Matrix Approach to Automata
Theory
Prashanth L A
Arun R
Agenda
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INTRODUCTION
NOTATION AND PRELIMINARIES
EXAMPLES
ULTIMATE PERIODICITY AND
REGULARITY PRESERVATION
CONCLUSION
Introduction
INTRODUCTION (1)
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
Boolean-matrix-based method to
automata theory (MAAT in short).
A new characterization of regularitypreserving functions derived in terms of
the property of ultimate periodicity with
respect to powers of Boolean matrices.
Notation and Preliminaries
NOTATION AND PRELIMINARIES (1)
Boolean matrix: a matrix (of size m x n) whose
elements are either 0 or 1. Boolean => matrix
operation follow boolean laws (1+1=1).
 M(Q): the set of square boolean matrices over Q i.e.,
f: QxQ --> [0, 1].
 Row Vector: A Boolean vector of dimension n i.e., an
n-tuple (b1 , b2 , ..., bn) of 0s and 1s.
 Column Vector: Transpose of the above.
 V(Q): vectors indexed over a set Q.
M(Q) and V(Q) are of interest to us.

NOTATION AND PRELIMINARIES (2)
More definitions:
 Adjacency Matrix: ∆(p,q)=1 if there is an edge
from p to q, 0 otherwise
 The characteristic vector of a subset A of Q is
the row vector IA such that IA( p)=1 iff if p ∈ A.
 ∆a (p,q)=1 if and only if ∆(p, a)=q.
 ∆ = sum of all members ∆a, a ∈
NOTATION AND PRELIMINARIES
(3)
Some Lemmas:
 Lemma 1: For every i≥0, ∆i(p, q)=1 if and only if there
is a length-i transition in M from state p to state q.
Here, ∆0 is defined to be the identity matrix.

Lemma 2 : Let M be a dfa and {∆a | a ∈ } its
Boolean matrix system. Then the language accepted
by M is the set {x | Iq0∆x IF’=1}, where q0 is the starting
state of M, F is the set of final states of M, and ( )t
stands for matrix transpose.
NOTATION AND PRELIMINARIES
(4)
Regularity-Preserving Functions:
 A function f on nonnegative integers is regularitypreserving if for any regular language L, the language
T(L, f ) is also regular, where
T(L, f ) := { x | ∃y, |y|= f(|x|) & xy ∈ L}.
 Second notion of regularity-preserving functions uses
T’(L, f ) := { x | ∃y, |y|= f(|x|) & y ∈ L}.
 Kozen showed that the two notions are equivalent
You have just seen the only
“definitions” in this talk…
…unfortunately
Examples
EXAMPLES [1]

Exponential Function:
M=(Q,∑ , , q0, F) with adjacency matrix ∆

Approach: Construct the DFA
M’=(Q’,∑ ,’, q0’ , F’)
where
EXAMPLES [2]
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
There are only finitely many Boolean matrices
in M(Q), so we do have a dfa.
M’ accepts T(L, x.2x)
It suffices to note that is the above is a transition
sequence in M’, ending with a final state if and
only if
• string a1a2……an drives M from q0 to qn and,
• from qn there is a path of length 2n in M leading
to an accepting state.
EXAMPLES [3]
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
Square Root of a Language L:
L’ = { w | w2 ∈ L }
M=(Q, ∑,  , q0, F) with adjacency matrix ∆
accepting regular language L
Approach: Similar to exponential function
example, we construct the DFA
M’=(Q’,∑,’, q0’ , F’) that accepts L’, where
EXAMPLES [4]

To see why this construction works, note that
a string x drives the machine M’ from state
(q0, I) to state (p, A) if and only if
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x drives the machine M from q0 to p and
A= ∆x.
However, Ip∆x IF’=1 means that, by Lemma 2,
x drives M from state p to a final state.
Therefore, x is accepted by M’ if and only if xx
is accepted by M.
EXAMPLES [5]
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
The two definitions of regularity-preserving
functions: T(L, f ) and T’(L, f ) can be shown
to be equivalent by a small variation of the
construction done in the previous examples.
For example, if we define the final states to
be F"={( p, A) | Iq0∆x IF’=1}, we get a dfa
accepting T’(L, x.x);
Ultimate Periodicity &
Regularity Preservation
Ultimate Periodicity & Regularity Preservation [1]
Regularity Preservation – Another
Characterization:
A function f is r.p. iff for any square Boolean
Matrix A,
∃ m ,∃ k , A f i= A f im , ∀ i≥k
A function that is u.p. w.r.t. powers of Boolean
Matrix will be abbreviated as m.u.p.
Examples of m.u.p. functions:
Identity Function :)
x
f

x
=2
Exponentiation:
Ultimate Periodicity & Regularity Preservation [2]
Proof: If f is m.u.p., then f is r.p
L = L(M), M = (Q, ∑, ∂, s, F), a DFA
M' = (Q', ∑, ∂', s', F')
Q' = [ p , i ,  f i : p∈Q , 0≤imk ]
f 0
[
s
,
0,

]
s' = f i
f [i1] 
m
[
p
,
i
,

]
[ p , a ,[i1]k , 
]
∂'( , a) = t
F' = { }
[ p , i , A]: I p , A , I F =1
M' accepts P(f, L)!
Note: [i]mk =i , i≤k ; ki−k  mod m , ik
m
k
Ultimate Periodicity & Regularity Preservation [3]
If f is r.p., then f is m.u.p.
Proof: Any Boolean Matrix ∆ can be interpreted as a directed graph (edges labeled, say, 1).
A ∆ and an index [i, j] defines an NFA N, with start state i and final state j. Call L(i,j) = L(N[i,j]).
f k 
So, { u.p. (A consequence of Q(f, L[i,j]) k :  i , } is j
being regular!), with period, say, m i , j
=1
That is, ∃ m ,∃ n , ∀ k≥n , if,lj =1 iff if,lm
j
The product of all such periods is the period for the m.u.p. function f.
Ultimate Periodicity & Regularity Preservation [4]
Theorem: If f is m.u.p., then so is  1≤i≤x f i
Proof: Let 1 , 2 ,... n be ALL boolean matrices of size n.
f ki
We can find m, k s.t. ∀ j∈{1,2, ... n }, ∀ i≥0,  j
 k 
 km
f kim
= j
 kt.m
Consider 
, and let ,  (a multiple ,
,... 
of m) be such that  = 
This is also the base for the inductive proof.
Assume the hypothesis ( )
 y= y
 y1= y f y1
 y1
  y f y1
 y f y1

=
=
since =n.m
 y1
 y1

=
Ultimate Periodicity & Regularity Preservation [5]
Consequence: Subtraction does not preserve
regularity.
Counter-example: Consider f(x) = x!, and
g(x) = x!, x is prime,
= 2(x!), x is not prime
Both are regularity preserving (Proof that g is
also r.p. is in the paper).
Please remember that, for a function to be r.p,
the inverse image of every singleton set under
−1

g−
f

0=PRIMES∪{1 }, which is not u.p !
it, should be u.p.
CONCLUSION
We have successfully applied MAAT, a
matrix-based approach to automata theory,to
the study of regularity-preserving functions.
 By virtue of the characterization of regularity
preserving functions in terms of m.u.p., many
properties of regularity-preserving functions:
both old and new, have been treated in an
uniform way.
References:
 Guo-Qiang Zhang: “Automata, Boolean
Matrices, and Ultimate Periodicity”. Submitted
to Information and Computation (Manuscript
available at http://www.cs.uga.edu/~gqz)

Thank You