1,155 research outputs found
Decision Problems For Convex Languages
In this paper we examine decision problems associated with various classes of
convex languages, studied by Ang and Brzozowski (under the name "continuous
languages"). We show that we can decide whether a given language L is prefix-,
suffix-, factor-, or subword-convex in polynomial time if L is represented by a
DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the
case that a regular language is not convex, we prove tight upper bounds on the
length of the shortest words demonstrating this fact, in terms of the number of
states of an accepting DFA. Similar results are proved for some subclasses of
convex languages: the prefix-, suffix-, factor-, and subword-closed languages,
and the prefix-, suffix-, factor-, and subword-free languages.Comment: preliminary version. This version corrected one typo in Section
2.1.1, line
Regular Languages meet Prefix Sorting
Indexing strings via prefix (or suffix) sorting is, arguably, one of the most
successful algorithmic techniques developed in the last decades. Can indexing
be extended to languages? The main contribution of this paper is to initiate
the study of the sub-class of regular languages accepted by an automaton whose
states can be prefix-sorted. Starting from the recent notion of Wheeler graph
[Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting
to labeled graphs-we investigate the properties of Wheeler languages, that is,
regular languages admitting an accepting Wheeler finite automaton.
Interestingly, we characterize this family as the natural extension of regular
languages endowed with the co-lexicographic ordering: when sorted, the strings
belonging to a Wheeler language are partitioned into a finite number of
co-lexicographic intervals, each formed by elements from a single Myhill-Nerode
equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with
states admits an equivalent Wheeler DFA (WDFA) with at most
states that can be computed in time. This is in sharp contrast with
general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper
superset of the WDFAs, a -time online algorithm to sort acyclic
WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By
contribution (i), our algorithms can also be used to index any WNFA at the
moderate price of doubling the automaton's size. (iii) We provide a
minimization theorem that characterizes the smallest WDFA recognizing the same
language of any input WDFA. The corresponding constructive algorithm runs in
optimal linear time in the acyclic case, and in time in the
general case. (iv) We show how to compute the smallest WDFA equivalent to any
acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version
with new results (W-MH theorem, linear determinization), added author:
Giovanna D'Agostin
Languages convex with respect to binary relations, and their closure properties
A language is prefix-convex if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefix-convex languages include prefix-closed languages at one end of the spectrum, and prefix-free languages, which include prefix codes, at the other. In a similar way, we define suffix-, bifix-, factor-, and subword-convex languages and their closed and free counterparts. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages. We generalize these notions to arbitrary binary relations on the set of all words over a given alphabet, and study the closure properties of such languages
Wheeler Languages
The recently introduced class of Wheeler graphs, inspired by the
Burrows-Wheeler Transform (BWT) of a given string, admits an efficient index
data structure for searching for subpaths with a given path label, and lifts
the applicability of the Burrows-Wheeler transform from strings to languages.
In this paper we study the regular languages accepted by automata having a
Wheeler graph as transition function, and prove results on determination,
Myhill_Nerode characterization, decidability, and closure properties for this
class of languages
On the interplay between Babai and Cerny's conjectures
Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with states in
this class, we prove that the reset thresholds are upper-bounded by
and can attain the value . In addition, we study diameters
of the pair digraphs of permutation automata and construct -state
permutation automata with diameter .Comment: 21 pages version with full proof
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