Languages Ordered by the Subword Order

Kuske, Dietrich; Zetzsche, Georg

doi:10.1007/978-3-030-17127-8_20

Cited by 22 publications

(19 citation statements)

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“…The theorem in fact holds for (a larger signature, and) a more expressive logic, first-order logic extended by a modulo counting quantifier [8]. The proof is by interpreting the structure in Presburger arithmetic, (N, +), which is known to be decidable in this logic.…”

Section: Languages Ordered By the Subword Ordermentioning

confidence: 90%

“…b. Words of a bounded, context-free language, under the subword order [8]. c. Basic process algebra under the subword order.…”

Section: Initial Limit Datalogmentioning

confidence: 99%

“…Unfortunately, this does mean that all elements of W must be constants in background theory, which makes it harder to obtain decidability (for example, the subword order is a WQO, but with constants, even the ∃-theory becomes undecidable [8,15]). Remark III.1.…”

Section: Higher-order Limit Datalogmentioning

confidence: 99%

“…The subword relation is a simple and important example of a WQO. [8] study the decidability of first-order theories (and fragments thereof) of languages with the subword order. Recall that a language L ⊆ ∆ * is bounded if L ⊆ w * 1 • • • w * n for some n ≥ 0, and w 1 , • • • , w n ∈ ∆ * .…”

Section: Languages Ordered By the Subword Ordermentioning

confidence: 99%

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Initial Limit Datalog: a New Extensible Class of Decidable Constrained Horn Clauses

Burn¹,

Ong²,

Ramsay³

et al. 2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We present initial limit Datalog, a new extensible class of constrained Horn clauses for which the satisfiability problem is decidable. The class may be viewed as a generalisation to higher-order logic (with a simple restriction on types) of the first-order language limit Datalog Z (a fragment of Datalog modulo linear integer arithmetic), but can be instantiated with any suitable background theory. For example, the fragment is decidable over any countable well-quasi-order with a decidable first-order theory, such as natural number vectors under componentwise linear arithmetic, and words of a bounded, context-free language ordered by the subword relation. Formulas of initial limit Datalog have the property that, under some assumptions on the background theory, their satisfiability can be witnessed by a new kind of term model which we call entwined structures. Whilst the set of all models is typically uncountable, the set of all entwined structures is recursively enumerable, and model checking is decidable.

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Section: Languages Ordered By the Subword Ordermentioning

confidence: 90%

“…b. Words of a bounded, context-free language, under the subword order [8]. c. Basic process algebra under the subword order.…”

Section: Initial Limit Datalogmentioning

confidence: 99%

Section: Higher-order Limit Datalogmentioning

confidence: 99%

Section: Languages Ordered By the Subword Ordermentioning

confidence: 99%

See 2 more Smart Citations

Initial Limit Datalog: a New Extensible Class of Decidable Constrained Horn Clauses

Burn¹,

Ong²,

Ramsay³

et al. 2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

View full text Add to dashboard Cite

show abstract

“…This versatility of scattered factors is also highlighted by the many contexts in which this concept appears. For instance, in [111,55,74], various theories of logic were developed around the notion of scattered factors which are analysed mostly with automata theory tools and discussed in connection to applications in formal verification.…”

Section: Chapter Introductionmentioning

confidence: 99%

On Special k-Spectra, k-Locality, and Collapsing Prefix Normal Words

Fleischmann¹

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The domain of Combinatorics on Words, first introduced by Axel Thue in 1906, covers by now many subdomains. In this work we are investigating scattered factors as a representation of non-complete information and two measurements for words, namely the locality of a word and prefix normality, which have applications in pattern matching. In the first part of the thesis we investigate scattered factors: A word u is a scattered factor of w if u can be obtained from w by deleting some of its letters. That is, there exist the (potentially empty) words u1, u2, . . . , un, and v0,v1,...,vn such that u = u1u2 ̈ ̈ ̈un and w = v0u1v1u2v2 ̈ ̈ ̈unvn. First, we consider the set of length-k scattered factors of a given word w, called the k-spectrum of w and denoted by ScatFactk(w). We prove a series of properties of the sets ScatFactk(w) for binary weakly-0-balanced and, respectively, weakly-c-balanced words w, i.e., words over a two- letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has c occurrences more than the other. In particular, we consider the question which cardinalities n = | ScatFactk (w)| are obtainable, for a positive integer k, when w is either a weakly-0- balanced binary word of length 2k, or a weakly-c-balanced binary word of length 2k ́ c. Second, we investigate k-spectra that contain all possible words of length k, i.e., k-spectra of so called k-universal words. We present an algorithm deciding whether the k-spectra for given k of two words are equal or not, running in optimal time. Moreover, we present several results regarding k-universal words and extend this notion to circular universality that helps in investigating how the universality of repetitions of a given word can be determined. We conclude the part about scattered factors with results on the reconstruction problem of words from scattered factors that asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word w P {a, b} ̊ can be reconstructed from the number of occurrences of at most min(|w|a, |w|b) + 1 scattered factors of the form aib, where |w|a is the number of occurrences of the letter a in w. Moreover, we generalise the result to alphabets of the form {1, . . . , q} by showing that at most ∑q ́1 |w|i (q ́ i + 1) scattered factors suffices to reconstruct w. Both results i=1 improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here. In the second part we consider patterns, i.e., words consisting of not only letters but also variables, and in particular their locality. A pattern is called k-local if on marking the pattern in a given order never more than k marked blocks occur. We start with the proof that determining the minimal k for a given pattern such that the pattern is k-local is NP- complete. Afterwards we present results on the behaviour of the locality of repetitions and palindromes. We end this part with the proof that the matching problem becomes also NP-hard if we do not consider a regular pattern - for which the matching problem is efficiently solvable - but repetitions of regular patterns. In the last part we investigate prefix normal words which are binary words in which each prefix has at least the same number of 1s as any factor of the same length. First introduced in 2011 by Fici and Lipták, the problem of determining the index (amount of equivalence classes for a given word length) of the prefix normal equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems.

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