smaller parsing tables compared to canonical LR(1) parsers, making it
practical for real-world applications.
2. Powerful Grammar Recognition:
o Handles most programming language grammars, including complex
constructs.
3. Widely Supported:
o Used in tools like YACC and Bison, making it a standard choice for many
compiler developers.
Disadvantages of LALR Parsing
1. Reduced Precision:
o Merging states can cause loss of lookahead precision, leading to possible
shift-reduce conflicts in rare cases.
2. Complexity:
o Constructing the canonical LR(1) items and merging states can be
computationally intensive.

Comparison with Other Parsers
Parser
Type
State
Size
Grammar
Power
Efficiency Use Case
LR(0) Small
Simple
Grammars
Fast, Limited
Lookahead
Simple grammars, quick parsing.
SLR(1)
Moderat
e
Common
Grammars
Efficient Suitable for many grammars.
LALR(1) Compact
Complex
Grammars
Efficient and
Practical
Widely used in real-world
applications.
LR(1) Large
All LR(1)
Grammars
Memory-
Intensive
Theoretically powerful but
impractical for large grammars.
Summary
 LALR parsing is a compact and efficient approach to bottom-up parsing,
capable of handling most programming language grammars.
 It strikes a balance between the power of canonical LR(1) parsers and the
efficiency of simpler parsers like SLR.
 LALR parsers are a practical choice for real-world compilers and parser
generators.
Would you like a detailed worked-out example of an LALR parser construction or
guidance on a specific aspect?

Q2: How to check
How to check if a grammar is an LALR?
To check if a grammar is LALR(1), you need to analyze its canonical LR(1) parsing table
and determine whether merging states results in any conflicts. A grammar is LALR(1) if
merging the states does not introduce new shift-reduce or reduce-reduce conflicts.
Here’s a step-by-step procedure:
Steps to Check if a Grammar is LALR(1)
1. Compute the Canonical LR(1) Items
 Start by constructing the canonical LR(1) item sets for the grammar.
 Steps:
1. Augment the grammar with a new start symbol S′ SS' to S, where SS is
→
the original start symbol.
2. Compute the closure of the augmented grammar for the initial state.
3. Construct the goto states by applying grammar rules and compute
closures for new states.
4. Each state contains LR(1) items, which are grammar rules with a position
marker (⋅cdot) and a lookahead symbol.
2. Identify Core Items
 The core of a state consists of its items without lookahead symbols.
o Example:
 LR(1) item: [A α
→ ⋅β,lookahead][A to alpha cdot beta,
text{lookahead}]
 Core: [A α
→ ⋅β][A to alpha cdot beta]
3. Merge States with the Same Core
 For an LALR parser, states with the same core items are merged into a single
state.
 Merge all lookahead symbols from states with identical cores.
o Example:

 State 1: [A α
→ ⋅β,a][A to alpha cdot beta, a]
 State 2: [A α
→ ⋅β,b][A to alpha cdot beta, b]
 Merged State: [A α
→ ⋅β,{a,b}][A to alpha cdot beta, {a, b}]
4. Construct the LALR(1) Parsing Table
 Build the LALR(1) parsing table using the merged states.
 Include shift, reduce, and goto actions for each state.
5. Check for Conflicts
 Examine the LALR(1) parsing table for conflicts:
o Shift-Reduce Conflict:
 Occurs when a state contains a shift action and a reduce action for
the same input symbol.
o Reduce-Reduce Conflict:
 Occurs when a state contains two different reduce actions for the
same input symbol.
 If merging introduces conflicts that were not present in the canonical LR(1)
table, the grammar is not LALR(1).
Example
Grammar:
S Aa
→
Steps:
1. Canonical LR(1) Items:
o Compute the canonical LR(1) item sets (closures and gotos).
2. Identify Core Items:
o Group states with the same core items (ignore lookaheads).
3. Merge States:
o Merge states with identical cores and union their lookahead symbols.
4. Check for Conflicts:
o Examine the resulting table. If no new conflicts arise due to merging, the
grammar is LALR(1).

Automated Approach
1. Parser Generators:
o Tools like YACC, Bison, or JFLAP can check whether a grammar is LALR(1)
by generating the parsing table.
o The tool will report if the grammar has conflicts.
2. Manual Verification:
o Follow the steps outlined above to compute the LALR(1) table and
analyze conflicts.
Key Points
 If a grammar has a canonical LR(1) table without conflicts, it is LR(1) and hence
LALR(1).
 If merging states in the canonical LR(1) table introduces conflicts, the grammar
is not LALR(1).
 LALR(1) grammars are a subset of LR(1) grammars, but they allow a smaller
parsing table.
Would you like a worked-out example with the steps fully detailed, or assistance in
checking a specific grammar?

