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We have seen that a lexical analyzer can identify tokens with the help of regular expressions and pattern rules.
It implies that every Regular Grammar is also context-free, but there exists some problems, which are beyond the scope of Regular Grammar.
In this section, we will first see the definition of context-free grammar and introduce terminologies used in parsing technology. One of the non-terminals is designated as the start symbol (S); from where the production begins. The strings are derived from the start symbol by repeatedly replacing a non-terminal (initially the start symbol) by the right side of a production, for that non-terminal.
We take the problem of palindrome language, which cannot be described by means of Regular Expression. This grammar describes palindrome language, such as: 1001, 11100111, 00100, 1010101, 11111, etc.
A syntax analyzer or parser takes the input from a lexical analyzer in the form of token streams. A derivation is basically a sequence of production rules, in order to get the input string. If the sentential form of an input is scanned and replaced from left to right, it is called left-most derivation. If we scan and replace the input with production rules, from right to left, it is known as right-most derivation. A grammar G is said to be ambiguous if it has more than one parse tree (left or right derivation) for at least one string.
If an operand has operators on both sides, the side on which the operator takes this operand is decided by the associativity of those operators.
Operations such as Addition, Multiplication, Subtraction, and Division are left associative. If two different operators share a common operand, the precedence of operators decides which will take the operand.
A grammar becomes left-recursive if it has any non-terminal ‘A’ whose derivation contains ‘A’ itself as the left-most symbol.
A top-down parser will first parse the A, which in-turn will yield a string consisting of A itself and the parser may go into a loop forever.
This does not impact the strings derived from the grammar, but it removes immediate left recursion. Second method is to use the following algorithm, which should eliminate all direct and indirect left recursions. If more than one grammar production rules has a common prefix string, then the top-down parser cannot make a choice as to which of the production it should take to parse the string in hand.
Then it cannot determine which production to follow to parse the string as both productions are starting from the same terminal (or non-terminal). This set is created to know what terminal symbol is derived in the first position by a non-terminal. Likewise, we calculate what terminal symbol immediately follows a non-terminal ? in production rules.

These tasks are accomplished by the semantic analyzer, which we shall study in Semantic Analysis.
This looks complicated, but it’s actually a one-to-one mapping of the information on the pedigree chart.
Exercise: The factorial of a number n (n!) is the product of all positive integers less than or equal to n. The most common practical problem solved with recursion is traversing a nested data structure. The program contains an arbitrary data structure (stuff) that has a number of nested dictionaries: dictionaries whose keys have values that are sometimes strings and sometimes other dictionaries (which themselves can have keys whose values are dictionaries).
Exercise 1: Make a similar program that prints out all elements from a data structure consisting of nested lists.
Exercise 2 (advanced): Make a similar program that prints out all elements from a data structure that consists of nested lists, dictionaries, and sets. Exercise 3 (super advanced): Make a similar program that prints out the string attribute for every tag in a BeautifulSoup document. A human being can understand the structure of these sentences, and understand what the sentences mean. From the examples above, we can see it takes more than just a knowledge of how words occur in sequence to understand (and generate) human language. A symbol that appears on the left side more than once (like VP, N and V above) can be rewritten with the right side of any of the given rules.
If the expansion of a symbol itself contains non-terminal symbols, replace each of those symbols with their expansions.
This isn’t a very sophisticated example, true, but the machinery is in place for producing very complex sentences. The contextfree module defines one function, generate_list, which takes a grammar definition and then recursively generates strings from the grammar.
The code in the __main__ section reads in a grammar in JSON format, passed as standard input. That is: rules are keys, and the values for those keys are all possible expansions (as a list of strings). The generate_list function is where the work gets done: taking an axiom and turning it into a full sentence. First, it creates an empty list (called s); this will contain the expansion for the given axiom. The diagram below illustrates the whole process (again, using our tiny sample grammar as an example).
Each attempt to expand NP would lead to another attempt to expand NP, which would lead to another attempt to expand NP, and so forth. One way to ensure that your context-free grammars will always completely expand is to check to make sure that every symbol has at least one expansion that will eventually lead to terminal symbols. Context Free is “a program that generates images from written instructions called a grammar. But a lexical analyzer cannot check the syntax of a given sentence due to the limitations of the regular expressions.

The productions of a grammar specify the manner in which the terminals and non-terminals can be combined to form strings. The parser analyzes the source code (token stream) against the production rules to detect any errors in the code. The sentential form derived by the left-most derivation is called the left-sentential form. The sentential form derived from the right-most derivation is called the right-sentential form.
The deepest sub-tree is traversed first, therefore the operator in that sub-tree gets precedence over the operator which is in the parent nodes. If the operation is left-associative, then the operand will be taken by the left operator or if the operation is right-associative, the right operator will take the operand. That is, 2+3*4 can have two different parse trees, one corresponding to (2+3)*4 and another corresponding to 2+(3*4).
In this technique, we make one production for each common prefixes and the rest of the derivation is added by new productions. We do not consider what the non-terminal can generate but instead, we see what would be the next terminal symbol that follows the productions of a non-terminal. Lexical analyzers are responsible for the validity of a token supplied by the syntax analyzer. For the purposes of this class, recursion is primarily useful as a strategy for dealing with nested data structures. The sum of a list of numbers can be calculated thusly: add the value of the first item of a list to the sum of the remaining items. A recursive function must, under some circumstance, return from the function without calling itself; otherwise, the recursion would continue forever.
Since a grammar is essentially a map (from rules to productions), it makes sense to use a dictionary to represent the data.
The non-terminals define sets of strings that help define the language generated by the grammar. No method can detect and remove ambiguity automatically, but it can be removed by either re-writing the whole grammar without ambiguity, or by setting and following associativity and precedence constraints. The advantage of recursion is it can elegantly cope with nested data structures of arbitrary depth.
Therefore, this phase uses context-free grammar (CFG), which is recognized by push-down automata. So, when the parser encounters the same non-terminal in its derivation, it becomes hard for it to judge when to stop parsing the left non-terminal and it goes into an infinite loop.
It then splits that expansion into individual tokens and calls generate_list on each of those tokens. Because generate_list returns a list, we use the extend method to add the result of expanding the token to the end of the list.

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