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q
tr1 tr2
p1 p2 …………… pn
Fig. 21.11: A closed connection.
A generic form of closed pattern thus is given by placei, transitionj,
placek, transitionr, placei. Formally, if we denote
a place by ‘a’,
the arc from a place to a transition by ‘b’,
the transition by ‘c’ and
the arc from a transition to a place by ‘d’,
then we could represent the closed connection of fig. 21.11 by (abcdadcb)
(abcdadcb)….(abcdadcb) where a sub-string in each pair of parenthesis
describes a closed connection in the pattern. It may, however, be added that a
certain order is implied here, when counting the closed connections.
Definition 21.2: Given two special strings α = abcd and β= adcb, the
resulting pattern,
i) (αβ) n , n>0 corresponds to redundant or subsumed rules,
ii) α n , n>0 corresponds to circular rules, and
iii) (α 2 β 2 ) n , n > 0 corresponds to conflicting rules,
where αβ denotes concatenation of α and β and x n denotes concatenation of
x with itself n times.
We now present a grammar for syntactic recognition of the different
patterns. Let G = (Vn, Vt, P, S) where the grammar G has four tuples; Vn = a
set of non-terminals, Vt = a set of terminals, P = a set of production/re-write
rules, S = a starting symbol. The following definitions of the grammar are
useful for recognizing four distinct types of patterns.