Lexical Analysis. DFA Minimization & Equivalence to Regular Expressions

Transcription

1 Lexical Analysis DFA Minimization & Equivalence to Regular Expressions Copyright 26, Pedro C. Diniz, all rights reserved. Students enrolled in the Compilers class at the University of Southern California have explicit permission to make copies of these materials for their personal use.

2 DFA State Minimization How to Reduce the Number of States of a DFA? Find unique minimum-state DFA (up to state names) Need to recognize the same language Normalization Assume every state has a transition on every symbol If not, just add missing transitions to a dead state Key Idea Find string w that distinguishes states s and t Algorithm Start with accepting vs. non-accepting states partition of states Refine state groups on all input sequences, i.e. by tracing all transitions Until no refinement is possible 2

3 Algorithm DFA State Minimization Start with accepting vs. non-accepting states partition of states Refine state groups on all input sequences, i.e. by tracing all transitions Until no refinement is possible Does this Terminate? Refinement will end; in the limit partition is state What to do When Refinement Terminates? Elect representative state for each partition Merge edges Remove unneeded states in each partition 3

4 Minimization Example s a b c d, e 4

5 Minimization Example P P 2 b a c e d 5

6 Minimization Example P P 2 b a c e d Label does not split any partition! 6

7 Minimization Example P P 2 b a c e d Label splits P and P 2 partitions! 7

8 Minimization Example P 3 P P 2 b a c e d Label splits P and P 2 partitions! 8

9 Minimization Example P 3 P P 2 b a c e d 9

10 Minimization Example P 3 P P 2 b a c e d Label does not splits any partition!

11 Minimization Example P 3 P P 2 b a c e d Label does not splits any partition!

12 Minimization Example P 3 P P 2 a b c d, e Elect Representative and Merge Edges 2

13 Minimization Example, s, p 3 p p 2 Elect Representative and Merge Edges 3

14 DFA State Minimization: Algorithm DFA = {D,, d, s, D F } P {D F, {D - D F }} while (P is still changing) T Ø for each set p P T T Split(p) P T Split(S) for each c if c splits S into s and s 2 then return {s, s 2 } return S 4

15 DFA to RE: Kleene Construction Path Problem over the DFA Starting from state s (numbering of states is N - important) Label all edges through all states to an accepting state What to do with cycles in the DFA, as they are infinite paths? Kleene Construction Iterate and merge path expressions for every pair of nodes i and j not going through any node with label higher then k Increase k up to N In the end do the union of all path expressions that start at s and end in a final state. 5

16 DFA to RE: Kleene Construction for i = to N for j = to N R ij = {a δ(s i,a) = s j } if (i = j) then R ij = R ij { ε } for k = to N for i = to N for j = to N i R k- ij R k- ik j k R k- kj R k- kk R k ij = R k- ik (R k- kk )* R k- kj R k- ij L = s j S F R N j 6

17 DFA to RE: Kleene Construction for i = to N Direct Path for j = to N R ij = {a δ(s i,a) = s j } if (i = j) then R ij = R ij { ε } for k = to N for i = to N for j = to N i R k- ij R k- ik j k R k- kj R k- kk R k ij = R k- ik (R k- kk )* R k- kj R k- ij L = s j S F R N j 7

18 DFA to RE: Kleene Construction for i = to N Direct Path for j = to N R ij = {a δ(s i,a) = s j } if (i = j) then R ij = R ij { ε } for k = to N Indirect Path for i = to N for j = to N i R k- ij R k- ik j k R k- kj R k- kk R k ij = R k- ik (R k- kk )* R k- kj R k- ij L = s j S F R N j 8

19 DFA to RE: Example [..9] r [..9] S S 2 S 3 R 2 = r R 23 = [..9] R 33 = [..9] ε 9

20 DFA to RE: Example [..9] r [..9] S S 2 S 3 R 2 = r R 23 = [..9] R 33 = [..9] ε R 3 = R (R )* R 3 R 3 = nil R 23 = R 2 (R )* R 3 R 23 = [..9] R 33 = R 3 (R )* R 3 R 3 = [..9] ε R kk = nil otherwise 2

21 DFA to RE: Example [..9] r [..9] S S 2 S 3 R 2 = r R 23 = [..9] R 33 = [..9] ε R kk = nil otherwise R 3 = R (R )* R 3 R 3 = nil R 23 = R 2 (R )* R 3 R 23 = [..9] R 33 = R 3 (R )* R 3 R 3 = [..9] ε R 2 3 = R 2 (R 22)* R 23 R 3 = r. ε* [..9] R 2 33 = R 32 (R 22)* R 23 R 33 = [..9] ε R 2 33 = R 32 (R 22)* R 23 R 33 = [..9] ε 2

22 DFA to RE: Example [..9] r [..9] S S 2 S 3 R 2 = r R 3 3 = R 2 3 (R 2 33)* R 2 33 R 2 3 R 23 = [..9] R 33 = [..9] ε R kk = nil otherwise 22

23 DFA to RE: Example [..9] r [..9] S S 2 S 3 R 2 = r R 23 = [..9] R 33 = [..9] ε R 3 3 = R 2 3 (R 2 33)* R 2 33 R 2 3 = (r. ε* [..9])([..9]*)([..9]) r. ε* [..9] = (r. [..9]+) r. [..9] = (r. [..9]+) R kk = nil otherwise 23

24 DFA to RE: Example [..9] r [..9] S S 2 S 3 L(M) = R 3 3 = r. [..9]+ 24

25 DFA, NFA and REs Kleene s Construction RE DFA Minimization DFA Code for a scanner Thompson s Construction NFA Subset Construction 25

26 DFA, NFA and REs Kleene s Construction RE DFA Minimization DFA Code for a scanner Thompson s Construction NFA Subset Construction Regular Expressions and FA are Equivalent 26

27 DFA Minimization Summary Find sequence w that discriminates states Iterate until no possible refinement DFA to RE Kleene construction Combine Path Expression for an increasingly large set of states DFA and RE are Equivalent Given one you can derive an equivalent representation in the other 27