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Introduction to Derivations
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Today, we're going to explore **derivations** in grammar, particularly leftmost and rightmost derivations. Can someone tell me what they think a derivation involves in grammar?
Isn't it about how we can generate strings from grammar rules?
Exactly! A derivation starts from the start symbol and applies production rules to generate strings. Now, do any of you know the difference between leftmost and rightmost derivation?
I think leftmost means you always expand the leftmost non-terminal first, while rightmost is the opposite?
That's spot on! In leftmost derivation, we always choose the leftmost non-terminal to expand. Let's remember this as **L for Left** and **L for Least** since we expand the least deep node first. What example of a grammar can illustrate this?
Could we use something simple like expressions? Like 'a + b'?
Yes, a great place to start. Weβll see how 'a + b' can be derived using both methods later in our sessions!
Leftmost Derivation Process
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Letβs proceed with a leftmost derivation using a simple grammar for expressions. We can use rules for expressions such as Expression β Term Expression'. Who wants to walk us through this?
Okay! We start with 'Expression'. Expanding that gives us 'Term Expression''.
Good! Now what is our next step?
We apply the rule for Term next, giving us 'Factor Term' for the leftmost non-terminal. Then we still have Expression'.
Excellent progression. What if we apply an ID token for 'Factor' next?
That gives us 'ID Term Expression''.
Fantastic! And if we substitute 'b' for ID, what is our current sentence?
It would be 'b Term Expression'' right now.
Perfect! So, our leftmost derivation builds up systematically this way. Let's iterate on this concept with rightmost derivation next!
Rightmost Derivation Process
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Now letβs tackle rightmost derivation on the same grammar we just used. Who remembers the first step?
We start with just 'Expression' again?
Correct! Then we expand to 'Term Expression'' as we have before. What's different in rightmost derivation?
We should choose to expand the rightmost non-terminal?
Exactly! So what happens next?
We apply the rule for Expression', which is 'Ξ΅' or a '+' leading term next.
Right! So if we were to say we opt for ID as 'c', what do we have now after we apply those?
The result should be 'Term + ID' or 'a + b'!
Exactly! Itβs important to note that both paths will construct the same final sentence, but the methods differ significantly.
Practical Example and Comparison
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Now let's visualize how both derivation types lead to the same parse tree. What string should we derive as an example?
Can we check 'a + b * c'?
Absolutely! So, recall our previous derivation methods. What would the initial expansion look like for both leftmost and rightmost at the beginning?
Well, it starts with 'Expression'.
Right, from there, leftmost would go to 'Term Expression'' and the rightmost would follow to 'Term + Term'.
Correct! And how would we summarize what each final sequence looks like?
Each gives the same final output 'a + b * c', but we just approached it differently.
Exactly! It shows how different paths through our grammar can still converge onto the same tree. Now we understand the power of the algorithmic process of derivations!
Ambiguity in Grammar
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Lastly, letβs look at ambiguity in grammar. What do we mean by ambiguity in the context of derivation?
I think it means a sentence can be derived in more than one way?
Precisely! This leads us to double interpretations of strings. Can someone provide an example?
Like the expression `A - B * C`, which could either mean `(A - B) * C` or `A - (B * C)`?
Right on target! Ambiguity is problematic in programming languages as it complicates the compiler's task. So how can we resolve these ambiguities?
By establishing precedence and associativity rules for operations, right?
Exactly! This ensures that while constructing parse trees, each string has a single interpretation, leading to consistent compiler behavior.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section highlights how leftmost and rightmost derivations lead to different paths in constructing the same parse tree. It explains the procedural steps for deriving expressions using these strategies and their significance in parsing algorithms, providing a clear approach to understanding the choice involved when multiple non-terminals appear in a sentential form.
Detailed
In the domain of syntax analysis, leftmost and rightmost derivations serve as foundational techniques for understanding how grammars generate strings and how parsers operate. At the core of derivation is the idea that a sentential form contains non-terminals that can be expanded based on production rules of a context-free grammar. A leftmost derivation selects the leftmost non-terminal at each step for expansion, while a rightmost derivation expands the rightmost non-terminal first. This section provides methods to illustrate these derivations through practical examples, demonstrating how they culminate in the formation of parse trees. Importantly, it highlights the fact that different sequences of derivations yield the same final string, resulting in identical parse trees but differing execution paths. Understanding these processes is vital for mastering grammar parsing techniques in compiler design.
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Introduction to Derivations
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When we perform a derivation, if a sentential form contains more than one non-terminal, we have a choice about which one to expand (replace with its production rule) next. This choice leads to different paths for constructing the same parse tree.
