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Introduction & Overview
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Quick Overview
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In this section, the practical implementation of context-free grammars is highlighted through the analysis of the CYK algorithm, demonstrating how it can determine string membership within a context-free language. Various examples and scenarios illustrate the application of grammar rules.
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Introduction to Example Trace
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Let's use a simple grammar G=(S,A,B,a,b,P,S) in CNF:
P:
1. StoAB
2. Atoa
3. Btob
Let the input string be w=βabβ. Length n=2.
Detailed Explanation
In this example trace, we are working with a simple Context-Free Grammar (CFG) represented as G, which consists of:
- Non-terminal symbols: S, A, B
- Terminal symbols: a, b
- Production rules that dictate how these symbols can be combined to produce strings. The production rules are:
1. S produces AB
2. A produces the terminal a
3. B produces the terminal b
We will use an input string, 'ab', which has a length of 2, to demonstrate how the parsing process works using the CYK algorithm.
Examples & Analogies
Think of this grammar like a recipe for making a sandwich. The grammar (G) tells us how to combine ingredients (symbols) to create the final sandwich (string). Just like a sandwich recipe might say: 'Combine bread (S) with lettuce (A) and tomato (B) to create a sandwich (w = βLTβ). In this case, 'ab' is like having a delicious sandwich made up of the ingredients specified in the grammar.
Initializing the Parse Table
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Table T (size 2times2):
Initialize with empty sets:
T[1,1]=emptyset, T[2,1]=emptyset
T[1,2]=emptyset
Detailed Explanation
We start by constructing a triangular table (or matrix) T, which will be used to track which non-terminals can derive each substring of the input string. The size of the table is determined by the length of the input string. At this point, we initialize all entries of the table with empty sets, indicating that we haven't derived any parts of the input string yet. The dimensions of the table for our input string 'ab' will be 2x2, as the length of 'ab' is 2. We will fill this table step by step as we parse the string.
Examples & Analogies
Imagine creating a chart where you keep track of inventory items in a store. At first, the shelves are empty, so your chart has all sections filled with empty boxes. As you receive and organize items, you will fill in those boxes with the type of items that correspond to each section, similar to how we will fill in the matrix as we find applicable non-terminals for each substring.
Processing Length 1 Substrings
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Step 1: Length 1 Substrings (j=1)
β T[1,1] (for substring w_1=βaβ):
β Check rules Xtoβaβ. Rule 2: Atoβaβ.
β So, T[1,1]=A.
β T[2,1] (for substring w_2=βbβ):
β Check rules Xtoβbβ. Rule 3: Btoβbβ.
β So, T[2,1]=B.
Detailed Explanation
In this step, we handle the substrings of length 1, which are individual characters from our input string 'ab'. We fill the first row of the table. For the first character 'a', we look for any production rules that can derive 'a'. We find that A can derive 'a', so we place A in T[1,1]. Similarly, for the second character 'b', we find that B can derive 'b', so we place B in T[2,1]. After this step, our table stores which non-terminals can derive each single-character substring.
Examples & Analogies
Think of this step as taking a list of individual items from your shopping list and checking them off against products available in a store. If you see that 'apples' (the letter 'a') can be bought as 'A', you write 'A' next to 'apples'. The same goes for 'bananas' (the letter 'b') and 'B'. By checking each item carefully, you identify which non-terminals correspond to which substrings.
Processing Length 2 Substring
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Step 2: Length 2 Substrings (j=2)
β T[1,2] (for substring w_1w_2=βabβ):
β Only one possible split point: k=1. (Splits into w_1 and w_2)
β First part: w_1=βaβ. Non-terminals from T[1,1]=A.
β Second part: w_2=βbβ. Non-terminals from T[2,1]=B.
β Consider pairs (P,Q) where PinT[1,1] and QinT[2,1]. Only pair is (A,B).
β Check grammar rules: Is there a rule XtoAB? Yes, Rule 1: StoAB.
β Add S to T[1,2].
β So, T[1,2]=S.
Detailed Explanation
Now, we move on to processing the whole substring 'ab', which has a length of 2. We examine possible ways to split the substring to determine if it can be generated by any non-terminals. We find only one valid split point, which breaks 'ab' into 'a' (w_1) and 'b' (w_2). Non-terminals capable of producing 'a' and 'b' are A and B, respectively. We can combine A and B to check our grammar rules for any rule that permits 'AB'. We find that S can be derived from A and B, so we place S in T[1,2].
Examples & Analogies
This step is like having a recipe for making a dish that requires combining two ingredients together. You first check the items you have (non-terminals youβve filled in before) and see how they combine to meet your recipe requirements (splitting and combining to create a full dish). When you recognize that A (lettuce) and B (tomato) can be combined to make a salad (S), you note that in your preparation chart, showing you can create a complete dish from your ingredients.
Acceptance Check
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Acceptance Check:
Is the start symbol S in T[1,2]? Yes, SinS.
Therefore, the string ab is accepted by the grammar.
Detailed Explanation
In the final step, we make the crucial check to determine if the input string 'ab' is accepted by our grammar. We do this by checking the topmost cell of our table, T[1,2]. If the start symbol S is present in that cell, it indicates that 'ab' is derivable from the grammar, meaning it belongs to the language defined by this CFG. In our case, S is indeed in T[1,2], confirming that the grammar accepts the string 'ab'.
Examples & Analogies
Think of this final check as a manager approving a project proposal based on a checklist of requirements. If all criteria (rules) are met and the proposal (string) includes the necessary components (start symbol), it gets the green light. In this case, since all rules align with your checklist, you confirm that the project can move forward successfully.