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Complexity Results for First-Order Two-Variable Logic with Counting

Leszek Pacholski, WiesL aw Szwast, Lidia Tendera
2000 SIAM journal on computing (Print)  
Let C 2 p denote the class of rst order sentences with two variables and with additional quanti ers \there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all  ...  The satis ability problem for logics with a bounded number of variables has applications in arti cial intelligence, notably in modal logics (see e.g. 22]), where counting comes in the context of graded  ...  We would, however, like to emphasize that it is easy to adapt all notions used in this paper and to modify the proofs in order to obtain the same results also for the full rst-order two-variable logic  ... 
doi:10.1137/s0097539797323005 fatcat:zdgkvn3wefbxvanmwopwyu7a3y

On Spatial Conjunction as Second-Order Logic [article]

Viktor Kuncak, Martin Rinard
2004 arXiv   pre-print
form of second-order quantification in two-variable logic is decidable.  ...  We construct an embedding from first-order logic with spatial conjunction into second-order logic, and more surprisingly, an embedding from full second order logic into first-order logic with spatial conjunction  ...  The consequences of the translation from second-order logic to spatial conjunction for two-variable logic with counting were crystallized in discussion with Greta Yorsh.  ... 
arXiv:cs/0410073v1 fatcat:ezlx6bafgzb5veyhymdgqr2rne

Counting of Teams in First-Order Team Logics

Anselm Haak, Juha Kontinen, Fabian Müller, Heribert Vollmer, Fan Yang, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of first-order logic in Tarski's  ...  class generated by dependence logic with respect to first-order reductions.  ...  #Σ 1 CNF − * is complete for #FO(=(. . . )) team with respect to first-order reductions. Having proven our results for dependence logic FO(=(. . . )), we turn to inclusion logic FO(⊆).  ... 
doi:10.4230/lipics.mfcs.2019.19 dblp:conf/mfcs/HaakKMVY19 fatcat:msle7mcok5cyndnisqgsnzi4ii

Elimination Ordering in Lifted First-Order Probabilistic Inference

Seyed Mehran Kazemi, David Poole
2014 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
In this paper, we show that these heuristics are inefficient for relational models, because they fail to consider the population sizes associated with logical variables.  ...  Many of these methods choose an order to eliminate (or branch on) the parameterized random variables.  ...  Thanks to David Buchman for valuable feedback.  ... 
doi:10.1609/aaai.v28i1.8847 fatcat:vqizgyofqfabbjbzehg2bwyldm

Weighted First-Order Model Counting in the Two-Variable Fragment With Counting Quantifiers [article]

Ondrej Kuzelka
2020 arXiv   pre-print
In this paper we extend this result to the two-variable fragment with counting quantifiers.  ...  It is known due to the work of Van den Broeck et al [KR, 2014] that weighted first-order model counting (WFOMC) in the two-variable fragment of first-order logic can be solved in time polynomial in the  ...  " and a donation from X-Order Lab.  ... 
arXiv:2007.05619v3 fatcat:nkrdkqwvprdopjqo7feu5ey76i

Counting Proportions of Sets: Expressive Power with Almost Order [chapter]

Argimiro Arratia, Carlos E. Ortiz
2006 Lecture Notes in Computer Science  
We present a second order logic of proportional quantifiers, , which is essentially a first order language extended with quantifiers that act upon second order variables of a given arity r, and count the  ...  We present a second order logic of proportional quantifiers, SOLP, which is essentially a first order language extended with quantifiers that act upon second order variables of a given arity r, and count  ...  The purpose of this paper is to introduce a second order counting logic with built-in order that contains fragments whose expressive power is meaningful for Complexity Theory, and where the replacement  ... 
doi:10.1007/11682462_14 fatcat:eyo4z4oitfg6df3xbbchve53ve

Weighted First-Order Model Counting in the Two-Variable Fragment With Counting Quantifiers

Ondrej Kuzelka
2021 The Journal of Artificial Intelligence Research  
In this paper we extend this result to the two-variable fragment with counting quantifiers.  ...  It is known due to the work of Van den Broeck, Meert and Darwiche that weighted first-order model counting (WFOMC) in the two-variable fragment of first-order logic can be solved in time polynomial in  ...  The author is grateful to the anonymous reviewers for their helpful comments and Guy Van den Broeck for pointing out the problem of inference in the two-variable fragment of FOL with counting quantifiers  ... 
doi:10.1613/jair.1.12320 fatcat:cs7nhec47vds5ex4atp5nit7gi

Finite Model Reasoning in Expressive Fragments of First-Order Logic

Lidia Tendera
2017 Electronic Proceedings in Theoretical Computer Science  
Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing  ...  the image of the standard translation of modal logic to first-order logic.  ...  Historical Remarks The observation that model logic can be seen as a fragment of the two-variable first-order logic was made in 1981 by Gabbay [10] .  ... 
doi:10.4204/eptcs.243.4 fatcat:47pmommxorc7hbjp7y2v3fxiuu

