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Lexical Analysis IV : NFA to DFA DFA Minimization Lecture 5 CS 4318/5331 Apan Qasem Texas State University Spring 2015 *some slides adopted from Cooper and Torczon

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Announcements REU programs in summer

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Review DFA For every RE there exists a DFA Cannot convert REs directly to DFAs NFA DFAs that allow non-determinism empty transitions multiple transitions on same symbol NFA and DFA recognize the same set of languages

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Review RE to NFA s0s1s0s1 a s3 a s2s4 b s5 s0 s1s3s0 ab s2 s1s3s0 a s2 a* ab a|b a

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Thompson’s Construction NFA properties Each NFA has a single start state and a single final state The only transition that enters the initial state is the initial transition No transitions leave the final state An empty transition always connects two states that were start or final states of a component NFA A state has at most two entering and two exiting empty transitions try to convince yourself that these properties hold

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Cycle of Construction RE Minimized DFA NFA Code Thompson’s Construction Subset Construction Hopcroft’s Algorithm

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Construct NFA for (a|b)*aa Step 1: Construct trivial NFAs s0s1 b s0s1 a

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Example : NFA for (a|b)*aa Step 2: Work inside parentheses a | b s0s1 b s0s1 a s0 s1

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Example : NFA for (a|b)*aa Step 2: Work inside parentheses a | b (rename states) s1s3 a s2s4 b s0 s5

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Example : NFA for (a|b)*aa Step 3: * (closure) s1s3 a s2s4 b s5s0s5s0

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Example : NFA for (a|b)*aa Step 3: * (closure) - renaming states s2s4 a s3s5 b s6s0s7s1

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Example : NFA for (a|b)*aa Step 4: concatenation a s2s4 a s3s5 b s6s7 s1s0 s8 s9 a

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Example : NFA for (a|b)*aa Step 5: concatenation a s2s4 a s3s5 b s6s7 s1s0 s10 s11 a s8 s9 a

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Example : NFA for (a|b)*aa Eliminating empty transitions for concatenation s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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NFA to DFA To convert NFAs to DFAs we need to get rid of non- determinism from NFAs Three cases of non-determinism in NFAs Transition to a state without consuming any input Multiple transitions on the same input symbol No transition on an input symbol

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Examples of Non-determinism in NFAs s3s4 s4s4 s2 a s3 a s2 s3s3 a s4 s2 E s3 a a s5 s2 a s4 s3 a a

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Examples of Non-determinism in NFAs s3s4 s3 s2 a s3 a s2 s3 a s4 s2 E s3 a a s2 a s4 s3 a a All we need to do is eliminate all transitions

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Subset Construction : Example s3s2 a s4 In state s2 on input a can go to either s3 or s4 s3s2 a s4 Create a state for the DFA that represents the combined state

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Subset Construction : Example s3s2 a s4 In state s2 on input a, can go to either s3 or s4. From s3, can go to s5 and s6. From s4 can go to S6. From S5 … and so on … s2 a s3s4 Follow the path for each state in the combined state to create new states s5s6 bc s5s6 c b

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NFA→DFA with Subset Construction Main Idea: For every state in the NFA, determine all reachable states for every input symbol The set of reachable states constitute a single state in the converted DFA Each state in the DFA corresponds to a subset of states in the NFA (hence the name) Find reachable states for each new DFA state, until no more new states can be found

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Finding Reachable States Two key functions Move(s i, a) is the set of states reachable from s i by a single hop only ε-closure(s i ) is the set of states reachable from s i by ε can follow multiple εhops (hence “closure”) Move(s 1, a) ? s3 Move(s 2, a) ? empty ε-closure(s 0 )? s0, s1, s2, s3, s5 ε-closure(s 2 ) ? s2 s1s3 a s2s4 b s5 s0

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Subset Construction : Algorithm // Start state, s 0 is derived from start state of NFA Take ε-closure of NFA start state, s 0 = ε-closure({n 0 }) s 0 represents all the possible states we can be in, at the very beginning For each state in s 0, Compute Move(s i, α) for each α ∈ Σ, and take its ε-closure // This step gives us the reachable states Iterate until no more states are added

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Subset Construction : Algorithm s 0 ← ε-closure({n 0 }) S ← {s 0 } W ← {s 0 } while ( W ≠ Ø ) select and remove s i from W for each α ∈ Σ t ← ε-closure(Move(s,α)) T[s,α] ← t if ( t ∉ S ) then add t to S add t to W The algorithm halts: 1.S contains no duplicates (test before adding) 2.2 {NFA states} is finite 3.while loop adds to S, but does not remove from S (monotone) ⇒ the loop halts S contains all the reachable NFA states Algorithm tries each character on each s i. It builds every possible NFA configuration ⇒ S and T form the DFA

