Regular Expressions (Pre Lecture)

Transcription

1 Regular Expressions (Pre Lecture) Dr. Neil T. Dantam CSCI-561, Colorado School of Mines Fall 2017 Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

2 Regular Expressions Outline Regular Expressions Regular Expressions to NFA NFA to Regular Expressions Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

3 Regular Expressions Regular Operator The regular set is closed under such an operator input One or more regular language(s), R... output A regular language, R op : R R... R Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

4 Regular Expressions Regular Language Basis empty set: defines the language {}, containing no members L ( ) = = {} empty string: defines the language {}, containing the empty string L () = {} = {()} single symbol: Any single symbol a Σ defines the language {a}, containing the string (a) L (a) = {a} Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

5 Regular Expressions Basis: Expression vs. Language Regular Expression Language L ( ) {} L () {()} a L (a) {a} Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

6 Regular Expressions Basic Regular Operators concatenation(α, β): αβ denotes L (α) followed by L (β) L (αβ) = {xy x L (α) y L (β)} union(α, β): α β denotes all members of L (α) or L (β) L (α β) = L (α) L (β) = {x x L (α) x L (β)} Kleene-closure(α): α denotes zero or more repetitions of L (α) L (α ) = {x 0... x n (n 0) (x i L (α))} Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

7 Regular Expressions Operators: Expression vs. Language Regular Expression Language αβ concatenate(α, β) L (αβ) {xy x L (α) y L (β)} α β union(α, β) L (α β) L (α) L (β) α kleene-closure(α, β) L (α ) {x 0... x n (n 0) (x i L (α))} Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

8 Regular Expressions Regex Precedence Convention 3: Kleene-Closure (highest/tightest) 2: Concatenation 1: Union (lowest/last) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

9 Regular Expressions Example: Regex Precedence (01) (10) 01 0(1 ) (1 ) (0(0 )) (1(1 )) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

10 Regular Expressions Example: Simple Regexes L (01) = {01} L (01 0) = {01, 0} L (0(1 0)) = {01, 00} L (0 ) = {, 0, 00, 000,...} Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

11 Regular Expressions Algebraic Properties Regex concatenation union not commutative L (αβ) L (βα) associative L ((αβ)γ) = L (α(βγ)) commutative L (α β) = L (β α) associative L ((α β) γ) = L (α (β γ)) distributive L (α(β γ)) = L (αβ αγ) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

12 Regular Expressions Example: Regex Algebra Simplify: L (α ) = L (α) Simplify: L (α) = L (α) Simplify: L (α ) = L (α ) Simplify: L ((0 ) ) = L (0 ) Factor: L ( ) = L ((00 11) 10) Expand: L ((a b)(b c)) = L (a(b c) b(b c)) = L (ab ac bb bc) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

13 Regular Expressions Historical Interlude Who s this Kleene guy? Alonzo Church Alan Turing Stephen Cole Kleene Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

14 Regular Expressions to NFA Outline Regular Expressions Regular Expressions to NFA NFA to Regular Expressions Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

15 Regular Expressions to NFA Regex to NFA Basis L (a) L () L ( ) a Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

16 Regular Expressions to NFA Concatenation concatenate N 1. N, 2. N 1. N 2. Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

17 Regular Expressions to NFA Concatenation (continued) concatenate N 1. N, 2. N 1 N 2.. Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

18 Regular Expressions to NFA Union union N 1., N 2. N 1 N 2.. Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

19 Regular Expressions to NFA Union (continued) union N 1., N 2. N 1 N 2.. Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

20 Regular Expressions to NFA Kleene-Closure kleene-closure N 1. N 1. Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

21 Regular Expressions to NFA Kleene-Closure (continued) kleene-closure N 1. N 1. Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

22 Regular Expressions to NFA Example: L (01) to NFA L (0) L (1) 0 1 Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

23 Regular Expressions to NFA Example: L (01) to NFA (continued) 0 1 Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

24 Regular Expressions to NFA Example: L (01 0) to NFA L (01) 0 1 L (0) 0 Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

25 Regular Expressions to NFA Example: L (01 0) to NFA (continued) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

26 Regular Expressions to NFA Example: L (0 ) to NFA L (0) 0 Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

27 Regular Expressions to NFA Example: L (0 ) to NFA (continued) 0 Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

28 Regular Expressions to NFA Example: L (01 0 ) to NFA Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

29 Regular Expressions to NFA Regular Expressions as Trees τ 1 τ 2... τ n CONCATENATION τ 1 τ 2... τ n UNION τ KLEENE-CLOSURE τ 1 τ 2... τ n τ 1 τ 2... τ n τ (:concatenation tau-1... tau-n) (:union tau-1... tau-n) (:kleene-closure tau) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

