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### Structure Theorem and Strict Alternation Hierarchy for FO2 on Words [chapter]

P. Weis, N. Immerman
Lecture Notes in Computer Science
For both languages, our structure theorems show exactly what is expressible using a given quantifier depth, n, and using m blocks of alternating quantifiers, for any m ≤ n.  ...  We prove precise structure theorems that characterize the exact expressive power of first-order logic with two variables on words.  ...  Alternation hierarchy for FO 2 [<] We define alternation rankers and prove our structure theorem (Theorem 4.5) for FO 2 m,n [<] .  ...

### On FO 2 Quantifier Alternation over Words [chapter]

Manfred Kufleitner, Pascal Weil
2009 Lecture Notes in Computer Science
We show that each level of the quantifier alternation hierarchy within FO^2[<] -- the 2-variable fragment of the first order logic of order on words -- is a variety of languages.  ...  hierarchy -- and conjecturally equal to it.  ...  Then M ∈ FO 2 m+1 \ FO 2 m−1 by Theorem 2.11. ⊓ ⊔ The fact that the R m and L m form strict hierarchies (Proposition 2.6), together with Theorem 2.11, proves that the F O 2 m hierarchy is infinite.  ...

### The Half-Levels of the FO2 Alternation Hierarchy

Lukas Fleischer, Manfred Kufleitner, Alexander Lauser
2016 Theory of Computing Systems
The alternation hierarchy in two-variable first-order logic FO 2 [<] over words was shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing.  ...  ., for every integer m one can decide whether a given regular language is definable in Σ 2 m . Among other techniques, the proof relies on an extension of block products to ordered monoids.  ...  It is known that the languages definable in FO 2 m form a strict hierarchy [43] .  ...

### The Regular Languages of First-Order Logic with One Alternation [article]

Corentin Barloy and Michaël Cadilhac and Charles Paperman and Thomas Zeume
2022 arXiv   pre-print
The regular languages with a neutral letter expressible in first-order logic with one alternation are characterized.  ...  This shows that the so-called Central Conjecture of Straubing holds for Σ_2 over languages with a neutral letter, the first progress on the Conjecture in more than 20 years.  ...  Acknowledgments We wish to thank Nikhil Balaji and Sébastien Tavenas. The last author acknowledges financial support by the DFG grant ZE 1235/2-1.  ...

### Block Products and Nesting Negations in FO2 [chapter]

Lukas Fleischer, Manfred Kufleitner, Alexander Lauser
2014 Lecture Notes in Computer Science
The alternation hierarchy in two-variable first-order logic FO 2 [<] over words was recently shown to be decidable by Kufleitner and Weil, and independently by Krebs and Straubing.  ...  Thus, the m th level of the FO 2 -alternation hierarchy is the Boolean closure of Σ 2 m .  ...  It is known that the languages definable in FO 2 m form a strict hierarchy, see e.g. [16] .  ...

### First-Order Logic with Two Variables and Unary Temporal Logic

Kousha Etessami, Moshe Y. Vardi, Thomas Wilke
1997 BRICS Report Series
We investigate the power of first-order logic with only two variables over<br />omega-words and finite words, a logic denoted by FO2.  ...  Our proof of the small model property for unary-TL is based on an analysis of unary-TL types.  ...  There are some remaining questions: (1) Is the FO 2 quantifier alternation hierarchy strict?  ...

### Understanding Semantic Web and Ontologies: Theory and Applications [article]

2010 arXiv   pre-print
Semantic Web is actually an extension of the current one in that it represents information more meaningfully for humans and computers alike.  ...  In fact, Ontology is a key technique with which to annotate semantics and provide a common, comprehensible foundation for resources on the Semantic Web.  ...  In other words, it is the process of change or modification of the structure of an ontology in order to make it suitable for purposes other than the original one. There are two types of translation.  ...

