4.7. chomskian hierarchy of languages
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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.
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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.
2.4 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.
2.3 context free grammars and languages
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.
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.
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- 6. 11/21/2017 Sampath Kumar S, AP/CSE, SECE 7
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Editor's Notes
- School of EECS, WSU