## Complexity of expanding a finite structure and related tasks (2006)

Venue: | The 8th Int. Workshop on Logic and Comput. Complexity (LCC |

Citations: | 4 - 4 self |

### BibTeX

@INPROCEEDINGS{Kolokolova06complexityof,

author = {Antonina Kolokolova and Yongmei Liu and David Mitchell and Eugenia Ternovska},

title = {Complexity of expanding a finite structure and related tasks},

booktitle = {The 8th Int. Workshop on Logic and Comput. Complexity (LCC},

year = {2006}

}

### OpenURL

### Abstract

The authors of [MT05] proposed a declarative constraint programming framework based on classical logic extended with non-monotone inductive definitions. In the framework, a problem instance is a finite structure, and a problem specification is a formula defining the relationship between an instance and its solutions. Thus, problem solving amounts to expanding a finite structure with new relations, to satisfy the formula. We present here the complexities of model expansion for a number of logics, alongside those of satisfiability and model checking. As the task is equivalent to witnessing the existential quantifiers in ∃SO model checking, the paper is in large part of a survey of this area, together with some new results. In particular, we describe the combined and data complexity of FO(ID), first-order logic extended with inductive definitions [DT04] and the guarded and k-guarded logics of [AvBN98] and [GLS01]. 1

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Citation Context ...results presented in this paper. 1sLogic Model checking Model expansion Satisfiability Combined Data Combined Data (finite) F O P SP ACE-c ≡BIT AC [Sto74] 0 NEXP -c ≡ NP [Fag74] undec [Tra50] [BIS90] =-=[Var82]-=- F O(LF P ) EXP -c =s P [Imm82, NEXP -c ≡ NP undec [Var82] Var82, Liv82] F O(ID) EXP -c =s P NEXP -c ≡ NP undec F O k P -c ∈ AC 0 NP -c NP -c, �≡ NP k ≥ 3: undec [Var95] k = 2: NEXP -c k = 1: EXP -c G... |

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Citation Context ... of F O(LF P ) and a structure A there is a formula φ ′ of ID-logic such that φ holds on A iff φ ′ holds on A extended by the relational variables interpreted as the names for the fixponts. Proof. By =-=[EF95]-=- theorem 9.4.2, every F O(LF P ) formula is equivalent to one of the form ∀u[LF P¯z,Zψ]ũ, where ψ ∈ ∆2. This can be written as an ID-logic formula {Z(¯z) ← ψ} ∧ ∀uZ(ũ). Now, whenever a structure A is ... |

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Citation Context ...lebrated theorem of Fagin [Fag74] stating that existential second order logic (∃SO) exactly captures the complexity class NP was the first result that led to the development of descriptive complexity =-=[Imm99]-=-, an area studying the relationship between logics and complexity classes. From the practical point of view, descriptive complexity results provide a way for logics to be viewed as “programming langua... |

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Citation Context ... and data complexity of FO(ID), first-order logic extended with inductive definitions [DT04] and the guarded and k-guarded logics of [AvBN98] and [GLS01]. 1 Introduction A celebrated theorem of Fagin =-=[Fag74]-=- stating that existential second order logic (∃SO) exactly captures the complexity class NP was the first result that led to the development of descriptive complexity [Imm99], an area studying the rel... |

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Citation Context ...ther with some new results. In particular, we describe the combined and data complexity of FO(ID), first-order logic extended with inductive definitions [DT04] and the guarded and k-guarded logics of =-=[AvBN98]-=- and [GLS01]. 1 Introduction A celebrated theorem of Fagin [Fag74] stating that existential second order logic (∃SO) exactly captures the complexity class NP was the first result that led to the devel... |

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Citation Context ...B |= φ iff Clique B is a set of vertices that forms a clique in B. For each of the problems (except satisfiability) we consider two notions of complexity (introduced by [Var82]; here we are following =-=[Lib04]-=- presentation). Let enc() denote some standard encoding of structures and formulae by binary strings. Definition 2.1. Let K be a complexity class and L a logic. • The data complexity of L is K if for ... |

121 | On uniformity within NC1
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Citation Context ...ibility results presented in this paper. 1sLogic Model checking Model expansion Satisfiability Combined Data Combined Data (finite) F O P SP ACE-c ≡BIT AC [Sto74] 0 NEXP -c ≡ NP [Fag74] undec [Tra50] =-=[BIS90]-=- [Var82] F O(LF P ) EXP -c =s P [Imm82, NEXP -c ≡ NP undec [Var82] Var82, Liv82] F O(ID) EXP -c =s P NEXP -c ≡ NP undec F O k P -c ∈ AC 0 NP -c NP -c, �≡ NP k ≥ 3: undec [Var95] k = 2: NEXP -c k = 1: ... |

