## Finite Variable Logics In Descriptive Complexity Theory (1998)

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Venue: | Bulletin of Symbolic Logic |

Citations: | 10 - 1 self |

### BibTeX

@ARTICLE{Grohe98finitevariable,

author = {Martin Grohe},

title = {Finite Variable Logics In Descriptive Complexity Theory},

journal = {Bulletin of Symbolic Logic},

year = {1998},

volume = {4},

pages = {345--398}

}

### OpenURL

### Abstract

this article.

### Citations

497 |
Finite Model Theory
- Ebbinghaus, Flum
- 1999
(Show Context)
Citation Context ...are presented in a rather condensed form in Sections 3 and 4. Most proofs are only sketched, if given at all. The reader who worries about the details here is referred to the literature (for example, =-=[12, 19, 64]-=-). Section 5 is concerned with the question of whether we can compute a model for a given L k -theory. More precisely, we study the complexity of inverting the L k -invariants. Furthermore, we discuss... |

270 |
Generalized first-order spectra and polynomial-time recognizable sets
- Fagin
- 1974
(Show Context)
Citation Context ...ach C F[ ], the class C is L-definable if, and only if, it is K-computable. The archetypal example is Fagin's Theorem: FINITE VARIABLE LOGICS IN DESCRIPTIVE COMPLEXITY THEORY 35 Theorem 8.3 (Fagin [2=-=1-=-]). Existential second-order logic 1 1 captures NPTIME. The proof of Fagin's Theorem also shows that second-order logic captures the polynomial hierarchy PH. Moreover, it is known that the extension ... |

264 |
The classical decision problems
- Börger, Grädel, et al.
- 1997
(Show Context)
Citation Context ...there is a projective plane of order 12. But of course there is no such algorithm; it is well-known that satisfiability of first-order sentences with three variables is undecidable (see, for example, =-=[8]-=-). However, the situation is different for complete C 3 -theories. Given a reasonable finite representation of a complete C k -theory (see below for such a representation), we can compute a model of t... |

244 | Structure and complexity of relational queries
- Chandra, Harel
- 1982
(Show Context)
Citation Context ... FINITE VARIABLE LOGICS IN DESCRIPTIVE COMPLEXITY THEORY 37 So it is no surprise that the question for a logic that effectively captures PTIME has first occurred in database theory: Chandra and Harel =-=[11]-=- posed it in a slightly different formulation. As we have seen in the previous subsection, Gurevich [35] made the notion of a logic effectively capturing PTIME precise. He conjectured that no logic ef... |

135 | An optimal lower bound on the number of variables for graph identification
- Cai, Fürer, et al.
- 1981
(Show Context)
Citation Context ...elements x". Counting is of a very low computational complexity. The logics L k can only count up to k, though. So it seems reasonable to extend them by counting quantifiers. Cai, Furer, and Imme=-=rman [9]-=- related the counting Appeared in Bulletin of Symbolic Logic 4:345--398, 1998. cs1998, Association for Symbolic Logic. 1 2 MARTIN GROHE logics to the so-called Lehman-Weisfeiler algorithm, a combinato... |

118 |
Generic computation and its complexity
- Abiteboul, Vianu
- 1991
(Show Context)
Citation Context ... a technical tool to obtain results concerning fixed-point logics, which are central in descriptive complexity. Once they had realized this possibility, Kolaitis, Vardi, and others (see, for example, =-=[1, 2, 17, 52, 54, 63]-=-) developed finite variable logics into the technical framework of a central part of finite model theory. Notably, Abiteboul and Vianu [2] introduced a machine model reflecting precisely the expressiv... |

96 |
Combinatorics: Topics, Techniques, Algorithms
- Cameron
- 1994
(Show Context)
Citation Context ...it is homogeneous. Proving these results requires game theoretic arguments, similar to those used in Example 6.8, combined with a few simple facts about designs, as they can, for example, be found in =-=[10]-=-. Similar results can be proved for graphs. 2.2. Isomorphism testing. An important algorithmic problem is to decide whether two given graphs are isomorphic. Rather than testing isomorphism directly, m... |

75 | Toward Logic Tailored for Computational Complexity
- Gurevich
- 1984
(Show Context)
Citation Context ...A j= '(a; X i ; b; C)g: We let X1 = S i0 X i . Then for a 2 A k we let A j= [IFP x;X '(x; X; b; C)]a () a 2 X1 : Inflationary fixed-point logic has been introduced to finite model-theory by Gurevich [=-=34]-=-, but has been studied in generalized recursion theory under the title non-monotone inductive definability long before (see [61]). Gurevich and Shelah [37] proved that it has the same expressive power... |