Q3: How different
How LALR(1) and LR(1) grammar is different?
Constructing LALR(1) parsing tables involves several steps, beginning with the
construction of canonical LR(1) items and ending with the creation of the optimized
LALR(1) table by merging compatible states. Here's a detailed guide:
Steps to Construct LALR(1) Parsing Tables
1. Augment the Grammar
 Add a new start symbol S′ SS' to S, where SS is the original start symbol.
→
 This ensures the parser can detect when parsing is complete.
2. Generate Canonical LR(1) Items
1. Construct the Initial State:
o Start with the augmented production S′→⋅S,$S' to cdot S, text{$},
where $text{$} represents the end-of-input symbol.
o Compute the closure for this state, which includes all items reachable via
non-terminal expansions.
2. Compute Goto Transitions:
o For each item in a state, compute the goto for every terminal and non-
terminal symbol, generating new states.
o Compute the closure for each new state.
3. Repeat Until All States Are Processed:
o Continue adding states and transitions until no new states can be
generated.
3. Identify Core Items
 The core of a state consists of its items without lookahead symbols.
o Example:
 Canonical LR(1) Item: [A α

 Core: [A α
→ ⋅β][A to alpha cdot beta]
4. Merge States with Identical Cores
 Group states that have identical cores, combining their lookahead symbols into
a single state.
o Example:
 State 1: [A α
 State 2: [A α
→ ⋅β,b][A to alpha cdot beta, b]
 Merged State: [A α
→ ⋅β,{a,b}][A to alpha cdot beta, {a, b}]
5. Construct the LALR(1) Parsing Table
1. Action Table:
o For each terminal in a state's lookahead set, define:
 Shift: If the next symbol is a terminal and there is a transition to a
new state.
 Reduce: If the item in the state is [A α
→ ⋅,lookahead][A to alpha
cdot, text{lookahead}] (completed production).
 Accept: If the item is [S′ S
→ ⋅,$][S' to S cdot, text{$}].
2. Goto Table:
o For each non-terminal, record the state transition based on the goto
computation.
3. Check for Conflicts:
o Identify any shift-reduce or reduce-reduce conflicts introduced by
merging states.
o If conflicts arise, the grammar is not LALR(1).
Example: Constructing an LALR(1) Table
Grammar:
S Aa
→
1. Canonical LR(1) Items

 Initial State:
o [S′→⋅S,$][S' to cdot S, $]
o Closure includes:
 [S→⋅Aa,$][S to cdot Aa, $]
 [S→⋅bAc,$][S to cdot bAc, $]
 [S→⋅dc,$][S to cdot dc, $]
 [S→⋅bda,$][S to cdot bda, $]
 [A→⋅d,{a,c}][A to cdot d, {a, c}]
 Goto Computation:
o Transitions on symbols like AA, aa, bb, dd, etc., generate new states.
2. Merge States
 Identify states with identical cores and merge their lookahead symbols.
o Example:
 [A d
→ ⋅,a][A to d cdot, a] and [A d
→ ⋅,c][A to d cdot, c] become
[A d
→ ⋅,{a,c}][A to d cdot, {a, c}].
3. Construct the LALR(1) Table
 Action Table:
o For each terminal and state, define shift/reduce actions.
o Example:
 In state I1I_1, dd leads to shift, while aa may lead to reduce.
 Goto Table:
o For non-terminals, record transitions between states.
4. Verify the Table
 Check for conflicts:
o If merging states introduces ambiguity (e.g., multiple actions for the
same terminal), the grammar is not LALR(1).
Tips for Constructing LALR(1) Tables
1. Automated Tools:

o Use parser generators like YACC, Bison, or JFLAP to simplify the process.
2. Conflict Detection:
o If conflicts arise, consider:
 Refactoring the grammar to remove ambiguity.
 Switching to a more powerful parser (e.g., full LR(1)).
3. Start Small:
o Practice constructing tables for small, simple grammars before tackling
complex ones.
Advantages of LALR(1) Parsing Tables
 Compactness: Smaller tables due to merged states.
 Efficiency: Reduces memory usage and computational overhead compared to
LR(1).
 Practicality: Suitable for most programming languages.
Would you like a fully worked-out example of constructing an LALR(1) table for a
specific grammar?