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Citation Context ...-c ∈ AC 0 NP -c NP -c, �≡ NP k ≥ 3: undec [Var95] k = 2: NEXP -c k = 1: EXP -c GFk P -c ∈ AC 0 NEXP -c k ≥ 2: ≡ NP k ≥ 2: undec [GO99, GLS01] k = 1: NP -c k = 1: 2EXP -c RGFk:NP -c RGFk: NP -c, �≡ NP =-=[Grä99]-=- µGF UP ∩ co-UP ∈ P NEXP -c NP -c 2EXP -c [GW99] GFk(ID) ∈ EXP ∈ P NEXP -c k ≥ 2: ≡ NP k ≥ 2: undec Table 1: Complexity of model checking, model expansion and satisfiability problems for some logics 2... |

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Citation Context ...ves an overview of the complexity and expressibility results presented in this paper. 1sLogic Model checking Model expansion Satisfiability Combined Data Combined Data (finite) F O P SP ACE-c ≡BIT AC =-=[Sto74]-=- 0 NEXP -c ≡ NP [Fag74] undec [Tra50] [BIS90] [Var82] F O(LF P ) EXP -c =s P [Imm82, NEXP -c ≡ NP undec [Var82] Var82, Liv82] F O(ID) EXP -c =s P NEXP -c ≡ NP undec F O k P -c ∈ AC 0 NP -c NP -c, �≡ N... |

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Citation Context ...P-c [GKV97] k = 1: EXP-c GFk P-c ∈ AC 0 NEXP-c ∗ k ≥ 2: ≡ NP ∗ k ≥ 2: undec [GO99, GLS01] k = 1: NP-c k = 1: 2EXP-c RGFk:NP-c ∗ RGFk:NP-c, �≡ NP ∗ [Grä99] µGF UP ∩ co-UP ∈ P NEXP-c NP-c 2EXP-c [GW99] =-=[Jur98]-=- GFk(ID) ∈ EXP ∈ P NEXP-c k ≥ 2: ≡ NP k ≥ 2: undec Table 1: Complexity of model checking, model expansion and satisfiability problems for some logics 2 Preliminaries and definitions In this section we... |

76 |
On the complexity of bounded-variable queries
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Citation Context ... [Fag74] undec [Tra50] [BIS90] [Var82] F O(LF P ) EXP -c =s P [Imm82, NEXP -c ≡ NP undec [Var82] Var82, Liv82] F O(ID) EXP -c =s P NEXP -c ≡ NP undec F O k P -c ∈ AC 0 NP -c NP -c, �≡ NP k ≥ 3: undec =-=[Var95]-=- k = 2: NEXP -c k = 1: EXP -c GFk P -c ∈ AC 0 NEXP -c k ≥ 2: ≡ NP k ≥ 2: undec [GO99, GLS01] k = 1: NP -c k = 1: 2EXP -c RGFk:NP -c RGFk: NP -c, �≡ NP [Grä99] µGF UP ∩ co-UP ∈ P NEXP -c NP -c 2EXP -c ... |

59 | Guarded fixed point logic
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(Show Context)
Citation Context ... k = 2: NEXP -c k = 1: EXP -c GFk P -c ∈ AC 0 NEXP -c k ≥ 2: ≡ NP k ≥ 2: undec [GO99, GLS01] k = 1: NP -c k = 1: 2EXP -c RGFk:NP -c RGFk: NP -c, �≡ NP [Grä99] µGF UP ∩ co-UP ∈ P NEXP -c NP -c 2EXP -c =-=[GW99]-=- GFk(ID) ∈ EXP ∈ P NEXP -c k ≥ 2: ≡ NP k ≥ 2: undec Table 1: Complexity of model checking, model expansion and satisfiability problems for some logics 2 Preliminaries and definitions In this section w... |

44 |
A hierarchy for nondeterministic time complexity
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Citation Context ... of MX for F O is in NP. (3): Since SAT can be decided in nondeterministic O(n 2 ) time, by Lemma 4.6, MX for RGFk can be decided in nondeterministic O(n 2k ) time. By Cook’s NT IME hierarchy theorem =-=[Coo73]-=-, for any i > 2k, there is a problem that can be solved in nondeterministic O(n i ) time but not nondeterministic O(n i−1 ) time. Thus there are infinitely many problems in NP that cannot be expressed... |

43 | On logics with two variables - Grädel, Otto - 1999 |

42 |
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(Show Context)
Citation Context ... of the input). Although data complexity of model expansion for full first-order logic is NP -complete, there are fragments of F O for which model expansion is feasible. In particular, the results of =-=[Grä92]-=- translate into the following result. Definition 3.3. A universal Horn formula is a first-order formula consisting of a conjunction of Horn clauses, preceded by universal first-order quantifiers. Here... |