74 |
E.: Canonical labeling of graphs
- Babai, Luks
- 1983
(Show Context)
Citation Context ...IME on C. Proof. It follows from results of Filotti and Mayer [22] and independently Miller [59] that there is a PTIME-canonization function for isomorphism on graphs of bounded genus. Babai and Luks =-=[4]-=- proved that there is a PTIME-canonization function for isomorphism on graphs of bounded valence. 2 8 Recall the definition on page 35, right before Subsection 8.1. 38 MARTIN GROHE 8.4. Definable Cano... |

64 |
On Moschovakis Closure Ordinals
- Barwise
- 1977
(Show Context)
Citation Context ... a very fruitful connection between fixedpoint logics and finite variable logics. It is best to make this connection through infinitary finite variable logics, 3 which have been introduced by Barwise =-=[6]-=-. Kolaitis and Vardi [51] were the first to realize their importance for finite model theory. As usual, L1! denotes the infinitary first-order logic allowing conjunctions and disjunctions over arbitra... |

58 | Two-variable logic with counting is decidable
- Grädel, Otto, et al.
- 1997
(Show Context)
Citation Context ...fiability problem (due to Mortimer [60]). Starting with a paper of Gradel, Kolaitis, and Vardi [24], the complexity of the satisfiability problem for 2-variable logics has been studied systematically =-=[27, 28, 65]-=-. See [26] for a survey on 2-variable logics. However, our emphasis is on logics with three or more variables. Central issues are the complexity of equivalence testing and the complexity of computing ... |

56 | Infinitary logic and inductive definability over finite structures
- Dawar, Lindell, et al.
- 1995
(Show Context)
Citation Context ... a technical tool to obtain results concerning fixed-point logics, which are central in descriptive complexity. Once they had realized this possibility, Kolaitis, Vardi, and others (see, for example, =-=[1, 2, 17, 52, 54, 63]-=-) developed finite variable logics into the technical framework of a central part of finite model theory. Notably, Abiteboul and Vianu [2] introduced a machine model reflecting precisely the expressiv... |

49 |
S.M.: Random graph isomorphism
- Babai, Erdős, et al.
- 1980
(Show Context)
Citation Context ...rmulas of the form ' 1 (x) = 9 m y(Exy ^ 9 n x Eyx): Continuing like this, we see that in each iteration step they get the same label. Thus they have the same stable color. Babai, Erdos, and Selkow [3=-=] have show-=-n that for almost all graphs all color classes have size one, thus for almost all graphs the color refinement algorithm yields a "canonical labelling" of the vertices. We are going to discus... |

48 | On the decision problem for two-variable first-order logic
- Grädel, Kolaitis, et al.
- 1997
(Show Context)
Citation Context ...variables. A distinguishing feature of L 2 compared to all L k for k > 2 is the decidability of its satisfiability problem (due to Mortimer [60]). Starting with a paper of Gradel, Kolaitis, and Vardi =-=[24]-=-, the complexity of the satisfiability problem for 2-variable logics has been studied systematically [27, 28, 65]. See [26] for a survey on 2-variable logics. However, our emphasis is on logics with t... |

43 | On logics with two variables
- Grädel, Otto
- 1999
(Show Context)
Citation Context ...(due to Mortimer [60]). Starting with a paper of Gradel, Kolaitis, and Vardi [24], the complexity of the satisfiability problem for 2-variable logics has been studied systematically [27, 28, 65]. See =-=[26]-=- for a survey on 2-variable logics. However, our emphasis is on logics with three or more variables. Central issues are the complexity of equivalence testing and the complexity of computing canonical ... |

37 | Fixpoint logics, relational machines, and computational conlplexity
- Abiteboul, Vardi, et al.
- 1992
(Show Context)
Citation Context ... a technical tool to obtain results concerning fixed-point logics, which are central in descriptive complexity. Once they had realized this possibility, Kolaitis, Vardi, and others (see, for example, =-=[1, 2, 17, 52, 54, 63]-=-) developed finite variable logics into the technical framework of a central part of finite model theory. Notably, Abiteboul and Vianu [2] introduced a machine model reflecting precisely the expressiv... |

