Introduction:
A context-free grammar (CFG) is a term used in formal languages theory to describe a certain type of formal grammar. A context-free grammar is a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the rule
A α
Context free grammars (CFGs) provide a formal way to describe the structure of languages by defining rewrite rules for replacing nonterminal symbols with strings of terminals and nonterminals. A CFG consists of variables, terminals, production rules, and a starting variable. Production rules take the form of replacing a single nonterminal with a string. CFGs can describe the recursive structure of natural languages but not agreement or reference. Examples are given for generating well-formed parentheses strings and showing how parse trees and derivations work. Ambiguous grammars that have multiple parse trees for some strings are discussed along with attempts to disambiguate grammars.
The document discusses context free grammars and related concepts. It defines context free grammars and provides examples. It also discusses Chomsky hierarchy, classifying grammars into types 0-3 (unrestricted to regular) based on production rules. Formal languages generated by each grammar type are described along with their properties and closure properties. Context free grammars are defined in more detail, covering derivation, Backus-Naur form, and leftmost and rightmost derivations.
In automata theory, a deterministic pushdown automaton (DPDA or DPA) is a variation of the pushdown automaton. The DPDA accepts the deterministic context-free languages, a proper subset of context-free languages. Machine transitions are based on the current state and input symbol, and also the current topmost symbol of the stack. Symbols lower in the stack are not visible and have no immediate effect. Machine actions include pushing, popping, or replacing the stack top. A deterministic pushdown automaton has at most one legal transition for the same combination of input symbol, state, and top stack symbol. This is where it differs from the nondeterministic pushdown automaton.
The document discusses various topics related to formal languages and automata theory including:
- Definitions of alphabets, strings, regular expressions, and formal languages. Regular expressions can be used to represent regular languages.
- Four types of grammars (Type-0 to Type-3) with Type-3 grammars generating regular languages and Type-2 grammars generating context-free languages.
- Components of a grammar including nonterminal symbols, terminal symbols, rules, and a starting symbol.
- Turing machines and their components including states, tape alphabet, transition function, initial/final states, and blank symbol.
- Decidability and reducibility. The halting problem is un
Context free grammars (CFGs) are formal systems that describe the structure of languages. A CFG consists of variables, terminals, production rules, and a start variable. Production rules take the form of a single variable producing a string of terminals and/or variables. CFGs can capture the recursive structure of natural languages while ignoring agreement and reference. They are used to define context-free languages and generate parse trees. Ambiguous grammars have sentences with multiple parse trees, and disambiguation aims to impose an ordering on derivations. While ambiguity cannot always be eliminated, simplifying and restricting grammars has theoretical and practical benefits.
Types of Language in Theory of ComputationAnkur Singh
This document discusses different types of formal languages in the Chomsky hierarchy:
1. Recursively enumerable languages are generated by Turing machines and include type-0 languages. They are closed under union, concatenation, and Kleene star but not difference or complement.
2. Context-sensitive languages are type-1 languages generated by linear-bounded automata. They are closed under union, intersection, concatenation, and Kleene star.
3. Context-free languages are type-2 languages generated by pushdown automata, including programming language grammars. They are closed under union, concatenation, Kleene star, and reversal.
4. Regular languages are type-3 languages generated by
Context-free languages can be described using context-free grammars, which are recursive rules that generate strings in a language. An example grammar is presented that generates strings of 1s and 0s separated by # symbols. Context-free grammars consist of variables, terminals, rules with variables on the left-hand side replacing with strings on the right-hand side, and a starting variable. Context-free languages can be recognized by pushdown automata using an extra stack. Regular languages are a subset of context-free languages. Context-free languages have closure properties including union, concatenation, and homomorphism. Derivation trees can represent grammar derivations and Backus-Naur form is a notation for compactly representing
Introduction:
A context-free grammar (CFG) is a term used in formal languages theory to describe a certain type of formal grammar. A context-free grammar is a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the rule
A α
Context free grammars (CFGs) provide a formal way to describe the structure of languages by defining rewrite rules for replacing nonterminal symbols with strings of terminals and nonterminals. A CFG consists of variables, terminals, production rules, and a starting variable. Production rules take the form of replacing a single nonterminal with a string. CFGs can describe the recursive structure of natural languages but not agreement or reference. Examples are given for generating well-formed parentheses strings and showing how parse trees and derivations work. Ambiguous grammars that have multiple parse trees for some strings are discussed along with attempts to disambiguate grammars.
