2.3 context free grammars and languages
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The document discusses context-free grammars and languages. It defines a context-free grammar as a quadruple consisting of non-terminal symbols, terminals, production rules, and a start symbol. Examples of context-free grammars are provided. Problems involving constructing context-free grammars for various languages are listed, along with their answers.
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2.4 derivations and languages
This document discusses derivations and languages in context-free grammars. It covers recursive inference versus derivation to determine if a string belongs to a language. It also discusses leftmost and rightmost derivations, derivation trees, sentential forms, and left and right recursive grammars. Several example problems are provided to demonstrate derivations, parse trees, and generating strings from given grammars.
2.5 ambiguity in context free grammars
The document discusses ambiguity in context-free grammars. It provides examples of ambiguous grammars, including one where the grammar G with production rules X → X+X | X*X |X|a is shown to be ambiguous because the string "a+a*a" has two different leftmost derivation trees. It then lists four other grammars and briefly states that each one is ambiguous without showing the derivations.
2.8 normal forms gnf & problems
Greibach Normal Form (GNF) is a type of context-free grammar where the right-hand side of each production rule consists of a single terminal symbol followed by zero or more non-terminal symbols. The document discusses GNF, provides an example grammar in GNF, describes two lemmas used to convert grammars to GNF, and shows the procedure and steps to convert an arbitrary context-free grammar into GNF. It also provides examples of converting several grammars to GNF and solving related problems.
Grammar generated by language iv
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's and c's.
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The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's and c's.
Grammar generated by language ii
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's, and c's.
Grammar generated by language iii
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's and c's.
Grammar generated by language
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's, and c's.
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This document discusses derivations and languages in context-free grammars. It covers recursive inference versus derivation to determine if a string belongs to a language. It also discusses leftmost and rightmost derivations, derivation trees, sentential forms, and left and right recursive grammars. Several example problems are provided to demonstrate derivations, parse trees, and generating strings from given grammars.
2.5 ambiguity in context free grammars
The document discusses ambiguity in context-free grammars. It provides examples of ambiguous grammars, including one where the grammar G with production rules X → X+X | X*X |X|a is shown to be ambiguous because the string "a+a*a" has two different leftmost derivation trees. It then lists four other grammars and briefly states that each one is ambiguous without showing the derivations.
2.8 normal forms gnf & problems
Greibach Normal Form (GNF) is a type of context-free grammar where the right-hand side of each production rule consists of a single terminal symbol followed by zero or more non-terminal symbols. The document discusses GNF, provides an example grammar in GNF, describes two lemmas used to convert grammars to GNF, and shows the procedure and steps to convert an arbitrary context-free grammar into GNF. It also provides examples of converting several grammars to GNF and solving related problems.
Grammar generated by language iv
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's and c's.
Grammar generated by language example
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's and c's.
Grammar generated by language ii
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's, and c's.
Grammar generated by language iii
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's and c's.
Grammar generated by language
The document discusses different types of grammars and languages:
- Type 3 grammars generate regular languages using rules of the form A → aB or A → a.
- Type 2 grammars generate context-free languages using rules of the form A → α.
- Type 1 grammars generate context-sensitive languages using rules of the form α → β where |α| ≤ |β|.
- Type 0 grammars generate recursively enumerable languages using rules of the form α → β.
The document provides two examples: one grammar generates the language of strings with a power of 2 a's, and another generates the language of strings with equal numbers of a's, b's, and c's.
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5.2 primitive recursive functions
The document discusses primitive recursive functions and predicates. It defines primitive recursive functions as those that can be constructed from initial functions using only composition and recursion. Some examples of primitive recursive functions given are addition, multiplication, factorial, power, predecessor, absolute value, and bounded quantification. Predicates like equality, less than, negation, conjunction, disjunction, division, and primality are also shown to be primitive recursive. The concept of bounded minimalization is introduced, where a function returns the least value t for which a primitive recursive predicate P(t,x1,...xn) is true.
4.7. chomskian hierarchy of languages
Noam Chomsky classified languages into four types (Type 0-3) based on the type of grammar and automaton that can describe them. Type 0 languages have the most expressive power and are recursively enumerable, while Type 3 languages have the least expressive power and are regular. The types differ based on the grammar and machines used to generate and recognize the language.
4.6 halting problem
The document discusses the halting problem and decidability of languages. It defines a decidable language as one where a Turing machine can accept or reject every input string and halt. The halting problem asks whether it is possible to determine if an arbitrary Turing machine will halt on a given input. The document proves the halting problem is undecidable by constructing a machine that contradicts itself, showing no algorithm can correctly determine if all Turing machines halt on all inputs.
4.3 techniques for turing machines construction
This document discusses techniques for constructing Turing machines including storing finite control in storage, using multiple tracks, checking off symbols, and implementing subroutines. It was presented by Sampath Kumar S on November 21, 2017 to discuss methods for building Turing machines.
