Language Recognition MSU CSE 260

Outline Introduction Regular Expressions & Regular Sets Kleene’s Theorem Regular Sets and Regular Grammars Examples More powerful Types of Machines Exercise 10.4

Introduction The sets (languages) recognized by FAs are those formed from: null (empty) set , empty string (), and singleton strings by taking (in arbitrary order): concatenations, unions, Kleene Closures. Regular languages

Regular Expressions Definition. The regular expressions over a set I are defined recursively by: the symbol  is a regular expression; the symbol  is a regular expression; the symbol x is a regular expression whenever x  I. the symbols (AB), (AB), and A* are regular expressions whenever A and B are regular expressions. Examples. The following are regular expressions over I={0,1}: 10*, (10)*, 001, 0(01)*, (0*1)* Note. AB is sometimes represented as A+B. Precedence rules: Kleene closure(*), concatenation, union

Regular Sets Each regular expression represents a (regular) set. Reg. Exp. Set represented  the empty set   the set {} x the set {x} (AB) the concatenation of the sets represented by A and by B (AB) the union of the sets represented by A and by B A* the Kleene closure of the set represented by A Examples. 10*: strings consisting of a 1 followed by any number of 0s (10)*: any arbitrary combinations of string 10 (including ) 0  01: the set consisting of the two strings 0 and 01 0(0  1)*: the set of strings which begin with 0

Kleene’s Theorem Kleene’s Theorem. A set is regular if and only if it is recognized by a finite-state automaton. Proof. The proof is done by construction of NFAs for sets represented by regular expressions.

NFAs for Basic Regular Sets recognizes  Start s0 recognizes {} Start s0 The NFA is easier to build and understand than the equivalent FA. a recognizes {a} Start s0 s1

NFA for Concatenation Let A be recognized by MA=(SA, I, fA, sA, FA), and B recognized by MB=(SB, I, fB, sB, FB) MAB f MA MB Start f i sA sB i The NFA is easier to build and understand than the equivalent FA. A transition to final state in MA produces a transition to sB. A transition from sB in MB produces a transition from sAB=sA if sA is a final state. Start state of MAB is sAB=sA, which is final if both sA and sB are final. Final states of MAB include all final states of MB.

NFA for Union Let A be recognized by MA=(SA, I, fA, sA, FA), and B recognized by MB=(SB, I, fB, sB, FB) i MA i sA MAB Start sAB MB The NFA is easier to build and understand than the equivalent FA. i sB i Start state of MAB is sAB, which is final if sA or sB is final. Final states of MAB are the final states of MA and MB.

NFA for Kleene Closure Let A be recognized by MA=(SA, I, fA, sA, FA). Start i MA* sA* sA i The NFA is easier to build and understand than the equivalent FA. Transitions from sA produce transitions from sA* and all final states of MA. Start state of MA* is sA*, which is a final state. Final states of MA* include all final states of MA.

NFA for a Set Generated by a Regular Grammar Theorem. A set generated by a regular grammar is regular. Let G = (V, T, S, P) be a regular grammar. Then the corresponding NFA is M = (S, I, f, s0, F): S contains a state sA for each nonterminal symbol A of G, and an additional (final) state sF. Start state is s0 = sS. Transitions are formed from the productions: A  aB produces a transition from sA to sB on input a A  a produces a transition from sA to sF on input a F also includes s0 if P includes production S  .

Example Let G=(V,T,S,P) where V={0,1,A,S}, T={0,1}, and the productions in P are: S1A, S0, S, A0A, A1A, A1 The corresponding NFA is M=(S, I, f, s0, F): S={sS, sA, sF}, s0=sS, F={sF}, I={0,1}, and state diagram 0,1 1 sA Start sS 1 sF

Regular Grammar Generating a Regular Set Theorem. If a set is regular, then there is a regular grammar that generates it. Let M = (S, I, f, s0, F) be the NFA recognizing this set. Then the regular grammar generating it is G = (V, T, S, P): Assign a nonterminal symbol As to each state s of S; Assign a terminal symbol to each input symbol in I; Start symbol S corresponds to start state s0. Productions are formed from the transitions: A transition from s to t on input a produces As  aAt A transition from s to a final state on input a produces As  a P also includes production S   if L(M).

Example Let NFA M=(S, I, f, s0, F), where: S={s0, s1, s2}, I={0,1}, F={s0, s2}, and state diagram: s1 Start s0 1 1 1 s2 The corresponding regular grammar G=(V,T,S,P) V={S=A0, A1, A2, 0, 1}, T={0, 1}, and productions in P: S0A1, S1A2, A10A1, A11A2, A20A1, A21A2, A11, A21, S1, S

More powerful Types of Machines The set {0n1n | n = 0, 1, 2, …} is not regular; has no FA. The limitation of FAs is their finite memory. More powerful models of computation: Pushdown automata Include all FA’s features, plus a stack for unlimited memory. Recognize languages generated by context-free grammars. Example: {0n1n | n=0, 1, 2, …} Linear bounded automata Context-sensitive languages. Example: {0n1n2n | n = 0, 1, 2,…} Turing machines Include FA’s features and a tape (infinite in both directions) for read and write. ►All languages generated by grammars.

Exercise 10.4