The document discusses context free grammars and related concepts. It defines context free grammars and provides examples. It also discusses Chomsky hierarchy, classifying grammars into types 0-3 (unrestricted to regular) based on production rules. Formal languages generated by each grammar type are described along with their properties and closure properties. Context free grammars are defined in more detail, covering derivation, Backus-Naur form, and leftmost and rightmost derivations.
In automata theory, a deterministic pushdown automaton (DPDA or DPA) is a variation of the pushdown automaton. The DPDA accepts the deterministic context-free languages, a proper subset of context-free languages. Machine transitions are based on the current state and input symbol, and also the current topmost symbol of the stack. Symbols lower in the stack are not visible and have no immediate effect. Machine actions include pushing, popping, or replacing the stack top. A deterministic pushdown automaton has at most one legal transition for the same combination of input symbol, state, and top stack symbol. This is where it differs from the nondeterministic pushdown automaton.
The document discusses various topics related to formal languages and automata theory including:
- Definitions of alphabets, strings, regular expressions, and formal languages. Regular expressions can be used to represent regular languages.
- Four types of grammars (Type-0 to Type-3) with Type-3 grammars generating regular languages and Type-2 grammars generating context-free languages.
- Components of a grammar including nonterminal symbols, terminal symbols, rules, and a starting symbol.
- Turing machines and their components including states, tape alphabet, transition function, initial/final states, and blank symbol.
- Decidability and reducibility. The halting problem is un
Context free grammars (CFGs) are formal systems that describe the structure of languages. A CFG consists of variables, terminals, production rules, and a start variable. Production rules take the form of a single variable producing a string of terminals and/or variables. CFGs can capture the recursive structure of natural languages while ignoring agreement and reference. They are used to define context-free languages and generate parse trees. Ambiguous grammars have sentences with multiple parse trees, and disambiguation aims to impose an ordering on derivations. While ambiguity cannot always be eliminated, simplifying and restricting grammars has theoretical and practical benefits.
Types of Language in Theory of ComputationAnkur Singh
This document discusses different types of formal languages in the Chomsky hierarchy:
1. Recursively enumerable languages are generated by Turing machines and include type-0 languages. They are closed under union, concatenation, and Kleene star but not difference or complement.
2. Context-sensitive languages are type-1 languages generated by linear-bounded automata. They are closed under union, intersection, concatenation, and Kleene star.
3. Context-free languages are type-2 languages generated by pushdown automata, including programming language grammars. They are closed under union, concatenation, Kleene star, and reversal.
4. Regular languages are type-3 languages generated by
Context-free languages can be described using context-free grammars, which are recursive rules that generate strings in a language. An example grammar is presented that generates strings of 1s and 0s separated by # symbols. Context-free grammars consist of variables, terminals, rules with variables on the left-hand side replacing with strings on the right-hand side, and a starting variable. Context-free languages can be recognized by pushdown automata using an extra stack. Regular languages are a subset of context-free languages. Context-free languages have closure properties including union, concatenation, and homomorphism. Derivation trees can represent grammar derivations and Backus-Naur form is a notation for compactly representing
Deterministic context free grammars &non-deterministicLeyo Stephen
Deterministic context-free grammars are always unambiguous, while there are non-deterministic unambiguous grammars. The problem of determining if a grammar is ambiguous is undecidable in general. Many languages can have both ambiguous and unambiguous grammars, but some languages only admit ambiguous grammars and are considered inherently ambiguous.
The document discusses context-free languages and context-free grammars. It defines context-free languages as languages generated by context-free grammars. Context-free grammars can be defined as a 4-tuple consisting of variables, terminals, production rules, and a start symbol. The document lists some properties of context-free languages, including that they are closed under union, concatenation, and Kleene star, but not intersection or complement. It also provides examples of languages that are and aren't context-free.