36 | Feasible Computation Through Model Theory
- Dawar
- 1993
(Show Context)
Citation Context ...ion of whether PTIME equals PSPACE is equivalent to the purely logical question of whether least fixed-point logic has the same expressive power as partial fixed-point logic. In his PhD thesis, Dawar =-=[12] give-=-s a coherent picture of the theory at this stage, putting some emphasis on model theoretic aspects. Immerman [45] proposed to extend the finite-variable logics by counting quantifiers such as "th... |

35 |
N.: A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus
- FILOTTI, MAYER
- 1980
(Show Context)
Citation Context ...aphs. 2 Corollary 8.11. Let C be a class of graphs of bounded genus or bounded valence. Then there is a logic that effectively captures PTIME on C. Proof. It follows from results of Filotti and Mayer =-=[22]-=- and independently Miller [59] that there is a PTIME-canonization function for isomorphism on graphs of bounded genus. Babai and Luks [4] proved that there is a PTIME-canonization function for isomorp... |

34 | Fixed-point logics on planar graphs, in
- Grohe
- 1998
(Show Context)
Citation Context ... and (c) are not isomorphic. The game goes on like this, and since the level decreases each round the spoiler eventually wins. 2 We state three further results without proof. Theorem 6.24. (i) (Grohe =-=[3-=-2]) There is a k 1 such that on the class of planar graphs, isomorphism and C k -equivalence coincide. (ii) (Grohe, Mari no [33]) Let k 1. On graphs of tree-width at most k, isomorphism and C k+3 -e... |

33 |
Extended logics: The general framework, in
- Ebbinghaus
- 1985
(Show Context)
Citation Context ...odel of a sentence (M; p) if M accepts L(C) for a class C of -structures that contains A in at most p(inputlength) steps. Classical definitions of a logic, such as the notion of a regular logic (see [=-=18]-=-) do not suffice for our purpose; in addition our logics are supposed to satisfy certain effectivity conditions. Gurevich [35] suggested the following definitions. Definition 8.7. A logic is a pair (L... |

29 | Inductive definability with counting on finite structures
- Grädel, Otto
- 1993
(Show Context)
Citation Context ... that this query is not definable in IFP # . Thus we have IFP # ae IFP+C. Fixed-point logic with counting has been introduced by Immerman [47]. In the form presented here it is due to Gradel and Otto =-=[25]. Tho-=-ugh it may seem artificial at first sight, the logic has turned out to be quite robust. Gradel and Otto [25] studied several different formalizations of the concept "inductive definability with c... |

28 |
Generalized quantifiers and logical reducibilities
- Dawar
- 1995
(Show Context)
Citation Context ...structures (see [35]). The notion of a logic effectively capturing a complexity class still seems to be far from what we expect a logic to be. However, it has been justified by a nice result of Dawar =-=[13]-=-, which says that if there is a logic that effectively captures a complexity class K then there is a logic of the form FO(Q ! C ) that effectively captures K. Here FO(Q ! C ) denotes the logic obtaine... |

26 | Undecidability results on twovariable logics
- Grädel, Otto, et al.
- 1997
(Show Context)
Citation Context ...fiability problem (due to Mortimer [60]). Starting with a paper of Gradel, Kolaitis, and Vardi [24], the complexity of the satisfiability problem for 2-variable logics has been studied systematically =-=[27, 28, 65]-=-. See [26] for a survey on 2-variable logics. However, our emphasis is on logics with three or more variables. Central issues are the complexity of equivalence testing and the complexity of computing ... |

26 | Definability and descriptive complexity on databases of bounded tree-width
- Grohe, Mariño
- 1999
(Show Context)
Citation Context ... We state three further results without proof. Theorem 6.24. (i) (Grohe [32]) There is a k 1 such that on the class of planar graphs, isomorphism and C k -equivalence coincide. (ii) (Grohe, Mari no [=-=33-=-]) Let k 1. On graphs of tree-width at most k, isomorphism and C k+3 -equivalence coincide. (iii) (Immerman, Lander [49]) On graphs of color class size at most 3, isomorphism and L 3 -equivalence coi... |

23 |
On the Decision Problem for Two-Variable First-Order Logic
- Kolaitis, Vardi
(Show Context)
Citation Context ...ariables. A distinguishing feature of L 2 compared to all L k for k > 2 is the decidability of its satisfiability problem (due to Mortimer [60]). Starting with a paper of Gr adel, Kolaitis, and Vardi =-=[24]-=-, the complexity of the satisfiability problem for 2-variable logics has been studied systematically [27, 28, 65]. See [26] for a survey on 2-variable logics. However, our emphasis is on logics with t... |

