COSE312: Compilers. Lecture 5 Lexical Analysis (4)

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1 COSE312: Compilers Lecture 5 Lexical Analysis (4) Hakjoo Oh 2017 Spring Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

2 Part 3: Automation Transform the lexical specification into an executable string recognizers: Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

3 From NFA to DFA Transform an NFA into an equivalent DFA (N, Σ, δ N, n 0, N A ) (D, Σ, δ D, d 0, D A ). Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

4 From NFA to DFA Transform an NFA into an equivalent DFA (N, Σ, δ N, n 0, N A ) (D, Σ, δ D, d 0, D A ). Running example: a start b 5 6 c Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

5 -Closures -closure(i): the set of states reachable from I without consuming any symbols. a start b 5 6 c closure({1}) = {1, 2, 3, 4, 6, 9} -closure({1, 5}) = {1, 2, 3, 4, 6, 9} {3, 4, 5, 6, 8, 9} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

6 Subset Construction Input: an NFA (N, Σ, δ N, n 0, N A ). Output: a DFA (D, Σ, δ D, d 0, D A ). Key Idea: the DFA simulates the NFA by considering every possibility at once. A DFA state d D is a set of NFA state, i.e., d N. Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

7 Running Example (1/5) The initial DFA state d 0 = -closure({0}) = {0}. start {0} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

8 Running Example (2/5) For the initial state S, consider every x Σ and compute the corresponding next states: -closure( δ(s, a)). s S Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

9 Running Example (2/5) For the initial state S, consider every x Σ and compute the corresponding next states: -closure( δ(s, a)). s S -closure( s {0} δ(s, a)) = {1, 2, 3, 4, 6, 9} -closure( s {0} δ(s, b)) = -closure( s {0} δ(s, c)) = start {0} a {1, 2, 3, 4, 6, 9} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

10 Running Example (3/5) For the state {1, 2, 3, 4, 6, 9}, compute the next states: -closure( s {1,2,3,4,6,9} δ(s, a)) = -closure( s {1,2,3,4,6,9} δ(s, b)) = {3, 4, 5, 6, 8, 9} -closure( s {1,2,3,4,6,9} δ(s, c)) = {3, 4, 6, 7, 8, 9} b {3, 4, 5, 6, 8, 9} start {0} a {1, 2, 3, 4, 6, 9} c {3, 4, 6, 7, 8, 9} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

11 Running Example (4/5) Compute the next states of {3, 4, 5, 6, 8, 9}: -closure( s {3,4,5,6,8,9} δ(s, a)) = -closure( s {3,4,5,6,8,9} δ(s, b)) = {3, 4, 5, 6, 8, 9} -closure( s {3,4,5,6,8,9} δ(s, c)) = {3, 4, 6, 7, 8, 9} b {3, 4, 5, 6, 8, 9} b start {0} a {1, 2, 3, 4, 6, 9} c c {3, 4, 6, 7, 8, 9} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

12 Running Example (5/5) Compute the next states of {3, 4, 6, 7, 8, 9}: -closure( s {3,4,6,7,8,9} δ(s, a)) = -closure( s {3,4,6,7,8,9} δ(s, b)) = {3, 4, 5, 6, 8, 9} -closure( s {3,4,6,7,8,9} δ(s, c)) = {3, 4, 6, 7, 8, 9} b {3, 4, 5, 6, 8, 9} b start {0} a {1, 2, 3, 4, 6, 9} c b c {3, 4, 6, 7, 8, 9} c Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

13 Subset Construction Algorithm Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

14 Running Example (1/5) a start b 5 6 c The initial state d 0 = -closure({0}) = {0}. Initialize D and W : D = {{0}}, W = {{0}} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

15 Running Example (2/5) Choose q = {0} from W. For all c Σ, update δ D : Update D and W : a b c {0} {1, 2, 3, 4, 6, 9} D = {{0}, {1, 2, 3, 4, 6, 9}}, W = {{1, 2, 3, 4, 6, 9}} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