The document discusses syntax analysis and parsing. It defines context-free grammars and different types of grammars. It also discusses derivation, parse trees, ambiguity in grammars and different parsing techniques like top-down and bottom-up parsing.
This document provides an introduction to formal language theory and defines several key concepts:
1) It defines formal languages and grammars, including Chomsky hierarchy which categorizes languages based on grammar complexity.
2) It introduces finite state automata as machines that read input sequences and transition between states, accepting or rejecting words based on reaching an accepting state.
3) It defines concepts like Kleene closure and regular expressions which are used to describe languages recognized by finite state automata.
This document provides an introduction to formal language theory and computational linguistics concepts. It defines key terms like formal grammars, automata, regular expressions, and Chomsky hierarchy. Finite state automata and regular grammars are discussed as the simplest types that can recognize formal languages. Context-free grammars allow more complex languages but are still parseable efficiently, unlike the most complex unrestricted grammars. Transition graphs and diagrams are presented as ways to visually represent automata and their state transitions.
This document introduces the key concepts in the theory of computation, including automata, formal languages, and grammars. It defines automata as abstract models that accept input, process it, and produce output. Formal languages are sets of strings formed from symbols according to rules, and grammars are sets of rules for generating the strings in a language. The document also reviews mathematical concepts needed to study computation and provides examples of operations on strings and languages.
The document summarizes key concepts about context-free grammars and parsing from the book "Compiler Construction: Principles and Practice" by Kenneth C. Louden. It covers notations like EBNF and syntax diagrams for representing grammars, properties of context-free languages, and provides grammar rules and diagrams for a sample TINY language as an example.
This document discusses context-free grammars and languages. It begins by introducing context-free grammars and their components. It then discusses different types of grammars based on production rules and derivation trees. Examples of context-free languages and grammars are provided. The document also covers derivations, derivation trees, simplifying grammars by removing useless symbols and productions. It concludes with discussing ambiguous grammars and normal forms for context-free grammars.
The document discusses the role and process of lexical analysis in compilers. It can be summarized as:
1) Lexical analysis is the first phase of a compiler that reads source code characters and groups them into tokens. It produces a stream of tokens that are passed to the parser.
2) The lexical analyzer matches character sequences against patterns defined by regular expressions to identify lexemes and produce corresponding tokens.
3) Common tokens include keywords, identifiers, constants, and punctuation. The lexical analyzer may interact with the symbol table to handle identifiers.
1) Assume the language is regular and pick a string w in the language that is longer than the pumping length k
2) Show that for any decomposition of w into xyz, there is a pumped string xyiz that is not in the language, violating the pumping lemma
3) Therefore, the language cannot be regular
The pumping lemma provides a structural property about strings in a regular language that can be used to prove a language is not regular by finding a string that cannot be pumped.
This document discusses regular expressions and regular languages. It defines regular expressions as patterns that define strings and notes their connection to regular languages. Regular languages are those that can be described by regular expressions and are accepted by finite automata. The document provides examples of regular expressions for different languages and discusses operations on regular languages like union and intersection. It also covers regular grammars and their properties.
This document summarizes Noam Chomsky's 1957 work defining the Chomsky hierarchy of formal languages. It introduces the four types of grammars - Type-3 (regular), Type-2 (context-free), Type-1 (context-sensitive), and Type-0 (recursively enumerable) - and describes their defining production rules. Context-free grammars, which generate context-free languages, are discussed in more detail. Examples are provided to illustrate context-free grammars and their ability to generate non-regular languages like {anbn}. Pushdown automata, which are equivalent to context-free grammars, are also introduced.
The document discusses formal languages and grammars. It defines key concepts such as alphabets, strings, languages, and regular expressions. Some key points:
- An alphabet is a set of symbols. A string is a finite sequence of symbols from an alphabet.
- A formal language is a set of strings over a given alphabet. Languages can be constructed using operations like union.
- Regular expressions are used to define regular languages recursively, using operators like concatenation and Kleene star.