18 |
The Classical Decision Problem
- örger, ädel, et al.
- 1997
(Show Context)
Citation Context ...tive plane of order 12. But of course there is no such algorithm; it is well-known 352 MARTIN GROHE that satisfiability of first-order sentences with three variables is undecidable (see, for example, =-=[8]-=-). However, the situation is di#erent for complete C 3 - theories. Given a reasonable finite representation of a complete C k -theory (see below for such a representation), we can compute a model of t... |

16 | A restricted second order logic for finite structures - Dawar - 1998 |

13 | A zero-one law for logic with a fixed-point operator, Information and Control 67
- Blass, Gurevich, et al.
- 1985
(Show Context)
Citation Context ...es as a complexity measure remained. In 1990, Kolaitis and Vardi [51] proved a 0-1 law for the infinitary finite variable logics. As a corollary, they re-proved a result of Blass, Gurevich, and Kozen =-=[7]-=- that there is a 0-1 law for least-fixed point logic. The fact that makes Kolaitis' and Vardi's paper so remarkable is that it uses finite variable logics as a technical tool to obtain results concern... |

10 | Equivalence in finite-variable logics is complete for polynomial time
- Grohe
- 1996
(Show Context)
Citation Context ...ary, but we can as well live without it. As an immediate consequence of Corollary 4.11 we obtain that L k -EQUIVALENCE is in PTIME. This result is due to Immerman and Lander [49]. Theorem 4.12 (Grohe =-=[29-=-]). For all k 2, L k -EQUIVALENCE is PTIME-complete under quantifier-free reductions. Quantifier-free reductions are logical reductions of a very low computational complexity; the result remains true... |

9 |
Inductive definability with counting on finite structures
- Gradel, Otto
- 1993
(Show Context)
Citation Context ...n that this query is not definable in IFP # . Thus we have IFP # IFP+C. Fixed-point logic with counting has been introduced by Immerman [47]. In the form presented here it is due to Gradel and Otto [=-=25]. Tho-=-ugh it may seem artificial at first sight, the logic has turned out to be quite robust. Gradel and Otto [25] studied several different formalizations of the concept "inductive definability with c... |

7 | Large finite structures with few L k -types
- Grohe
- 1997
(Show Context)
Citation Context ...sidered very unlikely that NPTIME is contained in PTIME=poly, in particular because Karp and Lipton [50] proved that if NPTIME PTIME=poly then the polynomial hierarchy collapses. Theorem 5.7 (Grohe [=-=30, -=-31]). Let k 3 and a vocabulary that contains at least one (k 1)-ary relation symbol. Then there is a (PTIME; PTIME=poly; PTIME)-translation from DIO to INV k [ ]. The complexity bounds in this theor... |

6 |
Types and indiscernibles in finite models
- Dawar
- 1998
(Show Context)
Citation Context ... the k-size of A only depends on Th L k(A). Thus we can define the k-size of a complete L k -theory to be the k-size of its models. It can be seen as a measure for the complexity of the theory. Dawar =-=[15-=-] observed that there is a function g : N ! N such that each complete L k [ ]- theory T has a model of size at most g(k-size(T)). (Remember that an L k -theory has a finite model by definition.) To se... |

6 | Implicit definability and infinitary logic in finite model theory - Kolaitis - 1995 |

5 |
Fixpoint logics, relational machines, and computational complexity
- Abiteboul, Vardi, et al.
- 1997
(Show Context)
Citation Context |

5 | Implicit definability and infinitary logic in finite model theory - Dawar, Hella, et al. - 1995 |

5 |
Private communication
- Ziegler
(Show Context)
Citation Context ...s a PTIME-computable inverse for I 2 . If we are only interested in a recursive rather than a PTIME-computable inverse of I 2 , we can proceed as follows, as has been observed by Flum and Ziegler [2=-=3]-=- and independently by Otto [64]: Given the invariant I 2 (A), we can clearly compute the L 2 -sentence ' 2 A that finitely axiomatizes Th L 2 (A) (see Corollary 4.4). By Mortimers Theorem [60], it is ... |