4.2 variantsof turing machines (types of tm)
This document discusses different types of Turing machines. It describes multi-tape/head Turing machines which have multiple tapes that each have their own independent read/write head. It also describes multi-track Turing machines which have a single tape head that can read and write to multiple tracks on a single tape simultaneously. The document is a presentation on Turing machines given by Sampath Kumar S on November 21, 2017.
4.1 turing machines
A Turing machine is a mathematical model of computation that consists of an infinite tape divided into cells, a head that reads and writes symbols on the tape, and a finite set of states. Alan Turing invented the Turing machine in 1936 to formalize the idea of algorithmic computation. A Turing machine operates by reading a symbol from the tape according to its transition function, then writing a symbol, changing its state, and moving the head left or right. If the machine reaches an accepting state, the input is accepted, otherwise it is rejected.
3.6 & 7. pumping lemma for cfl & problems based on pl
The document discusses the pumping lemma for context-free grammars (CFGs). It defines the pumping lemma, which states that for any CFG there is a pumping length n such that any string in the language of length >= n can be broken into five parts that satisfy certain properties related to pumping. The document gives examples of using the pumping lemma to determine whether languages are context-free or not, such as the language L={0n1n2n | n >= 1}. It concludes by listing additional problems involving using the pumping lemma to prove languages are not context-free.
3.5 equivalence of pushdown automata and cfl
The document discusses the equivalence between context-free grammars (CFGs) and pushdown automata (PDAs). It states that for any CFG, an equivalent PDA can be constructed to accept the language generated by the grammar, and vice versa. This allows a programming language to be specified by a CFG and implemented with a PDA in a compiler. The document also provides procedures for converting between CFGs and PDAs, including an example of constructing a PDA from a given CFG.
3.4 deterministic pda
The document discusses deterministic pushdown automata (DPDA) and provides examples of DPDAs that accept specific languages. It begins by defining DPDA as a PDA that has at most one choice of move in any state, as opposed to a non-deterministic PDA. It then gives an example of a DPDA that accepts the language L={anbn|n>1} and explains its transition function. Finally, it lists several other language recognition problems and invites the reader to design DPDAs to solve them.
3.1,2,3 pushdown automata definition, moves & id
A pushdown automaton (PDA) is a type of finite state machine that uses a stack to remember an infinite amount of information. A PDA has three components: an input tape, a control unit, and a stack with infinite size. It can perform two stack operations - push, which adds a new symbol to the top of the stack, and pop, which reads and removes the top symbol. A PDA is formally defined as a 7-tuple that specifies its states, alphabets, transition function, initial state, initial stack symbol, and accepting states. A PDA accepts a language either by entering an accepting state after consuming the input string or by having an empty stack when reading is complete.
2.7 normal forms cnf & problems
The document discusses Chomsky Normal Form (CNF) for context-free grammars. A grammar is in CNF if productions are of two forms: A → BC, with exactly two nonterminals on the right-hand side, or A → a, with a single terminal. The document outlines a procedure to convert any grammar into CNF by introducing new nonterminals to restrict the number and type of symbols on the right-hand side of productions. Several examples of converting grammars to CNF are provided.
2.1 & 2.2 grammar introduction – types of grammar
The document discusses different types of grammars according to Chomsky's classification. It defines a grammar as a 4-tuple consisting of non-terminal symbols, terminal symbols, a start symbol, and production rules. Type-3 grammars generate regular languages using productions with a single non-terminal and terminal. Type-2 grammars are context-free using productions of strings. Type-1 grammars are context-sensitive with restricted productions. Type-0 grammars are unrestricted and generate recursively enumerable languages.
1.10. pumping lemma for regular sets
The document discusses the pumping lemma for regular sets. It states that for any regular language L, there exists a constant n such that any string w in L of length at least n can be broken down into sections xyz such that y is not empty, xy is less than or equal to n, and xykz is in L for all k. The pumping lemma can be used to show a language is not regular by finding a string that does not satisfy the lemma conditions. Examples are provided to demonstrate how to use the pumping lemma to prove languages are not regular.
1.9. minimization of dfa
The document describes the process of minimizing a deterministic finite automaton (DFA) using the Myhill-Nerode theorem. It involves 4 steps: 1) creating a table of all state pairs, 2) marking pairs where one state is final and the other is not, 3) transitively marking additional pairs, 4) combining any remaining unmarked pairs into single states. An example is provided where the given 6-state DFA is minimized to a 3-state DFA using this process. Terminology used in DFA minimization is also defined.
1.8. equivalence of finite automaton and regular expressions
This document discusses Thompson's construction algorithm for converting regular expressions to equivalent nondeterministic finite automata (NFAs). It explains the rules for converting the operators in regular expressions, such as empty strings, symbols, union, concatenation, and Kleene star, into NFA transitions and states. Examples of converting regular expressions to NFAs are provided for practice. Thompson's construction allows regular expressions to be matched against strings using the resulting NFA.
