Renormalized measure of regular languages

Abstract

This paper modifies the signed real measure of regular languages, which has been reported in recent literature for analysis and synthesis of discrete event supervisory control laws. A new concept of renormalized measure is introduced to eliminate a user-selectable parameter in the present version of the language measure. The concept of measure renormalization is illustrated by an example.

Generalized language measure families of probabilistic finite state systems

“…0 þ , i.e., the probability of termination approaching zero. The resulting -parameterized model coincides with the desired non-terminating model in the limit (Chattopadhyay and Ray 2006). …”

“…Summary, conclusions, and future work This paper formulates and validates a concept of generalization of signed real measure of regular languages, which also leads to renormalization (Chattopadhyay and Ray 2006) of the normalized measure and eliminates the need for a user-selectable parameter in the original concept of language measure Ray (2005). These generalizations are achieved through a trace measure that is characterized by both initial and terminal states as well as the length of the trace and the choice of a vector norm for renormalization.…”

“…This implies that the preceding analysis is applicable to the case of terminating probabilistic languages (Garg 1992a, b) that have a non-zero probability of termination (arising from either intentional design or unmodelled dynamics of the plant automaton) at each state. If the probability of termination at each state, or equivalently the probability of transition to the (deadlock) dump state from each of the other states q i 2 Q, is set identically equal to 2 ð0, 1Þ, then the e &-matrix and the &-matrix can be -parameterized as follows (Chattopadhyay and Ray 2006): e &ðÞ ð1 À Þ e P and &ðÞ ð1 À ÞP,…”

“…Note that lim !0 þ l of the normalized language measure does not exist. This problem has been circumvented via renormalization (Chattopadhyay and Ray 2006) as explained below.…”

“…These generalizations are achieved through a new concept of trace measure that is characterized by both initiating and terminating states as well as the length of the trace and the choice of a vector norm (Naylor and Sell 1982). The concept of generalization can be viewed as renormalization (Chattopadhyay and Ray 2006) of the (normalized) language measure (Ray 2005).…”