5 |
Extended Logics: The General Framework, in Model-Theoretic Logics
- Ebbinghaus
- 1985
(Show Context)
Citation Context ...closed if for all A, B such that A # B we have A # B ## B # B. FINITE VARIABLE LOGICS IN DESCRIPTIVE COMPLEXITY THEORY 389 Classical definitions of a logic, such as the notion of a regular logic (see =-=[18]-=-) do not su#ce for our purpose; in addition our logics are supposed to satisfy certain e#ectivity conditions. Gurevich [35] suggested the following definitions. Definition 8.7. A logic is a pair (L, |... |

5 |
On logics with two variables’, Theoretical Computer Science 224, 73–113
- Grädel, Otto
- 1997
(Show Context)
Citation Context ...due to Mortimer [60]). Starting with a paper of Gr adel, Kolaitis, and Vardi [24], the complexity of the satisfiability problem for 2-variable logics has been studied systematically [27, 28, 65]. See =-=[26]-=- for a survey on 2-variable logics. However, our emphasis is on logics with three or more variables. Central issues are the complexity of equivalence testing and the complexity of computing canonical ... |

4 |
and forth through infinitary logic
- Back
- 1973
(Show Context)
Citation Context ... the game on a single structure. Here we prove a slightly different definability result. The proof uses ideas from the classical model theory of infinitary logics that go back to Scott [68] (also see [5]). Lemma 4.3 (Dawar, Lindell, Weinstein [17]). Let k 1, a vocabulary, A a -structure, and a 2 A k . Then there is an L k -formula ' k A;a (y) such that for all -structures B and b 2 B k we have ... |

3 | Canonization for L k -equivalence is hard
- Grohe
- 1998
(Show Context)
Citation Context ...sidered very unlikely that NPTIME is contained in PTIME=poly, in particular because Karp and Lipton [50] proved that if NPTIME PTIME=poly then the polynomial hierarchy collapses. Theorem 5.7 (Grohe [=-=30, -=-31]). Let k 3 and a vocabulary that contains at least one (k 1)-ary relation symbol. Then there is a (PTIME; PTIME=poly; PTIME)-translation from DIO to INV k [ ]. The complexity bounds in this theor... |

3 |
and forth through infinitary logic, Studies in model theory
- Barwise, Back
- 1973
(Show Context)
Citation Context ...n the game on a single structure. Here we prove a slightly di#erent definability result. The proof uses ideas from the classical model theory of infinitary logics that go back to Scott [68] (also see =-=[5]-=-). Lemma 4.3 (Dawar, Lindell, Weinstein [17]). Let k # 1, # a vocabulary, A a #-structure, andsa # A k . Then there is an L k -formula # k A,sa (sy) such that for all #-structures B and b # B k we hav... |

2 |
Moschovakis closure ordinals
- On
- 1977
(Show Context)
Citation Context ...a very fruitful connection between fixed-point logics and finite variable logics. It is best to make this connection through infinitary finite variable logics, 3 which have been introduced by Barwise =-=[6]-=-. Kolaitis and Vardi [52] were the first to realize their importance for finite model theory. As usual, L## denotes the infinitary first-order logic allowing conjunctions and disjunctions over arbitra... |

1 |
quantifiers and logical reducibilities
- Generalized
- 1995
(Show Context)
Citation Context ... structures (see [35]). The notion of a logic e#ectively capturing a complexity class still seems to be far from what we expect a logic to be. However, it has been justified by a nice result of Dawar =-=[13]-=-, which says that if there is a logic that e#ectively captures a complexity class K then there is a logic of the form FO(Q # C ) that e#ectively captures K. Here FO(Q # C ) denotes the logic obtained ... |

1 |
and indiscernibles in finite models, Logic colloquium '95
- Types
- 1998
(Show Context)
Citation Context ...the k-size of A only depends on Th L k (A). Thus we can define the k-size of a complete L k -theory to be the k-size of its models. It can be seen as a measure for the complexity of the theory. Dawar =-=[15]-=- observed that there is a function g : N # N such that each complete L k [#]-theory T has a model of size at most g(k-size(T)). (Remember that an L k -theory has a finite model by definition.) To see ... |

1 |
results on two-variable logics, Proceedings of the 14th annual symposium on theoretical aspects of computer science
- Undecidability
- 1997
(Show Context)
Citation Context ...iability problem (due to Mortimer [60]). Starting with a paper of Gr adel, Kolaitis, and Vardi [24], the complexity of the satisfiability problem for 2-variable logics has been studied systematically =-=[27, 28, 65]-=-. See [26] for a survey on 2-variable logics. However, our emphasis is on logics with three or more variables. Central issues are the complexity of equivalence testing and the complexity of computing ... |