13 research outputs found
Decidability in the logic of subsequences and supersequences
We consider first-order logics of sequences ordered by the subsequence
ordering, aka sequence embedding. We show that the \Sigma_2 theory is
undecidable, answering a question left open by Kuske. Regarding fragments with
a bounded number of variables, we show that the FO2 theory is decidable while
the FO3 theory is undecidable
Existential Definability over the Subword Ordering
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the ?? (i.e., existential) fragment is undecidable, already for binary alphabets A.
However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable.
We show that if |A| ? 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments ?_i: If |A| ? 3, then a relation is definable in ?_i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the ?_i-fragments for i ? 2 of the pure logic, where the words of A^* are not available as constants
Existential Definability over the Subword Ordering
We study first-order logic (FO) over the structure consisting of finite words
over some alphabet , together with the (non-contiguous) subword ordering. In
terms of decidability of quantifier alternation fragments, this logic is
well-understood: If every word is available as a constant, then even the
(i.e., existential) fragment is undecidable, already for binary
alphabets . However, up to now, little is known about the expressiveness of
the quantifier alternation fragments: For example, the undecidability proof for
the existential fragment relies on Diophantine equations and only shows that
recursively enumerable languages over a singleton alphabet (and some auxiliary
predicates) are definable. We show that if , then a relation is
definable in the existential fragment over with constants if and only if it
is recursively enumerable. This implies characterizations for all fragments
: If , then a relation is definable in if and
only if it belongs to the -th level of the arithmetical hierarchy. In
addition, our result yields an analogous complete description of the
-fragments for of the pure logic, where the words of
are not available as constants
Combinatorial Algorithms for Subsequence Matching: A Survey
In this paper we provide an overview of a series of recent results regarding
algorithms for searching for subsequences in words or for the analysis of the
sets of subsequences occurring in a word.Comment: This is a revised version of the paper with the same title which
appeared in the Proceedings of NCMA 2022, EPTCS 367, 2022, pp. 11-27 (DOI:
10.4204/EPTCS.367.2). The revision consists in citing a series of relevant
references which were not covered in the initial version, and commenting on
how they relate to the results we survey. arXiv admin note: text overlap with
arXiv:2206.1389
The Edit Distance to k-Subsequence Universality
A word u is a subsequence of another word w if u can be obtained from w by deleting some of its letters. In the early 1970s, Imre Simon defined the relation ?_k (called now Simon-Congruence) as follows: two words having exactly the same set of subsequences of length at most k are ?_k-congruent. This relation was central in defining and analysing piecewise testable languages, but has found many applications in areas such as algorithmic learning theory, databases theory, or computational linguistics. Recently, it was shown that testing whether two words are ?_k-congruent can be done in optimal linear time. Thus, it is a natural next step to ask, for two words w and u which are not ?_k-equivalent, what is the minimal number of edit operations that we need to perform on w in order to obtain a word which is ?_k-equivalent to u.
In this paper, we consider this problem in a setting which seems interesting: when u is a k-subsequence universal word. A word u with alph(u) = ? is called k-subsequence universal if the set of subsequences of length k of u contains all possible words of length k over ?. As such, our results are a series of efficient algorithms computing the edit distance from w to the language of k-subsequence universal words