DFA Minimization using Myphill-Nerode Theorem Algorithm. Input ? DFA. Output ? Minimized DFA. Step 1 ? Draw a table for all pairs of states (Q i, Q j) not necessarily connected directly [All are unmarked initially] Step 2 ? Consider every state pair (Q i, Q j) in the DFA where Q i ? F and Q j ? F or vice versa and mark them. [Here F is the set of final states]. PRACTICE PROBLEMS BASED ON MINIMIZATION OF DFA- Problem Minimize the given DFA- Solution- Step The given DFA contains no dead states and inaccessible states. Step Draw a state transition table-.
One set will contain all final states and other set will contain non-final states. This partition is called P 0. In each set of P k-1we will minimizze all possible pair of states. If two states of a set are distinguishable, we will split the sets into different sets in P k.
How to find whether two states in partition P k are distinguishable? Example Consider the following DFA shown in figure. Step 1. P0 will have two sets of states. One set will contain q1, q2, q4 which are final states of DFA and another set will contain remaining states. Step 2. To calculate P1, we will dfs whether sets of partition P0 can be partitioned or not:.
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Mar 26, · Minimization of DFA. Step 1: We will divide Q (set of states) into two sets. One set will contain all final states and other set will contain non-final states. This Step 2: Initialize k = 1. Step 3: Find P k by partitioning the different sets of P k In each set of P k-1, we will take all. TOC: Minimization of DFA - Examples (Part 1)This lecture shows how to minimize a DFA with an example. The steps are demonstrated using this example Contribut. How to Minimize a DFA? - The Myhill Nerode Method. If playback doesn't begin shortly, try restarting your device. Videos you watch may be added to the TV's watch history and influence TV.
In automata theory a branch of theoretical computer science , DFA minimization is the task of transforming a given deterministic finite automaton DFA into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same regular language.
Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory. For each regular language, there also exists a minimal automaton that accepts it, that is, a DFA with a minimum number of states and this DFA is unique except that states can be given different names. There are two classes of states that can be removed or merged from the original DFA without affecting the language it accepts to minimize it.
DFA minimization is usually done in three steps, corresponding to the removal or merger of the relevant states. Since the elimination of nondistinguishable states is computationally the most expensive one, it is usually done as the last step. Reachable states can be obtained with the following algorithm:. One algorithm for merging the nondistinguishable states of a DFA, due to Hopcroft , is based on partition refinement , partitioning the DFA states into groups by their behavior.
These groups represent equivalence classes of the Myhill—Nerode equivalence relation , whereby every two states of the same partition are equivalent if they have the same behavior for all the input sequences.
That is, for every two states p 1 and p 2 that belong to the same equivalence class within the partition P , and every input word w , the transitions determined by w should always take states p 1 and p 2 to equal states, states that both accept, or states that both reject.
It should not be possible for w to take p 1 to an accepting state and p 2 to a rejecting state or vice versa. The following pseudocode describes the algorithm:. The algorithm starts with a partition that is too coarse: every pair of states that are equivalent according to the Myhill—Nerode relation belong to the same set in the partition, but pairs that are inequivalent might also belong to the same set. It gradually refines the partition into a larger number of smaller sets, at each step splitting sets of states into pairs of subsets that are necessarily inequivalent.
The initial partition is a separation of the states into two subsets of states that clearly do not have the same behavior as each other: the accepting states and the rejecting states.
The algorithm then repeatedly chooses a set A from the current partition and an input symbol c , and splits each of the sets of the partition into two possibly empty subsets: the subset of states that lead to A on input symbol c , and the subset of states that do not lead to A.
Since A is already known to have different behavior than the other sets of the partition, the subsets that lead to A also have different behavior than the subsets that do not lead to A. When no more splits of this type can be found, the algorithm terminates. Given a fixed character c and an equivalence class Y that splits into equivalence classes B and C, only one of B or C is necessary to refine the whole partition.
Example: Suppose we have an equivalence class Y that splits into equivalence classes B and C. By the Lemma, we can choose either B or C as the distinguisher, let's say B. Then the states of D and E are split by their transitions into B. But F, which doesn't point into B, simply doesn't split during the current iteration of the algorithm; it will be refined by other distinguisher s.
The purpose of the outermost if statement if Y is in W is to patch up W, the set of distinguishers. We see in the previous statement in the algorithm that Y has just been split. If Y is in W, it has just become obsolete as a means to split classes in future iterations.
So Y must be replaced by both splits because of the Observation above. If Y is not in W, however, only one of the two splits, not both, needs to be added to W because of the Lemma above. Choosing the smaller of the two splits guarantees that the new addition to W is no more than half the size of Y; this is the core of the Hopcroft algorithm: how it gets its speed, as explained in the next paragraph.
The worst case running time of this algorithm is O ns log n , where n is the number of states and s is the size of the alphabet. This bound follows from the fact that, for each of the ns transitions of the automaton, the sets drawn from Q that contain the target state of the transition have sizes that decrease relative to each other by a factor of two or more, so each transition participates in O log n of the splitting steps in the algorithm.
The partition refinement data structure allows each splitting step to be performed in time proportional to the number of transitions that participate in it. Once Hopcroft's algorithm has been used to group the states of the input DFA into equivalence classes, the minimum DFA can be constructed by forming one state for each equivalence class.
If S is a set of states in P , s is a state in S , and c is an input character, then the transition in the minimum DFA from the state for S , on input c , goes to the set containing the state that the input automaton would go to from state s on input c.
Moore Like Hopcroft's algorithm, it maintains a partition that starts off separating the accepting from the rejecting states, and repeatedly refines the partition until no more refinements can be made. The algorithm terminates when this replacement does not change the current partition.
Its worst-case time complexity is O n 2 s : each step of the algorithm may be performed in time O ns using a variant of radix sort to reorder the states so that states in the same set of the new partition are consecutive in the ordering, and there are at most n steps since each one but the last increases the number of sets in the partition. The instances of the DFA minimization problem that cause the worst-case behavior are the same as for Hopcroft's algorithm.
The number of steps that the algorithm performs can be much smaller than n , so on average for constant s its performance is O n log n or even O n log log n depending on the random distribution on automata chosen to model the algorithm's average-case behavior. As Brzozowski observed, repeating this reversal and determinization a second time, again keeping only reachable states, produces the minimal DFA for the original language. We see that this holds for each pair of powerstates, and thus each powerstate is distinguishable from every other powerstate.
The worst-case complexity of Brzozowski's algorithm is exponential in the number of states of the input automaton. In the case of DFA, the exponential explosion can happen during determinization of the reversal of the input automaton; [note 5] in the case of NFA, it can also happen during the initial determinization of the input automaton. While the above procedures work for DFAs, the method of partitioning does not work for non-deterministic finite automata NFAs. However, there are methods of NFA minimization that may be more efficient than brute force search.
From Wikipedia, the free encyclopedia. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. September Learn how and when to remove this template message. Main article: NFA minimization. Leiss provides a ternary n -state DFA whose reversal requires 2 n states, the maximum possible. That is, the algorithm is at worst exponential, not doubly-exponential.
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