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### Two-Variable First Order Logic with Counting Quantifiers: Complexity Results [chapter]

Kamal Lodaya, A. V. Sreejith
2017 Lecture Notes in Computer Science
We extend this upper bound to the slightly stronger logic FO 2 [<, succ, ≡], which allows checking whether a word position is congruent to r modulo q, for some divisor q and remainder r.  ...  A more general counting quantifier, FOunC 2 [<, succ], makes the logic undecidable.  ...  We would like to thank four DLT referees and the DLT program committee for their suggestions to improve this paper.  ...

### Generalizing theorems in real closed fields

Matthias Baaz, Richard Zach
1995 Annals of Pure and Applied Logic
introduction of several quantifiers of the same type in one step, (3) LKB and the first-order schemata corresponding to Dedekind cuts and the supremum principle.  ...  Jan Krajihk posed the following problem: Is there is a generalization result in the theory of real closed fields of the form: If A(1 + ... + 1) (n occurrences of 1) is provable in length k for all n E  ...  Calculi, terms, uni~cation In the course of this paper we shall work with two logical calculi: The first one is Gentzen's  sequent calculus for classical logic LK.  ...

### Sharpened lower bounds for cut elimination

Samuel R. Buss
2012 Journal of Symbolic Logic (JSL)
We present sharpened lower bounds on the size of cut free proofs for first-order logic.  ...  Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas  ...  The pattern for ψ 3 generalizes to form skeletal trees of ψ i , i ≥ 1. The formation rules are as follows. The quantified variables in ψ i are x u , for u ∈ {0, 1} <i .  ...

### Faster Deciding MSO Properties of Trees of Fixed Height, and Some Consequences

Jakub Gajarsky, Petr Hlineny, Marc Herbstritt
2012 Foundations of Software Technology and Theoretical Computer Science
We prove, in the universe of trees of bounded height, that for any MSO formula with m variables there exists a set of kernels such that the size of each of these kernels can be bounded by an elementary  ...  In the second part of the paper we use this kernel structure to show that FO has the same expressive power as MSO 1 on the graph classes of bounded shrub-depth.  ...  only for graphs of bounded vertex cover, but also for those of bounded tree-depth.  ...

### Logical complexity of graphs: a survey [article]

Oleg Pikhurko, Oleg Verbitsky
2013 arXiv   pre-print
We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices.  ...  Also, we trace the behavior of the descriptive complexity of a graph as the logic becomes more restrictive (for example, only definitions with a bounded number of variables or quantifier alternations are  ...  We are grateful to Joel Spencer for the fruitful collaboration on the subject of this survey and for allowing us to use his unpublished ideas in the proof of Lemma 6.8 and in Section 7.1.1.  ...

### How to Universally Close the Existential Rule [chapter]

Kai Brünnler
2010 Lecture Notes in Computer Science
This paper introduces a nested sequent system for predicate logic. The system features a structural universal quantifier and a universally closed existential rule.  ...  The system for free logic is interesting because it has no need for an existence predicate.  ...  It binds variables in the same way as usual quantifiers, these variables are then called structurally bound.  ...

### Guarded Fragments with Constants

Balder ten Cate, Massimo Franceschet
2005 Journal of Logic, Language and Information
We prove ExpTime-completeness of the satisfiability problem for (loosely) guarded first-order formulas with a bounded number of variables and an unbounded number of constants.  ...  Guarded fragments with constants are interesting because of their connection to hybrid logic. 1 Marx  does explicitly state and prove the decidability of the satisfiability problem for loosely ∀-guarded  ...  Note that this will only polynomially increase the length of the formula, due to the fact that both the width and the quantifier depth of φ is bounded (keep in mind that φ is in normal form). 5 The number  ...

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

Kousha Etessami, Moshe Y. Vardi, Thomas Wilke
2002 Information and Computation
Our NEXP upper bound for FO 2 satisfiability has the advantage of being in terms of the quantifier depth of the input formula.  ...  An interesting and related aspect of our NEXP upper bound is that the time bound is only in terms of the quantifier depth of the FO 2 formula.  ...  First-Order Logic with Two Variables We will prove the following two upper bounds for the complexity of FO 2 satisfiability. Proof.  ...

