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Regular Grammars

Regular grammars are defined as quadruples G = (V, T, P, S), where V is a finite set of variables, T is a finite set of terminals, P is a finite set of production rules, and S is the start symbol. This guide provides clear examples of regular grammars, demonstrating how to derive strings using productions, showcasing tree derivations, and explaining the connection between non-deterministic finite automata (NFA) and regular grammars. Discover how non-terminals and terminals are utilized in various grammatical forms and the significance of the start symbol.

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Regular Grammars

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  1. Regular Grammars

  2. Definition • A Regular Grammar is a quadruple G= (V,T,P,S), where • V is a finite set of variables (nonterminals,syntactic categories) • T is a finite set of terminals (alphabet) • P is a finite set of productions : rules of the forms • V -> L (l) • V -> w (b) • V -> V (g rules) • V -> w V (d rules) where wÎT* • S, the start symbol, is an element of V

  3. Example 1 Non-terminals = [S,B] Terminals = [a,b] Start = S S -> S -> a S S -> B B -> b B -> b B

  4. Example 2 Non-terminals = [S,C] Terminals = [a,b,c] Start = S S -> a S S -> b C C -> C -> c C

  5. Example 3 Non-terminals = [A,B,C] Terminals = [a,b] Start = A A -> a A A -> a C A -> b B B -> a B C -> b B B ->

  6. Derivation • We say a grammar derives a string if • Start with any rule whose LHS is the start symbol. Write down the RHS. • Repeatedly, replace any Non-terminal, X, in the written down term, with rhs, where (X -> rhs) is one of the productions. • When there are no more Non-terminals, written down term is the derived string.

  7. Example Non-terminals = [S,C] Terminals = [a,b,c] Start = S S -> a S S -> b C C -> C -> c C • Rule • S -> a S • S -> a S • S -> b C • C -> c C • C -> c C • C -> • Right-Hand-Side • a S • a a S • a a b C • a a b c C • a a b c cC • a a b c c

  8. Tree Derivation S Non-terminals = [S,C] Terminals = [a,b,c] Start = S S -> a S S -> b C C -> C -> c C a S a S Derived string aabcc b C c C c C

  9. NFA to RegGram Non-terminals = [S0,S1,S2] Terminals = [a,b] Start = S0 S0 -> a S1 S0 -> b S2 S1 -> a S1 S1 -> b S1 S2 -> a S2 S2 -> b S2 S1 -> For every transition I –a-> J Add a production SI -> a SJ For every transition I -L-> J Add a production Si -> Sj For every final state K Add a production Sk->

  10. RegGram to GenNFA • The non-terminal become the states, but also invent a new final state F • For each kind of prod • V -> L (l) • V -> w (b) • V -> V (g rules) • V -> w V (d rules) • Add a transition • I -> L add I –L-> F • I -> w add I –w-> F • I -> w J add I –w-> J • I -> J add I –L-> J Terminals = [a,b,c,d] Start = S0 S0 -> a b S0 S0 -> c d S1 S1 -> S1 -> c S1

  11. Simplify GenNFA

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