Why empty strings? A bit like zero –you may think you can do without –but it makes definitions & calculations easier Definitions: –An alphabeth is a finite set of symbols assume is not a symbol –u is a string over an alphabet T iff u consists of only symbols in T –Hence is a string over any T allows us to define certain things more elegantly –Example: the notions substring, prefix, etc:

Definitions (from first lecture) A string u is a substring of w if there exist other strings x & y s.t. w = xuy. bab is a substring of babba. If u is a substring of w, and x above is, then u is a prefix of w. If u w, then u is a proper prefix. ba is a prefix of babba. If u is a substring of w, and y above is, then u is a suffix of w. If u w, then u is a proper suffix. bba is a suffix of babba.

some properties for any string u, u=u =u it follows that = = =...

some properties for any string u, u=u =u it follows that = = =...

edges We could easily have defined FSAs in such a way that edges dont exist Yet, our definition allowed FSAs to contain edges –We did this by allowing such moves explicitly:

FSA: Formal Definition A Finite State Automaton (FSA) is a 5-tuple (Q, I, F, T, E) where: Q = states = a finite set; I = initial states = a nonempty subset of Q; F = final states = a subset of Q; T = an alphabet; E = edges = a subset of Q (T + ) Q. FSA = labelled, directed graph =set of nodes (some final/initial) + directed arcs (arrows) between nodes + each arc has a label from the alphabet. Example: formal definition of A 1 Q = {1, 2, 3, 4} I = {1} F = {4} T = {a, b} E = { (1,a,2), (1,b,4), (2,a,3), (2,b,4), (3,a,3), (3,b,3), (4,a,2), (4,b,4) } A1A a b b b a a a,b

implications for strings accepted by FSA Recall –a string is accepted by an FSA iff the string is the label of a path from initial to final node Example: this NDFSA accepts the string 01 and the string 001. After reading the first 0, the FSA can either read a 0 or a λ i jkf m

usefulness of edges in FSAa Sometimes we want to prove things like Every so-and-so language can be accepted by an FSA One often proves such things by showing how FSAs can actually be constructed The resulting FSAs often contain edges Example: proof that every Regular Expression has an equivalent FSA.