### On Presburger arithmetic extended with non-unary counting quantifiers [article]

Peter Habermehl
2022 arXiv   pre-print
Further, the residue in modulo-counting quantifiers is given as a term. Our main result shows that satisfaction for this logic is decidable in two-fold exponential space.  ...  The bounds obtained this way allow to replace quantification in the original formula to integers of bounded size which then implies the first result mentioned above.  ...  The authors thank Christian Schwarz from TU Ilmenau for proofreading the paper and for correcting two mistakes.  ...

### On Kripke-style Semantics for the Provability Logic of Gödel's Proof Predicate with Quantifiers on Proofs

Rostislav Yavorskiy
2005 Journal of Logic and Computation
Kripke-style semantics is suggested for the provability logic with quantifiers on proofs corresponding to the standard Gödel proof predicate. It is proved that the set of valid formulas is decidable.  ...  The arithmetical completeness is still an open issue.  ...  Acknowledgements I am thankful to Sergei Artemov for drawing my attention to this problem, introducing me into the subject, and also for his regular support and encouragements.  ...

### Circuits, Logic and Games (Dagstuhl Seminar 15401)

Mikolaj Bojanczyk, Meena Mahajan, Thomas Schwentick, Heribert Vollmer, Marc Herbstritt
2016 Dagstuhl Reports
for circuits.  ...  complexity and circuit lower bounds inherently using methods from logic and algebra Proof systems with low circuit complexity Dynamic complexity: understanding the dynamic expressive power of small depth  ...  These talks covered results in two-variable first-order logic; dynamic complexity; graph colouring; database theory; circuit lower bounds; logics on words; and semigroup techniques.  ...

### Bounds for the quantifier depth in finite-variable logics: Alternation hierarchy [article]

Christoph Berkholz, Andreas Krebs, Oleg Verbitsky
2013 arXiv   pre-print
Given two structures G and H distinguishable in k (first-order logic with k variables), let A^k(G,H) denote the minimum alternation depth of a k formula distinguishing G from H.  ...  G from H with bounded quantifier depth, in Π_i this requires quantifier depth Ω(n^2).  ...  Acknowledgement The authors are thankful to Oleg Pikhurko for his useful suggestions.  ...

### On the Number of Quantifiers as a Complexity Measure [article]

Ronald Fagin and Jonathan Lenchner and Nikhil Vyas and Ryan Williams
2022 arXiv   pre-print
In 1981, Neil Immerman described a two-player game, which he called the "separability game" , that captures the number of quantifiers needed to describe a property in first-order logic.  ...  Immerman's paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the  ...  Then, if x 1 , . . . , x m are the names of the bound variables in φ, we can use these same variable names for the first m bound variables in ψ, and as well in φ • ψ, with no change in meaning.  ...

### On Gödel's theorems on lengths of proofs I: Number of lines and speedup for arithmetics

Samuel R. Buss
1994 Journal of Symbolic Logic (JSL)
This paper discusses lower bounds for proof length, especially as measured by number of steps (inferences).  ...  The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as axioms and allows all generalizations of axioms as axioms.  ...  From this it follows that the qb-depth of formulas in E is bounded by the total number of quantifier blocks in the proof skeleton P S .  ...

### Modular Quantifiers [chapter]

Howard Straubing
1994 Finite Automata, Formal Logic, and Circuit Complexity
A little background first, about the ordinary kind of quantifier: Properties of words over a finite alphabet Σ can be expressed in predicate logic by interpreting variables as positions {0, 1, . . . ,  ...  He showed that the levels of the dot-depth hierarchy corresponded precisely to level of the quantifier alternation hierarchy within first-order logic  , and applied Ehrenfeucht-Fraïssé games to prove  ...  Modular Quantifiers with a Bounded Number of Variables In  we investigated what happens when we bound the number of variables in sentences that contain modular quantifiers.  ...
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