State complexity of pattern matching in regular languages

Abstract

In a simple pattern matching problem one has a pattern w and a text t, which are words over a finite alphabet Σ. One may ask whether w occurs in t, and if so, where? More generally, we may have a set P of patterns and a set T of texts, where P and T are regular languages. We are interested whether any word of T begins with a word of P , ends with a word of P , has a word of P as a factor, or has a word of P as a subsequence. Thus we are interested in the languageswhere is the shuffle operation. The state complexity κ(L) of a regular language L is the number of states in the minimal deterministic finite automaton recognizing L. We derive the following upper bounds on the state complexities of our patternmatching languages, where κ(P ) m, and κ(T ) n: κ((P Σ * ) ∩ T ) mn; κ((Σ * P ) ∩ T ) 2 m−1 n; κ((Σ * P Σ * ) ∩ T ) (2 m−2 + 1)n; and κ((Σ * P ) ∩ T ) (2 m−2 + 1)n. We prove that these bounds are tight, and that to meet them, the alphabet must have at least two letters in the first three cases, and at least m − 1 letters in the last case. We also consider the special case where P is a single word w, and obtain the following tight upper bounds: κ((wΣ * ) ∩ T ) m + n − 1; κ((Σ * w) ∩ T ) (m − 1)n − (m − 2); κ((Σ * wΣ * ) ∩ T ) (m − 1)n; and κ((Σ * w) ∩ T ) (m − 1)n. For unary languages, we have a tight upper bound of m + n − 2 in all eight of the aforementioned cases.