Transparency No. 5-1 Formal Language and Automata Theory Chapter 5 Kleene Algebra and Regular Expressions (Supplementary Lecture A)

Nondeterministic Finite Automata Transparency No. 5-2 Kleene algebra (the algebra of regular sets) A Kleene algebra is a algebra K = (K, 0, 1,, +, *) where + is ACII:  (a + b) + c = a + (b + c) assoc.(A.1)  a + b = b + ccommutive(A.2)  a + a = aidempotent(A.3)  a + 0 = a0 is the identity for + (A.4) is A I An:  a(bc) = (ab) is associative(A.5)  a1 = 1a = a1 is the identity for (A.6)  a0 = 0a = 00 is an annihilator(A.7) is distributive w.r.t. +:  a(b+c) = ab + acleft distributive(A.8)  (a+b)c = ac + bcright distributive(A.9) The laws of *:

Nondeterministic Finite Automata Transparency No. 5-3 Kleene algebra (cont’d) Axioms involving *:  1 + aa* = a*(A.10)  1 + a*a = a*(A.11)  b + ac  c  a*b  c(A.12)  a + ca  c  ba*  c(A.13) where  refers to the order a  b  a + b = b in 2  *,  is the set inclusion  Examples of Kleene algebras: (2  *, {},  *, U,, *) (2 AxA, {}, {(x,x) | x in A}, U,, *) (the set of nxn boolean matrices, zero matrix, Identity matrix, +, x, * )

Nondeterministic Finite Automata Transparency No. 5-4 Matrices K: a Kleene algebra M(n,K): the set of nxn matrices over K, is also a Kleene algebra. Example: in M(2,K), the identities for + and are The operations +,, and * are given by: 

Nondeterministic Finite Automata Transparency No. 5-5 Matrices (cont’d) Find E* for a nxn matrix E over K: (by induction on n) n=1 => M(n,K) = K, and E* = E*. N > 1 => break E into E = s.t. A D are squares, say mxm and (n-m)x(n-m). Since A and D are squares, by ind. Hyp., A* and D* are meaningful, it then make sense to define E* = Problem: how about E* if n = 3, 6,... ? 

Nondeterministic Finite Automata Transparency No. 5-6 Matrices (cont’d) If E = ==> E* = ?

Nondeterministic Finite Automata Transparency No. 5-7