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Subset Construction : A fixed-point computation Example of a fixed-point computation Monotone construction of some finite set Halts when it stops adding to the set Proofs of halting and correctness are similar These computations arise in many contexts Other fixed-point computations Canonical construction of sets of LR(1) items Quite similar to the subset construction Classic data-flow analysis Differential Equation solvers Square root computation We will see more fixed-point computations later in this course

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Subset Construction : Final States Any DFA state containing an NFA final state becomes a final state of the DFA s3s2 a s4 s3s2 a s4

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Statesε-closure(move(∑,*)) DFANFAab q0 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 q3s4, s8, s9, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 s4, s8, s6, s7, s1, s2, s3 q3s4, s8, s9, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q3s4, s8, s9, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3 s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q3s4, s8, s9, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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Statesε-closure(move(∑,*)) DFANFAab q0s0, s1, s2, s3, s7s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q1s4, s8, s6, s7, s1, s2, s3 s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q2s5, s6, s7, s1, s2, s3, s4, s8, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 q3s4, s8, s9, s6, s7, s1, s2, s3 s5, s6, s7, s1, s2, s3 s2s4 a s3s5 b s6s7 s1s0 s9 a s8 a

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a b Statesε-closure(move(s,*)) DFANFAab q0s0, s1, s2, s3, s7q1q2 q1s4, s8, s6, s7, s1, s2, s3 q3q2 s5, s6, s7, s1, s2, s3,q1q2 q3s8, s4, s9, s6, s1, s2, s3, s7 q3q2 q0 q3 q1 a b b a b a DFA Transition Table Equivalent States

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DFA Minimization Goal Discover sets of equivalent states Represent each such set with just one state Definition of equivalence Two states are equivalent if and only if ∀ α ∈ Σ, transitions on α lead to identical (or equivalent) states i.e., both states do the same thing if we land on them Trick Easier to determine if two states are not equivalent α-transitions to distinct sets ⇒ states must be in distinct sets think about an algorithm for primality test

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Partition of a Set The DFA minimization algorithm is based on the notion of set partitions A partition P of S is a collection of sets P such that each s ∈ S is in exactly one p i ∈ P Not a partition Partition Not a partition

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Hopcroft’s Algorithm Proposed by John Hopcroft in 1971 Later improved efficiency to O(nlogn) Developed in the context of finite automaton but have found application in other areas alias analysis are the two variables referencing the same memory location? redundancy elimination are the values in two variables identical? Hopcroft also known for many other contributions to Computer Science The Cinderella book Hopcroft-Karp algorithm

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Hopcroft’s Algorithm Main idea Initially put all elements (states/variables/pointers) in a single partition At each step divide the current partition based on some distinguishing property or behavior of the elements Elements that remain grouped together are equivalent Find equivalent cars Initial partition? Subdivide by Make? Color?

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Algorithm for DFA Minimization Hopcroft’s algorithm applied to DFA Minimization Pick initial partition P 0 Two sets: final states and non-final states {F} and {S-F}, where D =(S,Σ,δ,s0,F) Iteratively split the sets based on the behavior of the the states state transitions States that remain grouped together are equivalent What should our initial partition be? How do we capture the behavior of the state?

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Splitting a Set pipi pkpk pjpj

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pkpk pjpj pmpm pnpn

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Splitting or partitioning a set by a Assume s a and s b ∈ p i, where p i is a subset of the original set of states (i) δ(s a,a) = s x and δ(s b,a) = s y (ii) s x ∈ p j, s y ∈ p k, j ≠ k

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Algorithm for DFA Minimization T ← {F, {S-F}} P ← { } while ( P ≠ T) P ← T T ← { } for each set p i ∈ P T ← T ∪ Split(p i ) Split(S) for each c ∈ Σ if c splits S into s1 & s2 then return {s1, s2} return S Partition P ∈ 2 S Start off with 2 subsets of S: {F} and {S-F} The while loop takes P i → P i+1 by splitting 1 or more sets P i+1 is at least one step closer to the partition with | S | sets Maximum of | S | splits Note that Partitions are never combined Initial partition ensures that final states remain final states

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DFA Minimization Refining the algorithm As written, it examines every p i ∈ P on each iteration This strategy entails a lot of unnecessary work Only need to examine p i if some T, reachable from p i, has split Reformulate the algorithm using a worklist Start worklist with initial partition, F and {S-F} When it splits P i into P 1 and P 2, place P 2 on worklist This version looks at each p i ∈ P many fewer times Hopcroft’s contribution