35 | Robbers, marshals, and guards: game theoretic and logical characterizations of hypertree width
- Gottlob, Leone, et al.
(Show Context)
Citation Context ...e new results. In particular, we describe the combined and data complexity of FO(ID), first-order logic extended with inductive definitions [DT04] and the guarded and k-guarded logics of [AvBN98] and =-=[GLS01]-=-. 1 Introduction Fagin’s theorem that existential second order logic (∃SO) exactly captures the complexity class NP [Fag74] was the first result that led to the development of descriptive complexity [... |

30 | A completeness result for reasoning with incomplete first-order knowledge bases
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(Show Context)
Citation Context ...ard. The combined complexity of MC for GFk is P -complete [GO99, GLS01]. In particular, MC for GFk can be done in time O(ln k ), where l is the size of the formula, and n is the size of the structure =-=[LL03]-=-. The finite satisfiability problem for GF is 2EXP -complete [Grä99]. In this section, we discuss complexity of MX for GFk: we show that the combined complexity of MX for GFk, k ≥ 1, is the same as th... |

29 |
A framework for representing and solving np search problems
- Mitchell, Ternovska
- 2005
(Show Context)
Citation Context ... “programming languages” for the corresponding classes. This, in turn, suggests taking the idea of logics as programming languages as the basis for practical tools. In particular, in the framework of =-=[MT05]-=-, search problems are cast as model expansion (abbreviated MX), which is the task of witnessing the (first block of) existential second order quantifiers in a SO sentence. Although even in the case of... |

28 | A logic of non-monotone inductive definitions
- Denecker, Ternovska
(Show Context)
Citation Context ...is in large part of a survey of this area, together with some new results. In particular, we describe the combined and data complexity of FO(ID), first-order logic extended with inductive definitions =-=[DT04]-=- and the guarded and k-guarded logics of [AvBN98] and [GLS01]. 1 Introduction A celebrated theorem of Fagin [Fag74] stating that existential second order logic (∃SO) exactly captures the complexity cl... |

23 |
On the Decision Problem for Two-Variable First-Order Logic
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(Show Context)
Citation Context ...5] FO(LFP) EXP-c ≡s P [Imm82, NEXP-c ≡ NP undec [Var82] Var82, Liv82] FO(ID) EXP-c ∗ ≡s P ∗ NEXP-c ∗ ≡ NP[MT05] undec FO k P-c ∈ AC 0 NP-c NP-c, �≡ NP k ≥ 3: undec [Var95, GO99] [Var95] k = 2: NEXP-c =-=[GKV97]-=- k = 1: EXP-c GFk P-c ∈ AC 0 NEXP-c ∗ k ≥ 2: ≡ NP ∗ k ≥ 2: undec [GO99, GLS01] k = 1: NP-c k = 1: 2EXP-c RGFk:NP-c ∗ RGFk:NP-c, �≡ NP ∗ [Grä99] µGF UP ∩ co-UP ∈ P NEXP-c NP-c 2EXP-c [GW99] [Jur98] GFk... |

10 |
The impossibility of an algorithm for the decision problem for finite domains
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Citation Context ... expressibility results presented in this paper. 1sLogic Model checking Model expansion Satisfiability Combined Data Combined Data (finite) F O P SP ACE-c ≡BIT AC [Sto74] 0 NEXP -c ≡ NP [Fag74] undec =-=[Tra50]-=- [BIS90] [Var82] F O(LF P ) EXP -c =s P [Imm82, NEXP -c ≡ NP undec [Var82] Var82, Liv82] F O(ID) EXP -c =s P NEXP -c ≡ NP undec F O k P -c ∈ AC 0 NP -c NP -c, �≡ NP k ≥ 3: undec [Var95] k = 2: NEXP -c... |

9 | Relational queries computable in polytime - Immerman - 1982 |

7 | Grounding for model expansion in k-guarded formulas with inductive definitions
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Citation Context ...T rue(x) ≡ ¬T rue(y))). Clearly, φ ∈ RGF1, and Γ is satisfiable iff A can be expanded to a model of φ. We quote the following result concerning polynomial-time grounding of RGFk sentences: Lemma 4.6 (=-=[PLTG06]-=-). There exists an algorithm that, given a structure A and a RGFk sentence φ, constructs in O(l 2 n k ) time a propositional formula ψ such that A can be expanded to a model of φ iff ψ is satisfiable,... |

3 |
guards: game theoretic and logical characterizations of hypertree width
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(Show Context)
Citation Context ...e new results. In particular, we describe the combined and data complexity of FO(ID), first-order logic extended with inductive definitions [DT04] and the guarded and k-guarded logics of [AvBN98] and =-=[GLS01]-=-. 1 Introduction A celebrated theorem of Fagin [Fag74] stating that existential second order logic (∃SO) exactly captures the complexity class NP was the first result that led to the development of de... |

2 | Languages for polynomial-time queries. In Computer-based modeling and optimization of heat-power and electrochemical objects - Livchak - 1982 |