Fall 2005 CSE 467/567 1 Formal languages regular expressions regular languages finite state machines

Fall 2005 CSE 467/567 2 Formal languages A string is a (perhaps empty) sequence of symbols.  denotes the empty string. A language is a (perhaps empty) set of strings.  denotes the empty set. There are many different classes of languages. Main ones in Chomsky hierarchy are regular, context free, context sensitive and unrestricted.

Fall 2005 CSE 467/567 3 Sets and set operations Examples:  is the empty set, a set with no members {a, b, c} is a set with three members Operations: Suppose A={a, b, c} and B={1, 2}, then A  B={a1, a2, b1, b2, c1, c2} A  ={(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)} A  B={a, b, c, 1, 2} A*={ , a, b, c, aa, ab, ac, ba, bb, bc, aaa, aab, aac, …}

Fall 2005 CSE 467/567 4 Regular languages The class of regular languages over an alphabet  can be defined recursively: base case 1:  is a regular language base case 2: {  } is a regular language base case 3: for each symbol  in , {  } is a regular language recursive cases: If S and T are regular languages, then so are: {st| s is in S and t is in T}, the concatenation of S and T {x| x is in S or x is in T}, the disjunction of S and T S*, the Kleene closure of S Nothing else is a regular language.

Fall 2005 CSE 467/567 5 Finite state automata Formally a 5-tuple (Q, ,q0,F,  ) where Q is a finite set of states  is a finite input alphabet of symbols q0  Q is the initial state F  Q is a set of final states  : Q   Q

Fall 2005 CSE 467/567 6 Examples FSA (5-tuple and diagram) accepting each of the following (assume  ={a,b,c,…z}): {a} {fred, wilma} {ball, bell, bill, boll, bull} { , a} { , a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, …}

Fall 2005 CSE 467/567 7 RE’s outside of Perl? Can always directly implement the DFA. –switch on state, switch on symbol –table-driven method (put  in an array) Example!

Fall 2005 CSE 467/567 8 Construction of FA Base cases  is a regular language 2.{  } is a regular language 3.for each symbol  in , {  } is a regular language Recursive cases If S and T are regular languages, then so are: 4.{st| s is in S and t is in T}, the concatenation of S and T 5.{x| x is in S or x is in T}, the disjunction of S and T 6.S*, the Kleene closure of S

Fall 2005 CSE 467/567 9 NFA vs. DFA Accept same set of languages. Simulation of NFA through search: depth-first (stack regime for next node) breadth-first (queue regime) Can mechanically convert NFA to DFA, with possible exponential increase in number of states.