- A formal grammar is a 4-tuple that can be used to generate a formal language. The language generated by a grammar is the set of strings derived from the start variable using the production rules.
Thoery of Computaion and Chomsky's ClassificationPrafullMisra
The document discusses Chomsky's hierarchy of formal languages and describes the four types of grammars:
1) Type-3 or regular grammars have productions with a single nonterminal on the left-hand side and a single terminal or terminal-nonterminal on the right-hand side.
2) Type-2 or context-free grammars have productions with a nonterminal on the left-hand side and a string of terminals and nonterminals on the right-hand side.
3) Type-1 or context-sensitive grammars have productions where the left-hand and right-hand sides are strings of terminals and nonterminals, but the right-hand side must be non-empty.
4) Type-
This document describes a Synchronized Alternating Pushdown Automaton (SAPDA) that accepts the language of reduplication with a center marker (RCM). The SAPDA utilizes recursive conjunctive transitions to check that the nth letter before the center marker '$' is the same as the nth letter from the end of the string, for all letters n. This allows the SAPDA to accept strings of the form w$w, where w is any string over the alphabet {a,b}. The construction of the SAPDA involves states that check specific letters at specific positions relative to the center marker.
This document discusses Chomsky normal form (CNF), a restricted form for context-free grammars (CFGs) where every production rule is either of the form A → BC or A → a, where A, B, C are variables and a is a terminal symbol. The key advantages of CNF include making the parse tree binary and allowing the determination of whether a string is in the language by exhaustive search. The document outlines the steps to convert any CFG to CNF, including removing epsilon productions, unit productions, and useless symbols. Placing a CFG in CNF allows calculating the depth of the longest branch in a parse tree derivation for a string.
Automata theory - describes to derives string from Context free grammar - derivation and parse tree
normal forms - Chomsky normal form and Griebah normal form
The document provides information about a course on the theory of automata. It includes details such as the course title, prerequisites, duration, lectures, laboratories, and topics to be covered. The topics include finite automata, deterministic finite automata, non-deterministic finite automata, regular expressions, properties of regular languages, context-free grammars, pushdown automata, and Turing machines. It also lists reference books and textbooks, and the marking scheme for the course.
This document provides information about a course on the theory of automata, including:
- The course title is Theory of Computer Science [Automata] and is intended for students pursuing a BCS degree in their fifth semester.
- Key topics that will be covered include finite automata, regular expressions, context-free grammars, pushdown automata, and Turing machines.
- Reference textbooks and materials are listed to support student learning in the course over 18 weeks of lectures and labs.
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Deterministic context-free grammars are always unambiguous, while there are non-deterministic unambiguous grammars. The problem of determining if a grammar is ambiguous is undecidable in general. Many languages can have both ambiguous and unambiguous grammars, but some languages only admit ambiguous grammars and are considered inherently ambiguous.
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1) Lexical analysis is the first phase of a compiler that reads source code characters and groups them into tokens. It produces a stream of tokens that are passed to the parser.
2) The lexical analyzer matches character sequences against patterns defined by regular expressions to identify lexemes and produce corresponding tokens.
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1) Assume the language is regular and pick a string w in the language that is longer than the pumping length k
2) Show that for any decomposition of w into xyz, there is a pumped string xyiz that is not in the language, violating the pumping lemma
3) Therefore, the language cannot be regular
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2) Type-2 or context-free grammars have productions with a nonterminal on the left-hand side and a string of terminals and nonterminals on the right-hand side.
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4) Type-
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This document provides information about a course on the theory of automata, including:
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𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐈𝐂𝐓 𝐢𝐧 𝐞𝐝𝐮𝐜𝐚𝐭𝐢𝐨𝐧:
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Decolonizing Universal Design for LearningFrederic Fovet
UDL has gained in popularity over the last decade both in the K-12 and the post-secondary sectors. The usefulness of UDL to create inclusive learning experiences for the full array of diverse learners has been well documented in the literature, and there is now increasing scholarship examining the process of integrating UDL strategically across organisations. One concern, however, remains under-reported and under-researched. Much of the scholarship on UDL ironically remains while and Eurocentric. Even if UDL, as a discourse, considers the decolonization of the curriculum, it is abundantly clear that the research and advocacy related to UDL originates almost exclusively from the Global North and from a Euro-Caucasian authorship. It is argued that it is high time for the way UDL has been monopolized by Global North scholars and practitioners to be challenged. Voices discussing and framing UDL, from the Global South and Indigenous communities, must be amplified and showcased in order to rectify this glaring imbalance and contradiction.