16 Running Example (3/5) Choose q = {1, 2, 3, 4, 6, 9} from W. For all c Σ, update δ D : a b c {0} {1, 2, 3, 4, 6, 9} {1, 2, 3, 4, 6, 9} {3, 4, 5, 6, 8, 9} {3, 4, 6, 7, 8, 9} Update D and W : D = {{0}, {1, 2, 3, 4, 6, 9}, {3, 4, 5, 6, 8, 9}, {3, 4, 6, 7, 8, 9}} W = {{3, 4, 5, 6, 8, 9}, {3, 4, 6, 7, 8, 9}} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

17 Running Example (4/5) Choose q = {3, 4, 5, 6, 8, 9} from W. For all c Σ, update δ D : a b c {0} {1, 2, 3, 4, 6, 9} {1, 2, 3, 4, 6, 9} {3, 4, 5, 6, 8, 9} {3, 4, 6, 7, 8, 9} {3, 4, 5, 6, 8, 9} {3, 4, 5, 6, 8, 9} {3, 4, 6, 7, 8, 9} D and W : D = {{0}, {1, 2, 3, 4, 6, 9}, {3, 4, 5, 6, 8, 9}, {3, 4, 6, 7, 8, 9}} W = {{3, 4, 6, 7, 8, 9}} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

18 Running Example (5/5) Choose q = {3, 4, 6, 7, 8, 9} from W. For all c Σ, update δ D : a b c {0} {1, 2, 3, 4, 6, 9} {1, 2, 3, 4, 6, 9} {3, 4, 5, 6, 8, 9} {3, 4, 6, 7, 8, 9} {3, 4, 5, 6, 8, 9} {3, 4, 5, 6, 8, 9} {3, 4, 6, 7, 8, 9} {3, 4, 6, 7, 8, 9} {3, 4, 5, 6, 8, 9} {3, 4, 6, 7, 8, 9} D and W : D = {{0}, {1, 2, 3, 4, 6, 9}, {3, 4, 5, 6, 8, 9}, {3, 4, 6, 7, 8, 9}} W = The while loop terminates. The accepting states: D A = {{1, 2, 3, 4, 6, 9}, {3, 4, 5, 6, 8, 9}, {3, 4, 6, 7, 8, 9}} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

19 Algorithm for computing -Closures The definition -closure(i) is the set of states reachable from I without consuming any symbols. is neither formal nor constructive. Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

20 Algorithm for computing -Closures The definition -closure(i) is the set of states reachable from I without consuming any symbols. is neither formal nor constructive. A formal definition: T = -closure(i) is the smallest set such that I δ(s, ) T. s T Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

21 Algorithm for computing -Closures The definition -closure(i) is the set of states reachable from I without consuming any symbols. is neither formal nor constructive. A formal definition: T = -closure(i) is the smallest set such that I δ(s, ) T. s T Alternatively, T is the smallest solution of the equation where F (X) (X) F (X) = I s X δ(s, ). Such a solution is called the least fixed point of F. Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

22 Fixed Point Iteration The least fixed point of a function can be computed by the fixed point iteration: T = repeat T = T T = T F (T ) until T = T Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

23 Example a start b 5 6 c closure({1}): Iteration T T 1 {1} 2 {1} {1, 2} 3 {1, 2} {1, 2, 3, 9} 4 {1, 2, 3, 9} {1, 2, 3, 4, 6, 9} 5 {1, 2, 3, 4, 6, 9} {1, 2, 3, 4, 6, 9} Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

24 Summary Key concepts in lexical analsis: Specification: Regular expressions Implementation: Deterministic Finite Automata Translation (homework 1) Next class: OCaml programming tutorial by TAs. Hakjoo Oh COSE Spring, Lecture 5 March 20, / 20

Computability. What can be computed?

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