## Model Expansion and the Expressiveness of FO(ID) and Other Logics

Citations: | 1 - 1 self |

### BibTeX

@MISC{Kolokolova_modelexpansion,

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

title = {Model Expansion and the Expressiveness of FO(ID) and Other Logics },

year = {}

}

### OpenURL

### Abstract

Model expansion problem is a question of determining, given a formula and a structure for a part of the vocabulary of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problem-solving paradigm based on model expansion by (Mitchell & Ternovska, 2005; Mitchell, Ternovska, Hach, & Mohebali, 2006) posed the question of complexity of this problem for logics used in the paradigm. We discuss the complexity of the model expansion problem for a number of logics, alongside that of satisfiability and model checking. As the task is equivalent to witnessing leading existential second-order quantifiers in a model checking setting, the paper is in large part a survey of this area together with some new results. In particular, we describe the combined and data complexity of model expansion for FO(ID) (Denecker & Ternovska, 2008), as well as guarded and k-guarded logics of (Andréka, van Benthem, & Németi, 1998) and (Gottlob, Leone, & Scarcello, 2001).

### Citations

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Citation Context ...er a vocabulary consisting of a single binary relation E (also called “graph structures”, since E is interpreted as an edge relation of a graph). For more background on second-order logic please see (=-=Enderton, 2001-=-). The existential fragment of SO, denoted ∃SO, is the fragment where all second-order quanfiers are existential. This fragment is of particular importance to us here. We need to introduce another log... |

538 |
The complexity of relational query languages
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Citation Context ...ded by existential quantifiers for all expansion predicates. 2.4 Data and Combined Complexity For the tasks of model checking and model expansion, we consider two notions of complexity introduced in (=-=Vardi, 1982-=-). Here, we are following the presentation from (Libkin, 2004). Let enc() denote some standard encoding of structures and formulas by binary strings. Definition 2.2. Let Comp be a complexity class and... |

509 |
Finite Model Theory
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Citation Context ...φ of FO(LFP) 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 fixpoints. Proof. By (=-=Ebbinghaus & Flum, 1995-=-) theorem 9.4.2, every FO(LFP) 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 a ... |

287 |
Descriptive Complexity
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(Show Context)
Citation Context ...niversality has been realized before, and proven for a fragment of ASP, for NP-Spec and ConSQL. An area which establishes a connection between logics and complexity classes is Descriptive Complexity (=-=Immerman, 1999-=-). A classic example of a descriptive complexity result is Fagin’s theorem, which tells us that, when the logic L is FO, we can describe precisely the search problems corresponding to the complexity c... |

283 |
Generalized First-Order Spectra and Polynomial-Time Recognizable Sets
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Citation Context ... all authors were supported in part by NSERC.Model Expansion and Expressiveness of FO(ID) and Other Logics Some of these results follow immediately from well-known theorems such as these of Fagin’s (=-=Fagin, 1974-=-) and Grädel’s (Grädel, 1992)); others are considered for the first time in this paper. We present these results in the context of the complexity of the closely related, but more well-studied, tasks o... |

217 | Modal languages and bounded fragments of predicate logic - Andréka, Benthem, et al. - 1998 |

211 |
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Citation Context ... 2.4 Data and Combined Complexity For the tasks of model checking and model expansion, we consider two notions of complexity introduced in (Vardi, 1982). Here, we are following the presentation from (=-=Libkin, 2004-=-). Let enc() denote some standard encoding of structures and formulas by binary strings. Definition 2.2. Let Comp be a complexity class and L a logic. • The data complexity of model checking problem f... |

124 | On uniformity within NC1
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Citation Context ... from the problem QBF (Stockmeyer, 1974). The data complexity of model checking for FO is complete for AC 0 ; moreover, FO captures AC 0 over structures with BIT predicate (or arithmetic structures) (=-=Barrington et al., 1990-=-). Theorem 3.1. The combined complexity of MX for FO is NEXP-complete; the data complexity is NP-complete. Moreover, MX for FO captures NP. Proof. The NEXP-completeness of combined complexity for MX o... |

124 | On the restraining power of guards
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Citation Context ...ecking 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 (Liu & Levesque, 2003). The finite satisfiability problem for GF is 2EXP -complete (=-=Grädel, 1999-=-b). In this section, we show that the combined complexity of MX for GFk, k ≥ 1, is the same as that for FO, and MX for GFk, k ≥ 2, captures NP just as MX for FO does. We also identify a fragment of GF... |

106 |
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Citation Context ...ic Complexities of model checking and satisfiability for FO have been known for several decades. The combined complexity of model checking for FO is PSPACE-complete by reduction from the problem QBF (=-=Stockmeyer, 1974-=-). The data complexity of model checking for FO is complete for AC 0 ; moreover, FO captures AC 0 over structures with BIT predicate (or arithmetic structures) (Barrington et al., 1990). Theorem 3.1. ... |

65 | Introducing ESRA, a Relational Language for Modelling Combinatorial Problems
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Citation Context ...tchell et al., 2006) is to build languages and systems explicitly based on model expansion for extensions of classical logic. Examples where the languages are not logics include ESSENCE (?) and ESRA (=-=Flener, Pearson, & Agren, 2004-=-), which have semantics specified denotationally, in the style of general programming languages. These languages both include fragments which are essentially fragments of FO and SO extended with a typ... |

64 | Extending classical logic with inductive definitions
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Citation Context ... & Ternovska, 2005), it was proposed to focus on a logic with inductive definitions for an MX-based constraint modeling framework. The formalism they chose for this purpose is ID-logic, developed in (=-=Denecker, 2000-=-; Denecker & Ternovska, 2004a, 2008); early ideas for using inductive definitions in Knowledge Representation appear already in (?), and, in combination with classical logic, in (?, ?). The induction ... |

