The DFAs of Finitely Different Languages

We show that a well-known family of deterministic finite automata can be used to distinguish distinct binary strings of the same length from every start state. Further, we establish almost matching lower and upper bounds on the number of states of such automata necessary to achieve this type of separation. Our result improves the currently best known linear upper bound for arbitrary DFA.

Let PT-DFA mean a deterministic finite automaton whose transition relation is a partial function. We present an algorithm for minimizing a PT-DFA in $O(m \lg n)$ time and $O(m+n+\alpha)$ memory, where $n$ is the number of states, $m$ is the number of defined transitions, and $\alpha$ is the size of the alphabet. Time consumption does not depend on $\alpha$, because the $\alpha$ term arises from an array that is accessed at random and never initialized. It is not needed, if tr...

We revisit the popular \emph{delayed deterministic finite automaton} (\ddfa{}) compression algorithm introduced by Kumar~et~al.~[SIGCOMM 2006] for compressing deterministic finite automata (DFAs) used in intrusion detection systems. This compression scheme exploits similarities in the outgoing sets of transitions among states to achieve strong compression while maintaining high throughput for matching. The \ddfa{} algorithm and later variants of it, unfortunately, require a...

It was shown before that the NP-hard problem of deterministic finite automata (DFA) identification can be effectively translated to Boolean satisfiability (SAT). Modern SAT-solvers can tackle hard DFA identification instances efficiently. We present a technique to reduce the problem search space by enforcing an enumeration of DFA states in depth-first search (DFS) or breadth-first search (BFS) order. We propose symmetry breaking predicates, which can be added to Boolean formu...

We examine questions involving nondeterministic finite automata where all states are final, initial, or both initial and final. First, we prove hardness results for the nonuniversality and inequivalence problems for these NFAs. Next, we characterize the languages accepted. Finally, we discuss some state complexity problems involving such automata.

This is a tutorial on finite automata. We present the standard material on determinization and minimization, as well as an account of the equivalence of finite automata and monadic second-order logic. We conclude with an introduction to the syntactic monoid, and as an application give a proof of the equivalence of first-order definability and aperiodicity.

This paper develops a free energy theory from physics including the variational principles for automata and languages and also provides algorithms to compute the energy as well as efficient algorithms for estimating the nondeterminism in a nondeterministic finite automaton. This theory is then used as a foundation to define a semantic similarity metric for automata and languages. Since automata are a fundamental model for all modern programs while languages are a fundamental ...

The separating words problem asks for the size of the smallest DFA needed to distinguish between two words of length <= n (by accepting one and rejecting the other). In this paper we survey what is known and unknown about the problem, consider some variations, and prove several new results.

These lecture notes are intended as a supplement to Moore and Mertens' The Nature of Computation or as a standalone resource, and are available to anyone who wants to use them. Comments are welcome, and please let me know if you use these notes in a course. There are 61 exercises. I emphasize that automata are elementary playgrounds where we can explore the issues of deterministic and nondeterministic computation. Unlike P vs. NP, we can prove that nondeterminism is equival...

Deterministic finite automata are one of the simplest and most practical models of computation studied in automata theory. Their conceptual extension is the non-deterministic finite automata which also have plenty of applications. In this article, we study these models through the lens of succinct data structures where our ultimate goal is to encode these mathematical objects using information-theoretically optimal number of bits along with supporting queries on them efficien...