30 Regular Expressions to NFA Example: Regex Trees (1 0) 0 CONCATENATION UNION CONCATENATION KLEENE-CLOSURE 0 1 CONCATENATION 0 0 UNION (:concatenation 0 1) (:union (:concatenation 0 1) 0) (:concatenation 0 (:union 0 1)) (:kleene-closure 0) Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

31 Regular Expressions to NFA McNaughton-Yamada-Thompson Algorithm Algorithm 1: Recursive McNaughton-Yamada-Thompson Algorithm Input: (Q, E, s), T ; // NFA states, edges, state, regex tree Output: (Q, E, a) ; // NFA states, edges, end state 1 if root(t) = CONCATENATION then 2 (Q, E, a) fold-left(myt, (Q, E, s), children(t )) ; 3 else if root(t) = UNION then 4 a newstate() ; // New state for accept 5 E E; 6 forall T children(t ) do 7 (Q, E, ã) MYT((Q, E, s), T ) ; // recurse on child { } 8 E E ã a ; // edge from child accept ã to current accept state a 9 else if root(t) = KLEENE-CLOSURE then 10 s newstate() ; // New state for of repetition 11 (Q, E, a) MYT((Q, E, s ), child(t )) ; // Recurse on child 12 E E (s s ) (s a) (a s ) ; 13 else // Base Case 14 a newstate(); 15 E E (s T a) ; Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

32 NFA to Regular Expressions Outline Regular Expressions Regular Expressions to NFA NFA to Regular Expressions Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

33 NFA to Regular Expressions Generalized NFA Intuition: an NFA with regular expressions as edge labels Ñ = (Q, Σ, δ, q, q accept ) Q is the finite set of states Σ is the input alphabet δ : (Q \ {qaccept }) (Q \ {q }) REGEX q Q is the state q Q is the accept state Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

34 NFA to Regular Expressions NFA to Regex: Convert-Rip rip E( q, q) E(q i, q) E( q, q j ) q i q q j E(q i, q j ) q i E(q i, q) (E( q, q)) E( q, q j ) E(q i, q j ) q j Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

35 NFA to Regular Expressions NFA to Regex Algorithm 2: NFA to Regex Input: N = (Q, Σ, E, q 0, F ) ; // NFA states, alphabet, edges,, accept Output: R ; // Regex /* Construct the initial GNFA */ 1 Q Q {q, q accept} ; // add new, accept states 2 E q0 q }{{ } } {q q accept ; edge to new q F }{{} edges to new accept // Merge multiple edges between nodes into union edges 3 forall q i Q do 4 forall q j Q do } 5 e {a σ b E a = q i b = q j ; // set of edges from qi to qj 6 if e = 1 then 7 E E e; 8 else if e > 1 then ( ) 9 l {σ} a b e σ 10 r regex (l 0... l } n); 11 E E r {q i qj ; ; // set of edge labels from qi to qj /* Call Convert() subroutine on the GNFA */ 12 R Convert(Q, E ); Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

36 NFA to Regular Expressions NFA to Regex: Convert Function Convert(Q,E) 1 if Q = 2 then 2 R E(q, q accept ) ; // Extract label of edge from GNFA to accept 3 return R; 4 else 5 q any state in (Q \ {q, q accept }); 6 Q Q \ { q}; 7 E E \ { q q}; 8 forall q i where E(q i, q) do // predecessors of q 9 forall q j where E( q, q j ) do // successors of q 10 r regex (E(q i, q) (E( q, q)) E( q, q j ) E(q i, q j )) ; 11 E E \ {(q i q), ( q q j ), (q i q j ), }; } 12 E E r {q i qj ; 13 return Convert(Q, E ); Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

37 NFA to Regular Expressions Example: NFA to Regex 0. Initial DFA b 1 2 a a,b Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

38 NFA to Regular Expressions Syntactic Sugar α + αα α? α [α 0 α 1... α n ] α 0 α 1... α n. {σ} σ Σ What about a complement operator? Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39

39 NFA to Regular Expressions Conversions subset construction NFA DFA McNaughton Yamada Thompson every DFA is an NFA GNFA conversion Regex also possible Dantam (Mines CSCI-561) Regular Expressions (Pre Lecture) Fall / 39