Detailed Explanation
When deriving strings from a context-free grammar (CFG), we may encounter situations where a sentential form contains multiple non-terminal symbols. This introduces a choice. For instance, if we have a non-terminal that can produce several different strings according to its production rules, we need to decide which specific rule to apply at that moment. This decision can lead down different paths in the derivation process, resulting in potentially different parse trees that yield the same final string.
Examples & Analogies
Imagine you are baking a cake and have several types of frosting to choose from. Each frosting represents a production rule. Depending on your choice, the cake's final appearance (which represents the derived string) may differ even though the core of the cake (the non-terminal) remains the same.
Leftmost Derivation Explained
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Leftmost Derivation: In a leftmost derivation, at each step, we always choose the leftmost non-terminal in the current sentential form to replace with its right-hand side. This is a common strategy for top-down parsers.
Detailed Explanation
In a leftmost derivation, the focus is on expanding the leftmost non-terminal first. This means that at each derivation step, you inspect the current string (or sentential form) from left to right and replace the first non-terminal you encounter with its corresponding production rule. This strategy aligns well with top-down parsing techniques, where the parser tries to build the structure of the input string by starting from the highest-level non-terminal and working downwards.
Examples & Analogies
Think of it as reading a book from beginning to end. When you encounter a chapter title (the leftmost non-terminal), you dive deeper into that chapter (expand it) before moving on to the next title.
Example of Leftmost Derivation
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Example: Deriving a + b * c from Expression (using our rewritten arithmetic rules, for clarity):
1. Grammar rules relevant for this example:
- Expression -> Term Expression'
- Expression' -> + Term Expression' | Ξ΅
- Term -> Factor Term'
- Term' -> * Factor Term' | Ξ΅
- Factor -> ID
2. Expression (Start)
3. Term Expression' (Applied Expression -> Term Expression', leftmost Expression)
4. Factor Term' Expression' (Applied Term -> Factor Term', leftmost Term)
5. ID Term' Expression' (Applied Factor -> ID, leftmost Factor) - Let's say ID becomes a
6. a Term' Expression'
7. a Expression' (Applied Term' -> Ξ΅, leftmost Term')
8. a + Term Expression' (Applied Expression' -> + Term Expression', leftmost Expression')
9. a + Factor Term' Expression' (Applied Term -> Factor Term', leftmost Term)
10. a + ID Term' Expression' (Applied Factor -> ID, leftmost Factor) - Let's say ID becomes b
11. a + b Term' Expression'
12. a + b * Factor Term' Expression' (Applied Term' -> * Factor Term', leftmost Term')
13. a + b * ID Term' Expression' (Applied Factor -> ID, leftmost Factor) - Let's say ID becomes c
14. a + b * c Term' Expression'
15. a + b * c Expression' (Applied Term' -> Ξ΅, leftmost Term')
16. a + b * c (Applied Expression' -> Ξ΅, leftmost Expression')
Detailed Explanation
The leftmost derivation for the expression 'a + b * c' begins with the start non-terminal 'Expression'. At each step, we replace the leftmost non-terminal with its production rule. For example, we first transform 'Expression' into 'Term Expression'' and then keep applying the production rules to expand 'Term' and 'Expression'' until we arrive at the final string. Each application of a rule follows our leftmost strategy, emphasizing how the structure of the derivation mirrors the final string output.
Examples & Analogies
Consider building a Lego model. You start with the largest block on the left (the leftmost non-terminal) and continue to attach smaller blocks to the left. Each time you make a new addition, you ensure the leftmost part is complete before moving towards the right.
Rightmost Derivation Explained
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Rightmost Derivation (Canonical Derivation): In a rightmost derivation, at each step, we always choose the rightmost non-terminal in the current sentential form to replace with its right-hand side.
Detailed Explanation
In contrast to the leftmost derivation, the rightmost derivation focuses on the rightmost non-terminal. Each derivation step processes from right to left, replacing the non-terminal that appears last. This approach is usually favored by bottom-up parsers and results in structurally similar, yet distinct parse trees compared to leftmost derivations.
Examples & Analogies
Imagine assembling a complex model, starting from the smallest pieces. You work from the end towards the beginning, ensuring that each piece fits before moving on to the next larger part. This is like replacing the rightmost non-terminal until the entire structure is complete.
