“A PROCEDURE FOR CHECKING EQUALITY OF REGULAR EXPRESSIONS” - A. GINZBURG Presented by Kenneth Burgess Ginzburg, A. "A Procedure for Checking Equality of Regular Expressions." Journal of the ACM 14.2 (1967): Web.

CONNECTION TO CLASS  FA -> TG -> RE -> FA  Polymorphism  In acquiescence to the insatiable demands of a ravening appetite, I require sustenance in the form of ingested nutrition.  Yes.  The Equality Dilemma

MOTIVE – WHY THIS MATTERS  Algorithmic  Which translates into…

FINALLY, THE PROCEDURE  Taking advantage of polymorphism  RE -> TG -> Finite set of left linear equations Y = aY a + bY b + ^ Y a = aY aa + bY ab Y b = aY ba + bY bb  If each possible pairing of derivatives share “^”, the RE’s are equal.

THE THEORY BEHIND  Taking derivatives of a RE  Definition of a derivative  R = ab+bab(a+b)*  An example derivative: R(b) = ab(a+b)*  You try: R(a) =

LINEAR EQUATIONS  Now we take the finite set of !!unequal!! derivatives  R = a + ba* R = aR a + bR b R a = aR aa + bR ab + ^ = aR aa + bR ab + ^ R b = aR ba + bR bb + ^ = aR b + bR aa + ^ R aa = aR aaa + bR aab = aR aa + bR aa

CHARACTERISTICS OF DERIVATIVES  Every RE has a finite number of unequal derivatives  Similar to our proposition -> every RE will produce a TG of finite states  Every derivative has a corresponding set of vertices in the graph  In short, R x is rep. by the same TG as R, but with A x as initial vertice

EXAMPLE – R ~ S

PAIRS  # of distinct derivatives = # of distinct sets of beginning inputs  To consider equality, we have to consider all possible beginning inputs for both regular expressions  u (the number of pairs of inputs we have to consider) <= mn  u (distinct pairs of R and S) <= m (distinct derivatives of R) n (distinct derivatives of S)

PAIRS OF R AND S Pair (R x,S x )InfersR has ^?S has ^? (^,^)(a,a)(b,b) (a,a)Yep (b,b)(ba,ba)Yep However, we do not continue, since ba is considered equal to b Since the columns R has ^? and S has ^? agree, the two RE’s are equivalent.

TABLE REPRESENTATION InputsVertices of REqual To^? 123 ^x axx bxx baxbx To streamline this process, we create tables of equivalencies – showing derivatives (or inputs), vertices corresponding, and whether they include ^.

HOW DOES THIS HELP US? Examine T 0 When 0 is inputted into the graph, 2 -> 4 3 -> 2

HOW DOES THIS HELP US? Examine T ^ When ^ is put into the graph, what happens?

HOW DOES THIS HELP US? A ^ are the starting vertices So A ^ T ^ = A ^ Relational functions… so 2 T 1 = 2 (does nothing) 2 T 0 = 4 (input 2, get out 4) 2 T ^ = 2,3

HOW DOES THIS HELP US? - A(10) can be computed by relational functions. - Computers can produce the derivatives. - Computers can check equivalency as long as they keep track of which states are accepting states

TO REVIEW  RE -> TG  Construct table representation of possible inputs  Represent TG as set of relational matrices  Determine U, the set of unique pairs of R and S inputs  For each pair in U, determine if both R and S agree on whether or not they accept