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The Power of Counting Logics on Restricted Classes of Finite Structures [chapter]

Anuj Dawar, David Richerby
Lecture Notes in Computer Science  
that the logic does capture P on specific classes of structures.  ...  Although Cai, Fürer and Immerman have shown that fixedpoint logic with counting (IFP + C) does not express all polynomialtime properties of finite structures, there have been a number of results demonstrating  ...  There is a growing body of work studying the finite model theory of restricted classes of structures, where the restrictions are essentially borrowed from graph structure theory.  ... 
doi:10.1007/978-3-540-74915-8_10 fatcat:vrbvnsu6gfbjjcn6abzya3ju6q

Reflections on Finite Model Theory

Phokion G. Kolaitis
2007 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)  
Main Finding: All major computational complexity classes, including P, NP, and PSPACE, can be characterized in terms of definability in various logics on classes of finite structures.  ...  Reinforces the unity of computation and logic. Yields machine-independent characterizations of computational complexity classes.  ...  Restricted Classes of Finite Structures Progressive shift of emphasis from the class of all finite structures to restricted classes of finite structures.  ... 
doi:10.1109/lics.2007.39 dblp:conf/lics/Kolaitis07 fatcat:3eulyaqcgbfdzihya6kzjxm2ru

On the expressive power of counting

Stéphane Grumbach, Christophe Tollu
1995 Theoretical Computer Science  
It is essential in the case of unordered structures. Our aim is to understand the expressive power gained with a limited counting ability.  ...  We investigate the expressive power of various extensions of first-order, inductive, and infinitary logic with counting quantifiers.  ...  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

On the Bisimulation Invariant Fragment of Monadic Σ1 in the Finite [chapter]

Anuj Dawar, David Janin
2004 Lecture Notes in Computer Science  
We investigate the expressive power of existential monadic second-order logic (monadic Σ1) on finite transition systems.  ...  We show that on finite unary transition systems the bisimulation invariant fragment of monadic Σ1 is equivalent to bisimulation-invariant monadic second order logic itself or, equivalently, the mu-calculus  ...  One reason why the question of the equivalence of these logics is so different in the finite is that, once we restrict ourselves to finite structures, we no longer have a tree model property.  ... 
doi:10.1007/978-3-540-30538-5_19 fatcat:v5ygznzd5zcprnovcpma4hm3xq

Page 4089 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
Results are given on formulas with a bounded number of variables over finite structures, and these results are applied to other ques- tions in finite model theory, such as the power of logics with fixed  ...  In particular, the class of planar graphs is not in Lix,(Q,).” 98g:03083 03C13 03C75 68Q15 Otto, Martin (D-AACH-GI; Aachen) The expressive power of fixed-point logic with counting.  ... 

Characterising Choiceless Polynomial Time with First-Order Interpretations

Erich Gradel, Wied Pakusa, Svenja Schalthofer, Lukasz Kaiser
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
It is a strict extension of fixed-point logic with counting, but to date the question is open whether it expresses all polynomial-time properties of finite structures.  ...  While this is very convenient and powerful for the design of abstract computations on structures, it makes the analysis of the expressive power of CPT rather difficult.  ...  If one imposes no further restriction on BGS-logic then every decidable class of structures can be defined in BGS-logic.  ... 
doi:10.1109/lics.2015.68 dblp:conf/lics/GradelPSK15 fatcat:o6zudj655jdxzk54icw22ulauu

Is Polynomial Time Choiceless? [chapter]

Erich Grädel, Martin Grohe
2015 Lecture Notes in Computer Science  
A long time ago, Yuri Gurevich made precise the problem of whether there is a logic capturing polynomial-time on arbitrary finite structures, and conjectured that no such logic exists.  ...  We survey some recent results on this logic.  ...  It follows that one-dimensional PIL, when evaluated on the expansions of finite structures by an ordered numerical sort, has precisely the expressive power of FPC.  ... 
doi:10.1007/978-3-319-23534-9_11 fatcat:n6rczztse5g4nezyzahgrmis3q

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

Argimiro Arratia, Carlos E. Ortiz
2006 Lecture Notes in Computer Science  
When restricted to monadic second order variables our logic of proportional quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting  ...  When restricted to monadic second order variables our logic of proportional quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting  ...  of the formulae on finite structures where all predicates are restricted to act subject to an integer modulo.  ... 
doi:10.1007/11682462_14 fatcat:eyo4z4oitfg6df3xbbchve53ve

