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### Decidability, Complexity, and Expressiveness of First-Order Logic Over the Subword Ordering [article]

Simon Halfon, Philippe Schnoebelen, Georg Zetzsche
2021 arXiv   pre-print
We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the Σ_1 theory is undecidable (already over two letters).  ...  entire first-order theory is decidable.  ...  The logic of subwords. In this paper we consider the first-order logic FO(A * , ⊑) of words over some alphabet A = {a, b, c, . . .} equipped with the subword relation ⊑.  ...

### Complexity of Counting First-Order Logic for the Subword Order

Dietrich Kuske, Christian Schwarz, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science
for first-order logic).  ...  This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word.  ...  M F C S 2 0 2 0 61:10 Complexity of Counting First-Order Logic for the Subword Order For first-order formulas ϕ, this result can be found in [7, Cor. 7.4 and Thm. 7.5] .  ...

### Graph Logics with Rational Relations and the Generalized Intersection Problem

Pablo Barcelo, Diego Figueira, Leonid Libkin
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science
Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword (factor) or subsequence.  ...  We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable  ...  The NLOGSPACE-data complexity matches that of RPQs, CRPQs, and ECRPQs  ,  ,  , and the combined complexity matches that of first-order logic, or ECRPQs without extra relations.  ...

### Page 1068 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews
This approach allows us to construct first-order theories in which lambda-abstraction and quantification can be easily expressed as terms of the language.”  ...  The present paper contains new results in the case of infinite words with linear subword complexity, and Sturmian words as a particular case.  ...

### Languages Ordered by the Subword Order [chapter]

Dietrich Kuske, Georg Zetzsche
2019 Green Chemistry and Sustainable Technology
For such structures, we consider the extension of first-order logic by threshold-and modulo-counting quantifiers.  ...  These results extend earlier ones where only the language of all words without the cover relation and fragments of first-order logic were considered.  ...  We are interested in logics over the subword order. Prior work on this has concentrated on first-order logic where the universe consists of all words over some alphabet.  ...

### Graph Logics with Rational Relations

Pablo Barcelo, Diego Figueira, Leonid Libkin, Nicole Schweikardt
2013 Logical Methods in Computer Science
Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword or subsequence.  ...  We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable  ...  of logical queries over graphs.  ...

### Graph logics with rational relations

Pablo Barceló, Pablo Muñoz
2014 Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) - CSL-LICS '14
Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword or subsequence.  ...  We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable  ...  of logical queries over graphs.  ...

### Languages ordered by the subword order [article]

Dietrich Kuske, Georg Zetzsche
2019 arXiv   pre-print
For such structures, we consider the extension of first-order logic by threshold- and modulo-counting quantifiers.  ...  These results extend earlier ones where only the language of all words without the cover relation and fragments of first-order logic were considered.  ...  Thus, to get a decidable theory, one has to restrict the expressiveness of first-order logic considerably.  ...

### Graph Logics with Rational Relations

Pablo Barceló, Pablo Muñoz
2017 ACM Transactions on Computational Logic
Such logics use conditions on paths expressed by regular languages and relations, but they often need to be extended by rational relations such as subword or subsequence.  ...  We prove that for several basic and commonly used rational relations, the intersection problem with regular relations is either undecidable (e.g., for subword or suffix, and some generalizations), or decidable  ...  of logical queries over graphs.  ...

### The complexity of first-order and monadic second-order logic revisited

Markus Frick, Martin Grohe
2004 Annals of Pure and Applied Logic
We establish a number of similar lower bounds for the model-checking problem for first-order logic, for example, on the class of all trees.  ...  In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic.  ...  A well-known result of Kamp  states that LTL and first-order logic have the same expressive power on words.  ...

### Two-variable logic on data words

Mikołaj Bojańczyk, Claire David, Anca Muscholl, Thomas Schwentick, Luc Segoufin
2011 ACM Transactions on Computational Logic
It should be noted however that full monadic second-order logic (and even first-order logic) over data words is known to be undecidable [NSV04] .  ...  In this sense, the decidability of EMSO 2 (∼, <, +1) can be seen as an extension of the classical decidability result of monadic second-order logic over words.  ...  Over graphs or over arbitrary relational structures, first-order logic is undecidable, while its two-variable fragment is decidable [Mor75] .  ...

### CaRet With Forgettable Past

Laura Bozzelli
2009 Electronical Notes in Theoretical Computer Science
The logic CaRet is less expressive than NVPA and is easily expressible in the first-order fragment FO μ of MSO μ . However, it is an open question whether CaRet is FO μ -complete .  ...  This situation is quite different from the logic PLTL, for which the separation property ensures that LTL has the same expressiveness as PLTL and, thus, corresponds to the class of sentences of the first-order  ...  Then, in Subsection 3.2, we show that NCaRet is expressively complete for the first order fragment FO μ of MSO μ  , which extends the classical monadic second order logic (MSO) over words with a binary  ...

### Separability by Short Subsequences and Subwords

Piotr Hofman, Wim Martens, Marc Herbstritt
2015 International Conference on Database Theory
We study the complexity of separability of regular languages by combinations of subsequences or subwords of a given length k.  ...  Typically, S comes from a simple, less expressive class of languages than I and E.  ...  We would like to thank the anonymous reviewers of ICDT 2015 for many helpful remarks. This work was supported by DFG grant MA4938/2-1.  ...

### The height of piecewise-testable languages and the complexity of the logic of subwords [article]

Prateek Karandikar, Philippe Schnoebelen
2019 arXiv   pre-print
As an application of these results, we show that FO^2(A^*,), the two-variable fragment of the first-order logic of sequences with the subword ordering, can only express piecewise-testable properties and  ...  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.  ...  Acknowledgments We thank the anonymous reviewers for their many helpful suggestions.  ...

### The height of piecewise-testable languages and the complexity of the logic of subwords

Prateek Karandikar, Philippe Schnoebelen
2019 Logical Methods in Computer Science
As an application of these results, we show that \$\mathsf{FO}^2(A^*,\sqsubseteq)\$, the two-variable fragment of the first-order logic of sequences with the subword ordering, can only express piecewise-testable  ...  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.  ...  Acknowledgments We thank the anonymous reviewers for their many helpful suggestions.  ...
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