This session represents an opportunity for the author to reflect on a volume he has just finished editing entitled Decolonizing UDL and to highlight and share insights into the key innovations, promising practices, and calls for change, originating from the Global South and Indigenous Communities, that have woven the canvas of this book. The session seeks to create a space for critical dialogue, for the challenging of existing power dynamics within the UDL scholarship, and for the emergence of transformative voices from underrepresented communities. The workshop will use the UDL principles scrupulously to engage participants in diverse ways (challenging single story approaches to the narrative that surrounds UDL implementation) , as well as offer multiple means of action and expression for them to gain ownership over the key themes and concerns of the session (by encouraging a broad range of interventions, contributions, and stances).
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8+8+8 Rule Of Time Management For Better ProductivityRuchiRathor2
This is a great way to be more productive but a few things to
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Lesson Outcomes:
- students will be able to identify and name various types of ornamental plants commonly used in landscaping and decoration, classifying them based on their characteristics such as foliage, flowering, and growth habits. They will understand the ecological, aesthetic, and economic benefits of ornamental plants, including their roles in improving air quality, providing habitats for wildlife, and enhancing the visual appeal of environments. Additionally, students will demonstrate knowledge of the basic requirements for growing ornamental plants, ensuring they can effectively cultivate and maintain these plants in various settings.
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2. Context-Free Grammar
In linguistics and computer science, a context-free grammar
(CFG) is a formal grammar in which every production rule is of
the form
V → w
where V is a “non-terminal symbol” and w is a “string” consisting
of terminals and/or non-terminals.
The term "context-free" expresses the fact that the non-terminal
V can always be replaced by w, regardless of the context in
which it occurs.
A formal language is context-free if there is a context-free
grammar that generates it.
3. Context-Free Grammar
Context-free grammars are powerful enough to
describe the syntax of most programming
languages; in fact, the syntax of most programming
languages is specified using context-free grammars.
On the other hand, context-free grammars are
simple enough to allow the construction of efficient
parsing algorithms which, for a given string,
determine whether and how it can be generated
from the grammar.
4. Context-Free Grammar
Not all formal languages are context-free.
A well-known counter example is
{ an bn cn : n >= 0 }
the set of strings containing some number of
a's, followed by the same number of b's and
the same number of c's.
5. Context-Free Grammar
Just as any formal grammar, a context-free
grammar G can be defined as a 4-tuple:
G = (Vt ,Vn ,P,S) where
Vt is a finite set of terminals
Vn is a finite set of non-terminals
P is a finite set of production rules
S is an element of Vn, the distinguished
starting non-terminal.
6. elements of P are of the form
Vn → ( Vt U Vn) *
A language L is said to be a Context-Free-Language
(CFL) if its grammar is Context-Free. More precisely,
it is a language whose words, sentences and
phrases are made of symbols and words from a
Context-Free-Grammar.
Usually, CFL is of the form L=L(G).
7. Example 1
A simple context-free grammar is given as:
S → a S b | ε
where | is used to separate multiple options
for the same non-terminal, and ε stands for
the empty string. This grammar generates the
language { an bn : n >= 0 } , which is not
regular.
8. Regular languages
A regular language is a formal language (i.e., a
possibly infinite set of finite sequences of symbols
from a finite alphabet) that satisfies the following
equivalent properties:
it can be accepted by a deterministic finite state
machine
it can be accepted by a nondeterministic finite state
machine
it can be accepted by an alternating finite automaton
it can be described by a regular expression
it can be generated by a regular grammar
it can be generated by a prefix grammar
9. Regular languages
The collection of regular languages over an
alphabet Σ is defined recursively as follows:
the empty language Ø is a regular language.
the empty string language { ε } is a regular
language.