The monadic theory of finite representations of infinite words

Anuj Dawar, David Janin
2007 Information Processing Letters  
classes of graphs coincide with classes of graphs definable by means of (an extension of) finite state ω-word automata.  ...  We show that, on these graphs, the bisimulation invariant fragment of monadic Σ1 equals the bisimulation invariant fragment of monadic second order logic itself, and that MSO-definable bisimulation closed  ...  We say that a class of graphs is (counting) bisimulation closed in the finite when the above property holds restricted to finite graphs only.  ... 
doi:10.1016/j.ipl.2007.02.014 fatcat:nvdw46e7wzfa3b5qbllnc76vve

Page 895 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
ISBN 3-540-62037-0 Descriptive complexity theory, a part of finite model theory, fo- cuses on studying the relationship between the computational complexity of a class of finite structures and its definability  ...  The author diagnoses this shortfall in the expressive power of first-order logic as being due to its lacking two features: recursion and counting.  ... 

Generalized quantifiers and pebble games on finite structures

Phokion G. Kolaitis, Jouko A. Väänänen
1995 Annals of Pure and Applied Logic  
First-order logic is known to have a severely limited expressive power on finite structures.  ...  quantifiers on finite structures.  ...  We allow the logic to be defined relative to a restricted class of structures, like the class of all finite structures, or the class of all ordered finite structures.  ... 
doi:10.1016/0168-0072(94)00025-x fatcat:awup6imfonb4tfu3ovybcfyvhe

Logical Definability of Counting Functions

Kevin J. Compton, Erich Grädel
1996 Journal of computer and system sciences (Print)  
For a logic L, *L is the class of functions on finite structures counting the tuples (T , cÄ ) satisfying a given formula (T , cÄ ) in L.  ...  The most well studied class of counting functions is *P, which consists of the functions counting the accepting computation paths of a nondeterministic polynomial-time Turing machine.  ...  THE POWER OF LOGICAL COUNTING CLASSES We now investigate the power of *FO and classes *L for logics L that extend FO.  ... 
doi:10.1006/jcss.1996.0069 fatcat:fksc6h5r2jdg5nohcko3pxeawq

Finite Model Reasoning in Expressive Fragments of First-Order Logic

Lidia Tendera
2017 Electronic Proceedings in Theoretical Computer Science  
the image of the standard translation of modal logic to first-order logic.  ...  This applies most notably to the guarded fragment, where quantifiers are appropriately relativized by atoms, and the fragment defined by restricting the number of variables to two.  ...  Restricted classes of structures In modal correspondence theory various conditions on the accessibility relations allow one to restrict the class of Kripke structures considered, e.g. to transitive structures  ... 
doi:10.4204/eptcs.243.4 fatcat:47pmommxorc7hbjp7y2v3fxiuu

Graded modal logic and counting bisimulation [article]

Martin Otto
2019 arXiv   pre-print
We focus on showing expressive completeness of graded multi-modal logic for those first-order properties of pointed Kripke structures that are preserved under counting bisimulation equivalence among all  ...  or among just all finite pointed Kripke structures.  ...  signature of ϕ; (ii) over the class of all finite pointed Kripke structures ϕ is expressible in graded modal logic CML: ϕ ≡ fin ϕ ′ for some ϕ ∈ CML.  ... 
arXiv:1910.00039v1 fatcat:v7htrysqvbep7kx7ayxdtnytte

Finite and Algorithmic Model Theory (Dagstuhl Seminar 17361)

Anuj Dawar, Erich Grädel, Phokion G. Kolaitis, Thomas Schwentick, Marc Herbstritt
2018 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 17361 "Finite and Algorithmic Model Theory".  ...  We formalize agent-based models as stochastic processes whose states are metafinite models, and we define a notion of abstraction.  ...  Our main results are conditions that imply an abstraction is sound, and further conditions that imply it preserves the Markov property.  ... 
doi:10.4230/dagrep.7.9.1 dblp:journals/dagstuhl-reports/DawarGKS17 fatcat:teyilwvdgrd6tch2gdb4d4dd5m
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