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 $\Sigma$. 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 languages $(P\Sigma^*)\cap T$, $(\Sigma^*P)\cap T$, $(\Sigma^* P\Sigma^*)\cap T$, and $$(\Sigma^* \mathbin{\operatorname{shu}} P)\cap T$$, where $\operatorname{shu}$ is the shuffle operation. The state complexity $\kappa(L)$ 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 pattern-matching languages, where $\kappa(P)\le m$, and $\kappa(T)\le n$: $\kappa((P\Sigma^*)\cap T) \le mn$; $$\kappa((\Sigma^*P)\cap T) \le 2<sup>{m-1}n$$; $$\kappa((\Sigma^{<em>P\Sigma^</em>)\cap} T) \le (2<sup>{m-2}+1)n$$; and $$\kappa((\Sigma^{*\mathbin{\operatorname{shu}}} P)\cap T) \le (2<sup>{m-2}+1)n$$. 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: $\kappa((w\Sigma^*)\cap T_n) \le m+n-1$; $$\kappa((\Sigma^*w)\cap T_n) \le (m-1)n-(m-2)$$; $$\kappa((\Sigma^{<em>w\Sigma^</em>)\cap} T_n) \le (m-1)n$$; and $$\kappa((\Sigma^{*\mathbin{\operatorname{shu}}} w)\cap T_n) \le (m-1)n$$. For unary languages, we have a tight upper bound of $m+n-2$ in all eight of the aforementioned cases.