For each a є Σ, the singleton language { a } is a
regular language.
If A and B are regular languages, then A ∩ B
(union), A ○ B (concatenation), and A* (Kleene star)
are regular languages.
No other languages over Σ are regular.
10. Finite languages
Finite languages are:
A specific subset within the class of regular
languages is the finite languages - those
containing only a finite number of words.
These are obviously regular as one can
create a regular expression that is the union
of every word in the language, and thus are
regular.
11. Example 2
A context-free grammar for the language consisting
of all strings over {a,b} which contain a different
number of a's to b's is
S → U | V
U → TaU | TaT
V → TbV | TbT
T → aTbT | bTaT | ε
Here, T can generate all strings with the same
number of a's as b's, U generates all strings with
more a's than b's and V generates all strings with
fewer a's than b's.
12. Example 3
Another example of a context-free language
is
This is not a regular language, but it is
context free as it can be generated by the
following context-free grammar:
S → b S bb | A
A → a A | ε
13. Normal forms
Every context-free grammar that does not generate the empty
string can be transformed into an equivalent one in Chomsky
normal form or Greibach normal form. "Equivalent" here means
that the two grammars generate the same language.
Because of the especially simple form of production rules in
Chomsky Normal Form grammars, this normal form has both
theoretical and practical implications.
For instance, given a context-free grammar, one can use the
Chomsky Normal Form to construct a polynomial-time algorithm
which decides whether a given string is in the language
represented by that grammar or not (the CYK algorithm).
14. Properties of context-free languages
An alternative and equivalent definition of context-
free languages employs non-deterministic push-
down automata: a language is context-free if and
only if it can be accepted by such an automaton.
A language can also be modeled as a set of all
sequences of terminals which are accepted by the
grammar. This model is helpful in understanding set
operations on languages.
The union and concatenation of two context-free
languages is context-free, but the intersection need
not be.
The reverse of a context-free language is context-
free, but the complement need not be.
15. Properties of context-free languages
Every regular language is context-free because it
can be described by a regular grammar.
The intersection of a context-free language and a
regular language is always context-free.
There exist context-sensitive languages which are
not context-free.
To prove that a given language is not context-free,
one may employ the pumping lemma for context-
free languages.
The problem of determining if a context-sensitive
grammar describes a context-free language is
undecidable.
16. Normal forms for Context-Free
Grammars
The goal is to show that every CFL (without ε)
is generated by a CFG in which all
productions are of the form A BC or A a,
where A, B, C are variables, and a is a
terminal.
17. Normal forms for Context-Free
Grammars
A number of simplifications is inevitable:
1. The elimination of useless symbols,
“variables or terminals that do not appear in
any derivation of a terminal string from the
start symbol”.
2. The elimination of ε-productions, those of the
form A ε for some variable A.
3. The elimination of unit productions, those of
the form A B for variables A and B.
18. Eliminating useless symbols
A symbol X is useful for Grammar
G = {V, T, P, S}, if there is some derivation of
the form S ═>* a X b ═>* w , where w є T*
X є V or X є T
The sentential form of a X b might be the
first or last derivation
If X is not useful, then X is useless
19. Eliminating useless symbols
Characteristics of useful symbols (for instance X):
1. X is generating if X ═>* w for some terminal
string w. Every terminal is generating since w can
be that terminal itself, which is derived by 0 steps.
2. X is reachable if there is a derivation
S ═>* a X b for some a and b.
A symbol which is useful is surely to be both
generating and reachable.
20. Eliminating useless symbols
Eliminating the symbols which are not
generating first followed by eliminating the
symbols which are not reachable from the
remaining grammar, this will generate a
grammar consisting of only useful symbols.
22. Eliminating useless symbols
Example 7.1
Notice that a and b generate themselves
“terminals”, S generates a, and A
generates b. B is not generating.
After eliminating B:
23. Eliminating useless symbols
Example 7.1
Notice that only S and a are reachable after
eliminating the non-generating B.
A is not reachable; so it should be eliminated.
