Minimization of dfa

Compiler Design

Optimization of DFA-Based Pattern Matchers

05/01/23 1ANKUR SRIVASTAVA ASSISTANT PROFESSOR JETGI

Optimization of DFA-Based Pattern Matchers

• It refers to detecting those states of a DFA whose presence or absence in a DFA does not affect the language accepted by the automata.

• These states can be eliminated from the automata without affecting the language accepted by the automata.

• SUCH STATES ARE:-• Unreachable States:• Dead States:• Nondistinguishable States:

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Contd…….

• Unreachable States: Unreachable states of a DFA are not reachable from the initial state of DFA on any possible input sequence.

• Dead States: A dead state is a nonfinal state of a DFA whose transitions on every input symbol terminates on itself. For example, q is a dead state if q is in Q F, and δ(q, a) = q for every a in Σ.

• Nondistinguishable States: Nondistinguishable states are those states of a DFA for which there exist no distinguishing strings; hence, they cannot be distinguished from one another.

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Contd……..

• Optimization entails:

• Detection of unreachable states and eliminating them from DFA;

• Identification of nondistinguishable states, and merging them together; and

• Detecting dead states and eliminating them from the DFA.

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Consider The Transition Diagram

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Fig: Partitioning down to a single state.

Contd……

• Initially, we have two groups, as shown below:

Group I Group IISince

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A, B, C, E, F D

Contd…..• Partitioning of Group I is not possible, because the

transitions from all the members of Group I go only to Group I. But since

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state F is distinguishable from the rest of the members of Group I. Hence, we divide Group I into two groups: one containing A, B, C, E, and the other containing F, as shown below:

Contd……

Group I Group II Group IIISince

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A, B, C, E F D

partitioning of Group I is not possible, because the transitions from all the members of Group I go only to Group I. But since

Contd…..

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states A and E are distinguishable from states B and C. Hence, we further divide Group I into two groups: one containing A and E, and the other containing B and C, as shown below:

Contd……

Group I Group II Group III Group IV Since

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A, E B, C F D

state A is distinguishable from state E. Hence, we divide Group I into two groups: one containing A and the other containing E, as shown below:

Contd……

Group I Group II Group III Group IV Group VSince

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A E B, C D

partitioning of Group III is not possible, because the transitions from all the members of Group III on a go to group III only. Similarly,

F

Contd…..

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partitioning of Group III is not possible, because the transitions from all the members of Group III on b also only go to Group III.

Hence, B and C are nondistinguishable states; therefore, we merge B and C to form a single state, B1, as shown in next Figure…

Contd……..

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Fig. Merging nondistinguishable states B&C into a single state B1.

THANKS

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