Q4: Examples
Can you provide examples for LALR parsing?
Certainly! Here are step-by-step examples of parsing using LALR(1) and shift-reduce
parsing techniques. These examples will illustrate how parsing tables are constructed
and used to analyze input strings.
Example 1: Parsing Using LALR(1) Table
Grammar:
S Aa
→
Input String:
bdabda
Step 1: Construct LALR(1) Table
The LALR(1) parsing table includes Action and Goto components.
Stat
e
bb dd aa cc $ AA
0
shift
3
goto
2
1
shift
4
reduce
1
reduce
1
accept
2 shift 5
3
shift
4
goto
6
4
reduce
2
reduce
2
5
reduce
3
6 shift 7
7
reduce
4

 Actions:
o Shift: Push the current input symbol onto the stack and transition to the
specified state.
o Reduce: Replace a sequence of symbols with the corresponding non-
terminal using a production rule.
o Accept: Parsing is complete.
 Goto:
o Specifies transitions for non-terminals.
Step 2: Parsing Process
Step Stack Input Action Explanation
1 0
bdabd
a
shift 3
Push bb and move to state
3.
2 0 3 dada shift 4
Push dd and move to state
4.
3 0 3 4 aa
reduce
2
Replace dd with AA.
4 0 3 A aa goto 6
Transition to state 6 on
AA.
5 0 3 A 6 aa shift 5
Push aa and move to state
5.
6
0 3 A 6
5
$
reduce
4
Replace bdabda with SS.
7 0 S $ accept Parsing is complete.
Example 2: Parsing Using Shift-Reduce
Grammar:
E E+T
→ ∣ TE to E + T ,|, T T T
→ ∗F ∣ FT to T * F ,|, F F (E)
→ ∣ idF to (E) ,|, id
Input String:
id+id∗idid + id * id
Step 1: Parsing Process

Step Stack Input Action Explanation
1
id+id∗idid + id *
id
shift
Push idid onto the
stack.
2 idid +id∗id+ id * id reduce F idF to id
→ Replace idid with FF.
3 FF +id∗id+ id * id reduce T FT to F
→ Replace FF with TT.
4 TT +id∗id+ id * id reduce E TE to T
→ Replace TT with EE.
5 EE +id∗id+ id * id shift Push ++ onto the stack.
6 E+E + id∗idid * id shift
Push idid onto the
stack.
7 E+idE + id ∗id* id reduce F idF to id
8 E+FE + F ∗id* id reduce T FT to F
→ Replace FF with TT.
9 E+TE + T ∗id* id shift Push ∗* onto the stack.
10 E+T∗E + T * idid shift
Push idid onto the
stack.
11
E+T∗idE + T *
id
reduce F idF to id
12
E+T∗FE + T *
F
reduce T T
→ ∗FT to T
* F
Replace T∗FT * F with
TT.
13 E+TE + T
reduce E E+TE to E
→
+ T
Replace E+TE + T with
EE.
Key Takeaways
 LALR(1) Parsing:
o Utilizes a compact parsing table and lookahead symbols.
o Efficiently parses strings without ambiguity for LALR(1) grammars.
 Shift-Reduce Parsing:
o Simulates bottom-up parsing using stack operations.
o Ideal for understanding the parsing mechanism without tables.
Would you like a detailed walkthrough of any specific grammar or additional examples?

Q5: RR Conlicts
How do reduce-reduce conflicts occur in LALR Parsing?
Reduce-reduce conflicts occur in bottom-up parsers (like LR or LALR parsers) when
the parsing table contains two or more possible reduce actions for the same input
symbol in the same state. This ambiguity means the parser cannot decide which
production rule to apply.
Why Do Reduce-Reduce Conflicts Occur?
Reduce-reduce conflicts typically arise due to:
1. Ambiguous Grammars:
o A grammar is ambiguous if there exists more than one valid parse tree
for the same input string. Ambiguity can lead to multiple reductions
being valid at a given point.
2. Grammar Design Issues:
o Poorly designed grammars may lead to multiple valid reductions at
intermediate stages, even if the grammar is unambiguous.
3. State Merging in LALR(1) Parsers:
o LALR(1) parsers merge states with identical core items (ignoring
lookaheads initially). If merging combines incompatible lookahead
symbols, it can create reduce-reduce conflicts.
How Reduce-Reduce Conflicts Arise in a Parsing Table
Consider the Grammar:
S A
→ ∣ BS to A ,|, B A aA to a B aB to a
→ →
 This grammar has two ways to derive aa (via AA or BB), even though it is
structurally unambiguous.
Canonical LR(1) Items:
1. S→⋅A,$S to cdot A, $
2. S→⋅B,$S to cdot B, $
3. A→⋅a,$A to cdot a, $

