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closure_properties_of_finite_state_machines [2007/05/11 17:11] ghid.maatouk |
closure_properties_of_finite_state_machines [2008/09/17 09:19] vkuncak |
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| Let $A = (\Sigma, Q_A, q_0^A, \Delta_A, F_A)$ and $B = (\Sigma, Q_B, q_0^B, \Delta_B, F_B)$.\\ | Let $A = (\Sigma, Q_A, q_0^A, \Delta_A, F_A)$ and $B = (\Sigma, Q_B, q_0^B, \Delta_B, F_B)$.\\ | ||
| Then the union of the two automata will be $C=(\Sigma, Q_A \cup Q_B, q_0, \Delta_A \cup \Delta_B \cup \{(q_0,\epsilon, q_0^A), (q_0,\epsilon, q_0^B)\}, F_A \cup F_B)$. | Then the union of the two automata will be $C=(\Sigma, Q_A \cup Q_B, q_0, \Delta_A \cup \Delta_B \cup \{(q_0,\epsilon, q_0^A), (q_0,\epsilon, q_0^B)\}, F_A \cup F_B)$. | ||
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| ==== Intersection ==== | ==== Intersection ==== | ||
| - | Product of state spaces for deterministic machines.\\ | + | Product of state spaces, even for non-deterministic machines.\\ |
| - | For nondeterministic machines, must use DeMorgan's laws with complementation and union. Note that every time we want to complement an machine, we must determinize it first. | + | |
| ==== Concatenation ==== | ==== Concatenation ==== | ||
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| The new accepting states are those of the second machine. | The new accepting states are those of the second machine. | ||
| - | ==== Iteration ==== | + | |
| + | ==== Iteration (Kleene star) ==== | ||
| Create a new initial state which is accepting and has an epsilon transition to the old start state.\\ | Create a new initial state which is accepting and has an epsilon transition to the old start state.\\ | ||
| Put a loop around the existing machine by adding epsilon transitions from the accepting states to the new start state. | Put a loop around the existing machine by adding epsilon transitions from the accepting states to the new start state. | ||
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| ==== Comments ===== | ==== Comments ===== | ||
| - | Deterministic versus non-deterministic machines in input and output. | + | If we start with deterministic machines, for which of the above constructions will we get a deterministic machine as a result? |
| - | Cost of complementation. Cost of determinization. | + | What is the cost of computing the machine for the complement of the language? |