Management Information Systems

Algorithms of informatics - download pdf or read online

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By Ivanyi A. (ed.)

ISBN-10: 9638759615

ISBN-13: 9789638759610

Ivanyi A. (ed.) Algorithms of informatics, vol.1.. foundations (2007)(ISBN 9638759615)

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5. The equivalent DFA with NFA A in Fig. 3. The empty set was excluded from the states, so we used here ∅ instead of {∅}. 10 to transform NFA A in Fig. 3. Introduce the following notation for the states of the DFA: S0 := {q0 , q1 }, S4 := {q0 , q2 }, S1 := {q0 }, S5 := {q1 , q2 }, S2 := {q1 }, S6 := {q0 , q1 , q2 } , S3 := {q2 }, where S0 is the initial state. Using the transition function we get the transition table: δ 0 1 S0 {S2 } {S3 } S1 S2 S3 S4 S5 {S2 } ∅ {S3 } {S5 } {S3 } ∅ {S3 } {S3 } {S3 } {S3 } S6 {S5 } {S3 } This automaton contains many inaccessible states.

The pushdown automaton accepts a word by empty stack when after reading it the automaton empties its stack. We show that these two acceptances are equivalent. 3. Pushdown automata and context-free languages 61 where • Q is the nite, non-empty set of states • Σ is the input alphabet, • W is the stack alphabet, • E ⊆ Q × Σ ∪ {ε} × W × W ∗ × Q is the set of transitions or edges, • q0 ∈ Q is the initial state, • z0 ∈ W is the start symbol of stack, • F ⊆ Q is the set of nal states. A transition (p, a, z, w, q) means that if pushdown automaton V is in state p, reads from the input tape letter a (instead of input letter we can also consider the empty word ε), and the top symbol in the stack is z , then the pushdown automaton enters state q and replaces in the stack z by word w.

14(b). For this operation it is also ∗ true that L(A∗1 ) = L(A1 ) . The denition of these tree operations proves again that regular languages are closed under the regular operations. 5. Minimization of nite automata A DFA A = (Q, Σ, E, {q0 }, F ) is called minimum state automaton if for any equivalent complete DFA A = (Q , Σ, E , {q0 }, F ) it is true that |Q| ≤ |Q |. We give an algorithm which builds for any complete DFA an equivalent minimum state automaton. States p and q of an DFA A = (Q, Σ, E, {q0 }, F ) are equivalent if for arbitrary word u we reach from both either nal or nonnal states, that is u u p −→ r, r ∈ F and q −→ s, s ∈ F or p ≡ q if for any word u ∈ Σ∗ u u p −→ r, r ∈ F and q −→ s, s ∈ F .

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Algorithms of informatics by Ivanyi A. (ed.)


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