52 research outputs found
The Height of Piecewise-Testable Languages with Applications in Logical Complexity
The height of a piecewise-testable language L is the maximum length of the words needed to define L by excluding and requiring given subwords. The height of L is an important descriptive complexity measure that has not yet been investigated in a systematic way. This paper develops a series of new techniques for bounding the height of finite languages and of languages obtained by taking closures by subwords, superwords and related operations.
As an application of these results, we show that FO^2(A^*, subword), the two-variable fragment of the first-order logic of sequences with the subword ordering, can only express piecewise-testable properties and has elementary complexity
Multi-Element Long Distance Dependencies: Using SPk Languages to Explore the Characteristics of Long-Distance Dependencies
In order to successfully model Long Distance Dependencies (LDDs) it is necessary to under-stand the full-range of the characteristics of the LDDs exhibited in a target dataset. In this paper, we use Strictly k-Piecewise languages to generate datasets with various properties. We then compute the characteristics of the LDDs in these datasets using mutual information and analyze the impact of factors such as (i) k, (ii) length of LDDs, (iii) vocabulary size, (iv) forbidden strings, and (v) dataset size. This analysis reveal that the number of interacting elements in a dependency is an important characteristic of LDDs. This leads us to the challenge of modelling multi-element long-distance dependencies. Our results suggest that attention mechanisms in neural networks may aide in modeling datasets with multi-element long-distance dependencies. However, we conclude that there is a need to develop more efficient attention mechanisms to address this issue
Matching Patterns with Variables Under Simon's Congruence
We introduce and investigate a series of matching problems for patterns with
variables under Simon's congruence. Our results provide a thorough picture of
these problems' computational complexity
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
Long-Distance Phonotactics as Tier-Based Strictly 2-Local Languages
This paper shows that the properties of locality observed for patterns of long-distance consonant agreement and disagreement belong to a well-defined and relatively simple class of subregular formal languages (stringsets) called the Tier-based Strictly 2-Local languages, and argues that analyzing them as such has desirable theoretical implications. Specifically, treating the two elements of a long-distance dependency as adjacent segments on the computationally defined notion of a tier allows for a unified account of locality that necessarily extends to the cross-linguistically variable behavior of neutral segments (transparency and blocking). This result is significant in light of the long-standing and persistent problems that long-distance dependencies have raised for phonological theory, with current approaches still predicting several pathological patterns that have little or no empirical support
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
Covering and separation for logical fragments with modular predicates
For every class of word languages, one may associate a decision
problem called -separation. Given two regular languages, it asks
whether there exists a third language in containing the first
language, while being disjoint from the second one. Usually, finding an
algorithm deciding -separation yields a deep insight on
.
We consider classes defined by fragments of first-order logic. Given such a
fragment, one may often build a larger class by adding more predicates to its
signature. In the paper, we investigate the operation of enriching signatures
with modular predicates. Our main theorem is a generic transfer result for this
construction. Informally, we show that when a logical fragment is equipped with
a signature containing the successor predicate, separation for the stronger
logic enriched with modular predicates reduces to separation for the original
logic. This result actually applies to a more general decision problem, called
the covering problem
- …