Timothy Ng

State Complexity of Neighbourhoods and Approximate Pattern Matching

The neighbourhood of a language L with respect to an additive distance consists of all strings that have distance at most the given radius from some string of L. We show that the worst case (deterministic) state complexity of a radius r neighbourhood of a language recognized by an n state nondeterministic finite automaton A is \((r+2)^n\). The lower bound construction uses an alphabet of size linear in n. We show that the worst case state complexity of the set of strings that contain a substring within distance r from a string recognized by A is \((r+2)^{n-2} + 1\).

Quasi-Distances and Weighted Finite Automata

We show that the neighbourhood of a regular language \(L\) with respect to an additive quasi-distance can be recognized by an additive weighted finite automaton (WFA). The size of the WFA is the same as the size of an NFA (nondeterministic finite automaton) for \(L\) and the construction gives an upper bound for the state complexity of a neighbourhood of a regular language with respect to a quasi-distance. We give a tight lower bound construction for the determinization of an additive WFA using an alphabet of size five. The previously known lower bound construction needed an alphabet that is linear in the number of states of the WFA.

State Complexity of Insertion

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

State Complexity of Prefix Distance

The prefix distance between strings x and y is the number of symbol occurrences in the strings that do not belong to the longest common prefix of x and y. The suffix and the substring distance are defined analogously in terms of the longest common suffix and longest common substring, respectively, of two strings. We show that the set of strings within prefix distance k from an n state DFA (deterministic finite automaton) language can be recognized by a DFA with \((k+1) \cdot n - \frac{k(k+1)}{2}\) states and this number of states is needed in the worst case. Also we give tight bounds for the nondeterministic state complexity of the set of strings within prefix, suffix or substring distance k from a regular language.

Descriptional Complexity of Error Detection

The neighbourhood of a language L consists of all strings that are within a given distance from a string of L. For example, additive distances or the prefix-distance are regularity preserving in the sense that the neighbourhood of a regular language is always regular. For error detection and error correction applications an important question is to determine the size of the minimal deterministic finite automaton (DFA) needed to recognize the neighbourhood of a language recognized by an n state DFA. This paper surveys recent work on the state complexity of neighbourhoods of regularity preserving distances.

Prefix Distance Between Regular Languages

The prefix distance between two words x and y is defined as the number of symbol occurrences in the words that do not belong to the longest common prefix of x and y. We show how to model the prefix distance using weighted transducers. We use the weighted transducers to compute the prefix distance between two regular languages by a transducer-based approach originally used by Mohri for an algorithm to compute the edit distance. We also give an algorithm to compute the inner prefix distance of a regular language.

Outfix-Guided Insertion

Motivated by work on bio-operations on DNA strings, we consider an outfix-guided insertion operation that can be viewed as a generalization of the overlap assembly operation on strings studied previously. As the main result we construct a finite language L such that the outfix-guided insertion closure of L is nonregular. We consider also the closure properties of regular and deterministic context-free languages under the outfix-guided insertion operation and decision problems related to outfix-guided insertion. Deciding whether a language recognized by a deterministic finite automaton is closed under outfix-guided insertion can be done in polynomial time.

Consensus String Problem for Multiple Regular Languages

The consensus string (or center string, closest string) of a set S of strings is defined as a string which is within a radius r from all strings in S. It is well-known that the consensus string problem for a finite set of equal-length strings is NP-complete. We study the consensus string problem for multiple regular languages. We define the consensus string of languages \(L_1, \ldots , L_k\) to be within distance at most r to some string in each of the languages \(L_1, \ldots , L_k\). We also study the complexity of some parameterized variants of the consensus string problem. For a constant k, we give a polynomial time algorithm for the consensus string problem for k regular languages using additive weighted finite automata. We show that the consensus string problem for multiple regular languages becomes intractable when k is not fixed. We also examine the case when the length of the consensus string is given as part of input.

State complexity of prefix distance

Abstract The prefix distance between strings x and y is the number of symbol occurrences in the strings that do not belong to the longest common prefix of x and y. The suffix and the substring distances are defined analogously in terms of the longest common suffix and longest common substring, respectively, of two strings. We show that the set of strings within prefix distance k from an n state DFA (deterministic finite automaton) language can be recognized by a DFA with ( k + 1 ) ⋅ n − k ( k + 1 ) 2 states and that this number of states is needed in the worst case. Also we give tight bounds for the nondeterministic state complexity of the set of strings within prefix, suffix or substring distance k from a regular language.

State Complexity of Prefix Distance of Subregular Languages

The neighbourhood of a regular language of constant radius with respect to the prefix distance is always regular. We give upper bounds and matching lower bounds for the size of the minimal deterministic finite automaton (DFA) needed for the radius k prefix distance neighbourhood of an n state DFA that recognizes, respectively, a finite, a prefix-closed and a prefix-free language. For prefix-closed languages the lower bound automata are defined over a binary alphabet. For finite and prefix-free regular languages the lower bound constructions use an alphabet that depends on the size of the DFA and it is shown that the size of the alphabet is optimal.