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DFA Minimization : Example b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b Stateab S0S0 S1S1 S2S2 S1S1 S1S1 S3S3 S2S2 S1S1 S2S2 S3S3 S1S1 S4S4 S4S4 S1S1 S2S2 DFA for (a | b)*abbTransition Table

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3} b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4} b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}noneNone b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}none P0P0 {s4} {s0,s1,s2,s3}{s0,s1,s2,s3} b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}noneNone P0P0 {s4} {s0,s1,s2,s3}{s0,s1,s2,s3}none{s0,s1,s2} {s3} b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}none P0P0 {s4} {s0,s1,s2,s3}{s0,s1,s2,s3}none{s0,s1,s2} {s3} P1P1 {s4} {s0,s1,s2} {s3}{s0,s1,s2} b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}noneNone P0P0 {s4} {s0,s1,s2,s3}{s0,s1,s2,s3}none{s0,s1,s2} {s3} P1P1 {s4} {s0,s1,s2} {s3}{s0,s1,s2}none{s0,s2} {s1} b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}none P0P0 {s4} {s0,s1,s2,s3}{s0,s1,s2,s3}none{s0,s1,s2} {s3} P1P1 {s4} {s0,s1,s2} {s3}{s0,s1,s2}none{s0,s2} {s1} P2P2 {s4} {s0,s2} {s1} {s3}{s0,s2}none b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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DFA Minimization : Example Current Partitionpipi Split on aSplit on b P0P0 {s4} {s0,s1,s2,s3}{s4}none P0P0 {s4} {s0,s1,s2,s3}{s0,s1,s2,s3}none{s0,s1,s2} {s3} P1P1 {s4} {s0,s1,s2} {s3}{s0,s1,s2}none{s0,s2} {s1} P2P2 {s4} {s0,s2} {s1} {s3}{s0,s2}none b b S 0, S 2 s1s1 s3s3 s4s4 a b a a b b b s0s0 s1s1 s2s2 s3s3 s4s4 a a b a a b a b

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Example : Putting it together … Construct regular expression for language that contains all strings that start with an a, followed by any number of b ’s and c’s a(b|c)*

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Example : RE to NFA a(b|c)* Step 1: Compute trivial NFAs s0s1 c s0s1 b s0s1 a

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Example : RE to NFA a(b|c)* Step 2: Work inside parentheses b | c s0s1 c s0s1 b s0 s5

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Example : RE to NFA a(b|c)* Step 2: Work inside parentheses b | c s1s3 b s2s4 c s0 s5

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Example : RE to NFA a(b|c)* Step 3: * (closure) s1s3 b s2s4 c s5s0s5s0

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Example : RE to NFA a(b|c)* Step 3: * (closure) s2s4 b s3s5 c s6s0s7s1

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Example : RE to NFA a(b|c)* Step 4: concatenation s4s5 b s6s7 c s8 s1 s9 s3s2 s0 a

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Statesε-closure(move(s,*)) DFANFAabc s0q0 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0q1, q2, q3 q4, q6, q9 none s1q1, q2, q3 q4, q6, q9 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0q1, q2, q3 q4, q6, q9 none s1q1, q2, q3 q4, q6, q9 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0q1, q2, q3 q4, q6, q9 none s1q1, q2, q3 q4, q6, q9 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0q1, q2, q3 q4, q6, q9 none s1q1, q2, q3 q4, q6, q9 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 s2q5, q8, q9 q3, q4, q6 s3q7, q8, q9 q3, q4, q6 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0q1, q2, q3 q4, q6, q9 none s1q1, q2, q3 q4, q6, q9 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 s2q5, q8, q9 q3, q4, q6 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 s3q7, q8, q9 q3, q4, q6 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0q1, q2, q3 q4, q6, q9 none s1q1, q2, q3 q4, q6, q9 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 s2q5, q8, q9 q3, q4, q6 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 s3q7, q8, q9 q3, q4, q6 none q5, q8, q9 q3, q4, q6 q7, q8, q9 q3, q4, q6 q4q5 b q6q7 c q8 q1 q9 q3q2 q0 a NFA to DFA with Subset Construction

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Statesε-closure(move(s,*)) DFANFAabc s0q0s1none s1q1, q2, q3 q4, q6, q9 none s2s3 s2q5, q8, q9 q3, q4, q6 none s2s3 q7, q8, q9 q3, q4, q6 none s2s3 NFA to DFA with Subset Construction b c s0s0 s1s1 s2s2 s3s3 b a bc c

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b c s0s0 s1s1 s2s2 s3s3 b a bc c DFA Minimization Already minimized!

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Homework 1 Homework 1 is out, due by March 9

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