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

2017
*
Lecture Notes in Computer Science
*

We extend this upper

doi:10.1007/978-3-319-62809-7_19
fatcat:yoiajoyy55apxbgktpjc5a53ni
*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

1995
*
Annals of Pure and Applied Logic
*

introduction of several

doi:10.1016/0168-0072(94)00054-7
fatcat:yiyg3tc35zhqln6lqitptdac7e
*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 [7] sequent calculus*for*classical*logic*LK. ...##
###
Sharpened lower bounds for cut elimination

2012
*
Journal of Symbolic Logic (JSL)
*

We present sharpened lower

doi:10.2178/jsl/1333566644
fatcat:s4cuzlrwrfeqply23rh2jc5y6m
*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

2012
*
Foundations of Software Technology and Theoretical Computer Science
*

We prove,

doi:10.4230/lipics.fsttcs.2012.112
dblp:conf/fsttcs/GajarskyH12
fatcat:wit3in442fe4pn23tikkr6d54y
*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]

2013
*
arXiv
*
pre-print

We discuss

arXiv:1003.4865v4
fatcat:pxpwl4gumbgfpmvsrrdx3z7u4i
*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]

2010
*
Lecture Notes in Computer Science
*

This paper introduces a nested sequent system

doi:10.1007/978-3-642-16242-8_13
fatcat:ajmlwu7k7ja2vbd4m4v6wjcyui
*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

2005
*
Journal of Logic, Language and Information
*

We prove ExpTime-completeness of

doi:10.1007/s10849-005-5787-x
fatcat:5s4v5fm4jnhh5j2jssnnc7jx7m
*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 [9] 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

2002
*
Information and Computation
*

Our NEXP upper

doi:10.1006/inco.2001.2953
fatcat:xjveochdqvagjn6kwqvzk5fg3u
*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]

2022
*
arXiv
*
pre-print

Further,

arXiv:2204.03903v1
fatcat:embmrmipmjdhhec4iebjuulnru
*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

2005
*
Journal of Logic and Computation
*

Kripke-style semantics is suggested

doi:10.1093/logcom/exi037
fatcat:dd22qypiknhudfc4cr2avhadvy
*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)

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]

2013
*
arXiv
*
pre-print

Given

arXiv:1212.2747v4
fatcat:lzati3rb4nbqfe63k5zp53gdfq
*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]

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

1994
*
Journal of Symbolic Logic (JSL)
*

This paper discusses lower

doi:10.2307/2275906
fatcat:et43wqbhunfrhied2ethd2neby
*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]

1994
*
Finite Automata, Formal Logic, and Circuit Complexity
*

A little background first, about

doi:10.1007/978-1-4612-0289-9_7
fatcat:jn5teolrhrf3tfysd3dnrku7qu
*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*[26] , and applied Ehrenfeucht-Fraïssé games to prove ... Modular*Quantifiers*with a*Bounded*Number of*Variables**In*[21] we investigated what happens when we*bound**the*number of*variables**in*sentences that contain modular*quantifiers*. ...
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