On the expressive power of counting

Stéphane Grumbach, Christophe Tollu
1995 Theoretical Computer Science  
We investigate the expressive power of various extensions of first-order, inductive, and infinitary logic with counting quantifiers.  ...  Our aim is to understand the expressive power gained with a limited counting ability. We consider two problems: (i) unnested counters, and (ii) counters with no free variables.  ...  Acknowledgements We are greatly indebted to Guy Fayolle for the collaboration on the proof of Theorem 5.3. We also thank Jianwen Su for comments on an earlier draft.  ... 
doi:10.1016/0304-3975(95)00026-s fatcat:vqzgtgau2ne65dtgzzv3hcbexu

Finite and Algorithmic Model Theory (Dagstuhl Seminar 17361)

Anuj Dawar, Erich Grädel, Phokion G. Kolaitis, Thomas Schwentick, Marc Herbstritt
2018 Dagstuhl Reports  
Our main results are conditions that imply an abstraction is sound, and further conditions that imply it preserves the Markov property.  ...  and the m-variable fragment of first-order logic.  ...  logic with counting quantifiers for large values of k.  ... 
doi:10.4230/dagrep.7.9.1 dblp:journals/dagstuhl-reports/DawarGKS17 fatcat:teyilwvdgrd6tch2gdb4d4dd5m

On logics with two variables

Erich Grädel, Martin Otto
1999 Theoretical Computer Science  
A classical result due to Mortimer says that the two-variable fragment of first-order logic, denoted FO*, has the finite model property and is therefore decidable for satisfiability.  ...  It has turned out that the satisfiability problems for two-variable logics with full first-order quantification are usually much harder (and indeed highly undecidable in many cases) than the satisfiability  ...  These properties involve some very basic forms of counting. Two-variable first-order with counting C2 extends F02 by allowing first-order counting quantifiers ZIGM, Slam , El="' for all m > 1.  ... 
doi:10.1016/s0304-3975(98)00308-9 fatcat:krqnks7mfbdyfamtm4m7swdkry

Just Count the Satisfied Groundings: Scalable Local-Search and Sampling Based Inference in MLNs

Deepak Venugopal, Somdeb Sarkhel, Vibhav Gogate
2015 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
of groundings of a first-order formula that are true given a truth assignment to all of its ground atoms.  ...  The main computational bottleneck in various sampling based and local-search based inference algorithms for Markov logic networks (e.g., Gibbs sampling, MC-SAT, MaxWalksat, etc.) is computing the number  ...  Acknowledgments This work was supported by the DARPA Probabilistic Programming for Advanced Machine Learning Program under AFRL prime contract number FA8750-14-C-0005.  ... 
doi:10.1609/aaai.v29i1.9676 fatcat:3c2gm4tjwjcshexuoajygqsxu4

Counting of Teams in First-Order Team Logics [article]

Anselm Haak, Juha Kontinen, Fabian Müller, Heribert Vollmer, Fan Yang
2020 arXiv   pre-print
These team-based logics are closely related to existential second-order logic and its fragments, hence our results also shed light on the complexity of counting for extensions of FO in Tarski's semantics  ...  In this paper we extend this study to classes beyond #P and extensions of first-order logic with team semantics.  ...  Let us now recall relevant previous results on the descriptive complexity of counting problems. We begin by defining the most important complexity classes for counting problems. Definition 3.  ... 
arXiv:1902.00246v3 fatcat:gisjetmrvvguhpsd5zq2farqzu

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

Kamal Lodaya, A. V. Sreejith
2017 Lecture Notes in Computer Science  
Etessami, Vardi and Wilke [5] showed that satisfiability of two-variable first order logic FO 2 [<] on word models is Nexptime-complete.  ...  If we allow the more powerful modulo counting quantifiers of Straubing, Thérien and Thomas [22] (we call this two-variable fragment FOmod 2 [<, succ]), satisfiability becomes Expspace-complete.  ...  We would like to thank four DLT referees and the DLT program committee for their suggestions to improve this paper.  ... 
doi:10.1007/978-3-319-62809-7_19 fatcat:yoiajoyy55apxbgktpjc5a53ni

Complex Markov Logic Networks: Expressivity and Liftability [article]

Ondrej Kuzelka
2020 arXiv   pre-print
We then observe that discrete Fourier transform can be computed using weighted first order model counting (WFOMC) with complex weights and use this observation to design an algorithm for computing relational  ...  We introduce complex MLNs, which use complex-valued weights, and we show that, unlike standard MLNs with real-valued weights, complex MLNs are fully expressive.  ...  " and a donation from X-Order Lab.  ... 
arXiv:2002.10259v2 fatcat:yhp4qxrx3zg2jfqkidkajqsm3q
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