1.7. eqivalence of nfa and dfa
The document discusses the conversion of non-deterministic finite automata (NFA) to deterministic finite automata (DFA). It states that an NFA with epsilon transitions can first be converted to an NFA without epsilon transitions, which can then be converted to an equivalent DFA. The conversion of NFA to DFA involves constructing a DFA with up to 2^n states for an n-state NFA, with the same input alphabet and transition and acceptance functions based on the NFA. Examples are provided of converting epsilon NFAs to NFAs without epsilon transitions and NFAs to equivalent DFAs.
1.5 & 1.6 regular languages & regular expression
This document discusses regular languages and regular expressions. It defines regular expressions as notations used to represent regular languages, which are sets of strings that are either finite or can be generated using simple recursive rules. The document then provides the formal definitions and operations of regular expressions, including empty languages, single characters, unions, concatenations, and closures. It concludes by listing 37 problems involving writing regular expressions for specific languages over given alphabets.
1.4. finite automaton with ε moves
1) An epsilon-NFA (ε-NFA) allows instantaneous transitions between states without reading an input symbol, represented by epsilon (ε) labels on arcs.
2) An ε-NFA is formally defined as a 5-tuple (Q, Σ, δ, q0, F) where Q is states, Σ is the alphabet, δ is the transition function, q0 is the initial state, and F are final states. Epsilon is not a member of the alphabet.
3) The epsilon-closure of a state p is the set of all states reachable from p through epsilon transitions.
1.3.2 non deterministic finite automaton
This document defines and provides examples of non-deterministic finite automata (NFA). It states that in an NFA, the transition to the next state for an input symbol is non-deterministic, meaning it can be any combination of states, unlike a DFA which transitions to a single state. An NFA is formally defined as a 5-tuple with states, input symbols, transition function, initial state, and final states. The document provides an example NFA and compares NFAs to DFAs. It concludes with practice problems for designing NFAs for different languages.
1.3.1 deterministic finite automaton
This document defines and provides examples of deterministic finite automata (DFAs). It begins by defining a DFA as a finite state machine where the transition function is deterministic. Formally, a DFA is defined as a 5-tuple consisting of a finite set of states, a finite input alphabet, a transition function, an initial state, and a set of final/accepting states. The document provides examples of representing DFAs using state diagrams and transition tables and discusses how to represent the extended transition function. It concludes by posing several problems about designing DFAs with certain language definitions.
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2.3 context free grammars and languages
- 1. Context Free Grammars and Languages -Sampath Kumar S, AP/CSE, SECE
- 2. Definition of CFG A context-free grammar (CFG) consisting of a finite set of grammar rules is a quadruple G= (N, T, P, S) where N is a set of Non-Terminal symbols. T is a set of Terminals where N ∩ T = NULL. P is a set of Production rules, P: N → (N ∪ T)* S is the Start symbol. 11/21/2017 Sampath Kumar S, AP/CSE, SECE 2
- 3. Example The grammar G= ({A}, {a, b, c}, P, A), P : A → aA, A → abc. The grammar G=({S}, {a, b}, P, S), P: S → aSa, S → bSb, S → ε The grammar G=({S, F}, {0, 1}, P, S), P: S → 00S | 11F, F → 00F | ε 11/21/20173 Sampath Kumar S, AP/CSE, SECE
- 4. Problems to discuss: 52. What is language of CFG G=({A,B}, {a,b}, P, A) where P: A → Ba, B → b. 53. Construct the CFG for Regular Expression (0+1)*. 54. Construct the CFG for defining a palindrome over {a, b}. 55. Construct the CFG for the set of strings with equal number of a’s and b’s. 56. Construct the CFG for the language L(G)={anb2n where n>1}. 11/21/20174 Sampath Kumar S, AP/CSE, SECE
- 5. Problems to discuss: 57. Construct the CFG for the language containing all string of different 1st and last symbol over Σ = {0,1}. 58. Give CFG for R.E (a+b)*cc(a+b)*. 59. What is language of CFG G=({S,C}, {a,b},P,S) where P: S → aCa, C → aCa|b. 60. What is language of CFG G=({S}, {0,1},P,S) where P: S → 0S1 | ε. 11/21/2017 Sampath Kumar S, AP/CSE, SECE 5
- 6. Answers: 52. L(G) = {ab} 53. P = { S → 0S | 1S |ε } 54. P = { S → aSb | bSa | a | b |ε } 55. P = { S → SaSbS | SbSaS |ε } 56. P = { S → aSbb |abb } 57. P = { S → 0A1 | 1A0 , A → 0A | 1A |ε } 58. P = { S → AccA, A → aA | bA |ε } 59. L(G) = {anban where n>1} 60. L(G) = {0n1n where n>1} 11/21/2017 Sampath Kumar S, AP/CSE, SECE 6
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Editor's Notes
- School of EECS, WSU