The result :
This production itself is a grammar that has the
same result, which is {a}, as the original
grammar.
24. Computing the generating and reachable
symbols
Basis: Every Symbol of T is obviously generating; it
generates itself.
Induction: If we have a production A → a, and every
symbol of a is already known to be generating,
then A is generating; because it generates all and
only generating symbols, even if a = ε ; since all
variables that have ε as a production body are
generating.
Theorem: The previous algorithm finds all and only
the Generating symbols of G
25. Computing the generating and reachable
symbols
Basis : For a grammar G = {V, T, P, S}
S is surely reachable.
Induction: If we discovered that some variable A is
reachable, then for all productions with A in the
head (first part of the expression), all the symbols of
the bodies (second part of the expression) of those
productions are also reachable.
Theorem: The above algorithm finds all and only the
Reachable symbols of G
26. Eliminating useless symbols
So far, the first step, which is the elimination
of useless symbols is concluded.
Now, for the second part, which is the
elimination of ε-productions.
27. Eliminating ε-productions
The strategy is to have the following:
if L is CFG, then L – {ε} is also CFG
This is done through discovering the nullable
variables. A variable for instance A, is
nullable if: A ═>* ε .
Whenever A appears in a production body, A
might or might not derive ε
28. Eliminating ε-productions
Basis: If A ε is a production of G, then A
is nullable
Induction: If there is a production
B C1 C2 … Ck such that each C is a
variable and each C is nullable, then B is
nullable
29. Eliminating ε-productions
Theorem: For any grammar G, the only nullable symbols
are the variables that derive ε in previous algorithm
Proof:
for one step : A ε must be a production, then this implies
that A is discovered as nullable (as in basis).
for N > 1 steps: the first step is A C1 C2 … Ck ε , each
Ci derives ε by a sequence < N steps.
By the induction, each Ci is discovered by the algorithm to
be nullable. So by the inductive step, A is eventually found
to be nullable.
30. Eliminating ε-productions
If a grammar G1 is constructed by the
elimination of ε-productions “ using the
previous method ” of grammar G, then
L(G1) = L(G) - {ε}
31. Eliminating unit productions
The last part concerns the eliminating of unit
productions
Any production of the form A B , where A
and B are variables, is called a unit
production.
These production introduce extra steps in the
derivations that obviously are not needed in
there.
32. Eliminating unit productions
Basis: (A, A) is a unit pair of any variable A, if
A ═>* A by 0 steps.
Induction: Let’s (A, B) be a unit pair, and let B C
is a production, where A, B, and C are variables,
then we can conclude that
(A, C) is also a unit pair.
Theorem: The previous algorithm (basis and
induction) finds exactly all the unit pairs for any
grammar G.
35. Eliminating unit productions
Example 7.12
After eliminating the unit productions, the generated
grammar is:
This grammar has no unit productions and still
generates the same expressions as the previous one.
36. Chomsky Normal Form
Conclusion of all three elimination stages:
Theorem: If G is a CFG which generates a
language that consists of at least one string
along with ε, then there is another CFG G1
such that:
L{G1} = L{G} – {ε} , “no ε-productions”, and G1
has neither unit productions nor useless
symbols
37. Chomsky Normal Form
Proof: Start by performing the elimination of
ε-productions. Then perform the elimination
of unit productions, so the resulting grammar
won’t introduce any ε-productions since the
new bodies are still identical to some bodies
of the old grammar. Finally, perform the
elimination of useless symbols, and since this
eliminates productions and symbols, it will
never reintroduce any ε-productions nor unit
productions
38. Chomsky Normal Form
Every nonempty CFL without ε has grammar G in
which all productions are in one of the following
forms:
A BC , where A, B, and C are variables or
A a , where A is a variable and a is a terminal
Also G doesn’t contain any useless symbols
A grammar complying to these forms is called a
Chomsky Normal Form (CNF).
39. Chomsky Normal Form
The construction of CNF is performed
through:
1. Arrangement of all bodies of length 2 or
more to contain only variables.
2. Breaking bodies of length 3 or more into a
cascade productions, where each one has a
body consisting of 2 variables.