### The Regular Languages of First-Order Logic with One Alternation

Corentin Barloy, Michael Cadilhac, Charles Paperman, Thomas Zeume
2022 Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science
The regular languages with a neutral letter expressible in firstorder logic with one alternation are characterized.  ...  This shows that the so-called Central Conjecture of Straubing holds for Σ 2 over languages with a neutral letter, the first progress on the Conjecture in more than 20 years.  ...  ACKNOWLEDGMENTS We wish to thank Nikhil Balaji and Sébastien Tavenas for fruitful discussions.  ...

### Ontology revision [chapter]

Norman Foo
1995 Lecture Notes in Computer Science
This paper is partly a summary of early results, and partly an account of ongoing work in this area.  ...  A number of sub-disciplines in artificial intelligence, philosophy and recursion theory have results that are relevant to ontology expansion even though their motivations were quite different.  ...  ACKNOWLEDGEMENTS I am indebted to my colleagues in the Knowledge Systems Group for many discussions pertinent to this work.  ...

### Around Dot Depth Two [chapter]

Kamal Lodaya, Paritosh K. Pandya, Simoni S. Shah
2010 Lecture Notes in Computer Science
Automata for this class are not known, nor is its precise placement within the dot-depth hierarchy of starfree languages.  ...  Using ideas from Straubing's "delay theorem", we extend our earlier work [LPS08] to propose partially-ordered two-way deterministic finite automata with look-around (po2dla) and a new interval temporal  ...  KL would like to thank Manfred Kufleitner, Howard Straubing and Pascal Weil for discussions about the status of the logic FO 2 [<, S] within the dot-depth hierarchy during a visit in May-June '07 to LaBRI  ...

### Deterministic Logics for UL [article]

Paritosh K. Pandya, Simoni S. Shah
2014 arXiv   pre-print
We revisit this language class with emphasis on notion of unambiguity and develop on the concept of Deterministic Logics for UL.  ...  Logics such as TL[F,P], FO2[<] are not deterministic and have been shown to characterize UL using algebraic methods.  ...  Since then, several diverse and unexpected characterizations have emerged for this language class: ∆ 2 [<] in the quantifier-alternation hierarchy of first-order definable languages [PW97] , the two variable  ...

### Formal language theory of logic fragments [article]

Alexander Lauser, Universität Stuttgart, Universität Stuttgart
2015
A combinatorial description in terms of so-called rankers is given for all full levels as well as all half levels of the quantifier alternation hierarchy of FO2 over the order predicate, both with and  ...  Also in the second part, effective algebraic criteria describing all full levels as well as all half levels of the quantifier alternation hierarchy of FO2 over several signatures are given, yielding in  ...  The structure of the quantifier alternation hierarchy is depicted in Figure 4 .2.  ...

### Abstracts of Working Papers in Economics

1997 Abstracts of Working Papers in Economics
TI Report on the Investment Performance of Defaulted Bonds for 1994 and 1987-1994. AU Altman, Edward I.;  ...  We prove the conjecture for n less than or equal to 3, and provide bounds on the number of NE's in m x n nondegenerate games when min (m,n) is less than or equal to 4. Money.  ...  TI The Term Structure of Interest Rates: Alternative Approaches and Their Implications for the Valuation of Contingent Claims. AA New York University.  ...

### Membership and separation problems inside two-variable first order logic

Viktor Henriksson
2022
variables, a fragment known as FO2[For languages of finite words, membership for these fragments has already been solved.  ...  This thesis studies the expressive power of restricted fragments of first order logic on words with the order predicate.  ...  FO 2 quantifier alternation hierarchy, in other words, for all levels of the ranker comparison hierarchy.  ...

### An Introduction to Model Versioning [chapter]

Petra Brosch, Gerti Kappel, Philip Langer, Martina Seidl, Konrad Wieland, Manuel Wimmer
2012 Lecture Notes in Computer Science
Thereby, we aim at equipping students and researchers alike that are new to this domain with enough information for commencing to contribute to this challenging research area. P. Brosch et al.  ...  In this tutorial, we give an introduction to the foundations of model versioning, the underlying technologies for processing models and their evolution, as well as the state of the art in model versioning  ...  Merging textual languages with a strict syntactic structure (such as XML) might be more appropriate than merging languages which mix several properties of potentially independent concepts into one line  ...
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