4. B→⋅a,$B to cdot a, $
Parsing Table:
Stat
e
aa $
0 shift 1
1
reduce A aA to a, reduce B aB
→ →
to a
 Conflict:
o In state 1, the parser sees aa and can either reduce A aA to a or B aB
→ →
to a. This is a reduce-reduce conflict because the parser cannot decide
which reduction to apply.
Detecting Reduce-Reduce Conflicts
1. Parsing Table Analysis:
o Examine the Action Table for a state with multiple reduce actions for the
same lookahead symbol.
2. State Merging in LALR(1):
o In LALR(1) parsers, merging states with identical cores but different
lookaheads can cause conflicting reductions.
Resolving Reduce-Reduce Conflicts
1. Refactor the Grammar:
o Rewrite the grammar to remove ambiguity or avoid situations where
multiple reductions are possible.
o Example:
 For the ambiguous grammar above, introduce a precedence rule
to prioritize AA over BB, or merge AA and BB into a single non-
terminal.
2. Use a More Powerful Parser:
o Switch to a canonical LR(1) parser, which may handle the conflicts more
effectively by avoiding state merging.
3. Adjust Grammar Design:

o Add context or lookahead symbols explicitly to disambiguate between
competing reductions.
o Example:
 For the grammar: S A
→ ∣ BS to A ,|, B Add a new rule to
distinguish AA and BB: S aX,X ϵ
→ → ∣ bS to aX, quad X to
epsilon ,|, b
4. Use Semantic Rules:
o If the grammar is inherently ambiguous but intended to prioritize one
reduction, apply semantic rules during parsing to resolve conflicts.
Example: Grammar Causing Reduce-Reduce Conflict
Ambiguous Grammar:
S A
→ ∣ BS to A ,|, B A aA to a B aB to a
→ →
 Conflict: Both AA and BB reduce aa.
Refactored Grammar:
S A,A aX,X ϵ
→ → → ∣ bS to A, quad A to aX, quad X to epsilon ,|, b
 Now, A aXA to aX explicitly resolves ambiguity between AA and BB.
→
Summary
 Reduce-reduce conflicts occur when a parser cannot decide which reduction to
apply due to ambiguity or state merging.
 They can be resolved by refactoring the grammar, improving parser power, or
using semantic disambiguation.
Would you like an example of resolving a specific reduce-reduce conflict in a grammar?

Q6: Non shift-reduce parser
Is there any bottom-up parser that does not use shift-reduce?
Yes, there are bottom-up parsers that do not rely on the traditional shift-reduce
mechanism. While shift-reduce parsing is the most common bottom-up approach
(e.g., SLR, LALR, and LR parsers), other bottom-up parsing techniques exist:
1. CYK Parser (Cocke-Younger-Kasami)
 Overview: The CYK algorithm is a dynamic programming approach for parsing
context-free grammars in Chomsky Normal Form (CNF).
 How it Works: It builds a triangular table where each cell represents a substring
of the input. The parser fills the table bottom-up by combining smaller
substrings into larger ones based on the grammar rules.
 Notable Features:
o Does not use the shift-reduce mechanism.
o More of a general-purpose parser for CNF grammars.
o Used primarily in computational linguistics and natural language
processing.
 Time Complexity: O(n3)O(n^3), where nn is the length of the input string.
2. Chart Parsing (Earley Parser)
 Overview: The Earley parser is a versatile parsing algorithm that works for any
context-free grammar, not restricted to shift-reduce techniques.
 How it Works: It uses a dynamic programming table called a "chart" to track
partially parsed grammar rules, maintaining states for predictions, scans, and
completions.
o Can handle ambiguous grammars and left-recursive rules.
o Operates in a bottom-up or top-down manner depending on
implementation.
 Time Complexity:
o O(n3)O(n^3) in the general case.
o O(n2)O(n^2) for unambiguous grammars.

o O(n)O(n) for simple LL(1) grammars.
3. GLR Parser (Generalized LR)
 Overview: GLR parsing extends traditional LR parsing to handle all context-free
grammars, including ambiguous ones.
 How it Works: Instead of a strict shift-reduce approach, it uses a graph-
structured stack to explore multiple parsing possibilities simultaneously.
o Processes grammars that traditional LR parsers cannot handle.
o Works well in cases of ambiguity, as it can follow multiple parse trees in
parallel.
 Time Complexity: Can vary depending on the grammar, but potentially
O(n3)O(n^3) in the worst case.
4. Parsing Expression Grammar (PEG) Parsers
 Overview: PEG-based parsers use a deterministic approach and are different
from traditional context-free grammar parsers.
 How it Works: Parsing expressions are evaluated in a recursive-descent manner,
often using memoization (Packrat Parsing) to simulate a bottom-up approach.
o Avoids ambiguity by prioritizing alternatives.
o Does not rely on the shift-reduce mechanism.
 Time Complexity: O(n)O(n) with memoization (Packrat).
Key Differences from Shift-Reduce
 No Stack for Tokens: Some parsers like CYK and Earley don't rely on an explicit
stack for shift/reduce operations but instead maintain parsing states or tables.
 Focus on Grammar Rules: Many of these parsers operate directly on the
grammar rules using dynamic programming or state representation.
 Ambiguity Handling: Approaches like GLR and Earley can handle ambiguities
naturally, unlike standard shift-reduce parsers.
These alternatives showcase the diversity in parsing techniques beyond the shift-
reduce paradigm, catering to different needs and grammar types.

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