Example of Rightmost Derivation
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Example: Deriving a + b * c from Expression (using the same grammar rules as above):
1. Expression (Start)
2. Term Expression' (Applied Expression -> Term Expression', rightmost Expression')
3. Term + Term Expression' (Applied Expression' -> + Term Expression', rightmost Expression')
4. Term + Term * Factor Term' Expression' (Applied Term' -> * Factor Term', rightmost Term')
5. Term + Term * ID Term' Expression' (Applied Factor -> ID, rightmost Factor) - Let's say ID becomes c
6. Term + Term * c Term' Expression'
7. Term + Factor * c Term' Expression' (Applied Term -> Factor Term', rightmost Term')
8. Term + ID * c Term' Expression' (Applied Factor -> ID, rightmost Factor) - Let's say ID becomes b
9. Term + b * c Term' Expression'
10. Factor + b * c Term' Expression' (Applied Term -> Factor Term', rightmost Term')
11. ID + b * c Term' Expression' (Applied Factor -> ID, rightmost Factor) - Let's say ID becomes a
12. a + b * c Term' Expression'
13. a + b * c Expression' (Applied Term' -> Ξ΅, rightmost Term')
14. a + b * c (Applied Expression' -> Ξ΅, rightmost Expression')
Detailed Explanation
The rightmost derivation for 'a + b * c' follows a similar path as the leftmost version but with distinct choices. Starting from 'Expression', we focus on the rightmost non-terminal. We apply the relevant production rules until only terminal symbols remain. The key aspect of this derivation is that despite the different order of substitutions compared to the leftmost derivation, we end up with the same final string.
Examples & Analogies
Think of crafting a two-part ornament. You first complete the detail on the back and only then add the more prominent features on the front, building from the back to the front. This is akin to working with the rightmost parts first in a derivation.
Final Thoughts on Derivations
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Notice that both derivations produce the exact same final string and would result in the same parse tree, even though the order of rule applications is different.
Detailed Explanation
Both leftmost and rightmost derivations ultimately produce the same final string, demonstrating that even with different paths and orders of rules applied, the context-free grammar ensures these strings are equivalent. The choice between leftmost and rightmost derivations primarily affects the parse trees generated rather than the final outcome.
Examples & Analogies
Consider following two recipes that yield the same dish but differ in the preparation steps. One recipe might list tasks in a left-to-right order, while the other works right-to-left. Regardless of the method used, the ultimate goal of serving a delicious dish remains unchanged, just as both derivation methods achieve the same resulting string.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Derivation: The core process of generating strings from a grammar.
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Leftmost Derivation: Expanding the leftmost non-terminal first.
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Rightmost Derivation: Expanding the rightmost non-terminal first.
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Sentential Form: A production generated in the process of derivation.
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Parse Tree: Represents the structure and generation of a string from a grammar.
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Ambiguous Grammar: A grammar that allows multiple derivations for the same string.
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Precedence and Associativity: Rules that dictate operation order and grouping in expressions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example using grammar rules: For 'a + b * c', leftmost derivation may start as 'Expression -> Term Expression' while the rightmost starts as 'Expression -> Term + Term'.
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Ambiguous grammar example: The expression 'A - B * C' can be derived as '(A - B) * C' or 'A - (B * C)', highlighting ambiguity.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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For leftmost, we go left, for rightmost we go right, each brings us to the same end, within grammar's light.
π Fascinating Stories
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Imagine a tree where each branch is a choice, leftmost means you pick the branch on the left, and rightmost takes the rightβeach path leads to fruit, but which one will be ripe?
π§ Other Memory Gems
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To remember LEFT vs RIGHT, think about 'Left is Less Independent, Right is Ready to Go'.
π― Super Acronyms
DIR
- **D**erive using **I**ndependent **R**ules for parsing.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Derivation
Definition:
The process of generating a string from a grammar starting from the start symbol and applying production rules.
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Term: Leftmost Derivation
Definition:
A type of derivation where the leftmost non-terminal is expanded first in each step.
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Term: Rightmost Derivation
Definition:
A type of derivation where the rightmost non-terminal is expanded first in each step.
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Term: Sentential Form
Definition:
A string derived from the start symbol which may contain both non-terminals and terminals.
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Term: Parse Tree
Definition:
A hierarchical representation that shows how a string is generated based on a grammar.
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Term: Ambiguous Grammar
Definition:
A grammar that can generate the same string in more than one way, leading to multiple valid parse trees.
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Term: Precedence Rules
Definition:
Rules that define the order of operations in expressions to resolve ambiguities in grammar.
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Term: Associativity Rules
Definition:
Rules that define how operators of the same precedence are grouped in expressions.