63 | Guarded fixed point logic
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Citation Context ...ecking 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 (Liu & Levesque, 2003). The finite satisfiability problem for GF is 2EXP -complete (=-=Grädel, 1999-=-b). In this section, we show that the combined complexity of MX for GFk, k ≥ 1, is the same as that for FO, and MX for GFk, k ≥ 2, captures NP just as MX for FO does. We also identify a fragment of GF... |

45 | On Logics with Two Variables
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Citation Context ...responds to the complexity of MX for GF . In particular, they show that GSO is strictly more powerful than monadic second-order logic. The combined complexity of model checking for GFk is P-complete (=-=Grädel & Otto, 1999-=-; Gottlob et al., 2001). In particular, model checking 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 (Liu & Levesque, 2003). The finite s... |

44 |
A hierarchy for nondeterministic time complexity
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Citation Context ...MX for FO is in NP. For (3), since SAT can be decided in nondeterministic O(n 2 ) time, by Lemma 4.7, MX for RGFk can be decided in nondeterministic O(n 2k ) time. By Cook’s NT IME hierarchy theorem (=-=Cook, 1973-=-), 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 expresse... |

44 |
Capturing complexity classes by fragments of second-order logic
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Citation Context ... in part by NSERC.Model Expansion and Expressiveness of FO(ID) and Other Logics Some of these results follow immediately from well-known theorems such as these of Fagin’s (Fagin, 1974) and Grädel’s (=-=Grädel, 1992-=-)); others are considered for the first time in this paper. We present these results in the context of the complexity of the closely related, but more well-studied, tasks of finite satisfiability and ... |

42 | NP-SPEC: an executable specification language for solving all problems in NP
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- 2000
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Citation Context ...G. Many of the existing approaches to representing search problems explicitly use logical languages. Examples include answer set programming (ASP) languages, such as those of LPARSE and DLV, NP-Spec (=-=Cadoli, Ianni, Palopoli, Schaerf, & Vasile, 2000-=-), ConSQL (Cadoli & Mancini, 2002) and ASPPS (East & Truszczynski, 2006), and it is easy to see that the underlying task is model expansion1 . The logic of NP-Spec is an extension of Datalog with nega... |

36 | A framework for representing and solving NP search problems
- Mitchell, Ternovska
- 2005
(Show Context)
Citation Context ... for a part of the vocabulary of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problem-solving paradigm based on model expansion by (=-=Mitchell & Ternovska, 2005-=-; Mitchell, Ternovska, Hach, & Mohebali, 2006) posed the question of complexity of this problem for logics used in the paradigm. We discuss the complexity of the model expansion problem for a number o... |

35 | Robbers, marshals, and guards: game theoretic and logical characterizations of hypertree width
- Gottlob, Leone, et al.
(Show Context)
Citation Context ...icular, we describe the combined and data complexity of model expansion for FO(ID) (Denecker & Ternovska, 2008), as well as guarded and k-guarded logics of (Andréka, van Benthem, & Németi, 1998) and (=-=Gottlob, Leone, & Scarcello, 2001-=-). 1. Introduction In model theory, expanding a structure means adding new functions and relations to the structure, while keeping the same domain. Model expansion (abbreviated MX), for a logic L, is ... |

33 | A logic of non-monotone inductive definitions and its modularity properties
- Denecker, Ternovska
(Show Context)
Citation Context ...05), it was proposed to focus on a logic with inductive definitions for an MX-based constraint modeling framework. The formalism they chose for this purpose is ID-logic, developed in (Denecker, 2000; =-=Denecker & Ternovska, 2004-=-a, 2008); early ideas for using inductive definitions in Knowledge Representation appear already in (?), and, in combination with classical logic, in (?, ?). The induction construct of IDlogic allows ... |

33 | A logic of nonmonotone inductive definitions
- Denecker, Ternovska
(Show Context)
Citation Context ...n a model checking setting, the paper is in large part a survey of this area together with some new results. In particular, we describe the combined and data complexity of model expansion for FO(ID) (=-=Denecker & Ternovska, 2008-=-), as well as guarded and k-guarded logics of (Andréka, van Benthem, & Németi, 1998) and (Gottlob, Leone, & Scarcello, 2001). 1. Introduction In model theory, expanding a structure means adding new fu... |

33 | A completeness result for reasoning with incomplete first-order knowledge bases - Levesque - 1998 |

30 | Inductive situation calculus
- Denecker, Ternovska
(Show Context)
Citation Context ...05), it was proposed to focus on a logic with inductive definitions for an MX-based constraint modeling framework. The formalism they chose for this purpose is ID-logic, developed in (Denecker, 2000; =-=Denecker & Ternovska, 2004-=-a, 2008); early ideas for using inductive definitions in Knowledge Representation appear already in (?), and, in combination with classical logic, in (?, ?). The induction construct of IDlogic allows ... |

22 | Predicatecalculus based logics for modeling and solving search problems
- East, Truszczyński
(Show Context)
Citation Context ...uages. Examples include answer set programming (ASP) languages, such as those of LPARSE and DLV, NP-Spec (Cadoli, Ianni, Palopoli, Schaerf, & Vasile, 2000), ConSQL (Cadoli & Mancini, 2002) and ASPPS (=-=East & Truszczynski, 2006-=-), and it is easy to see that the underlying task is model expansion1 . The logic of NP-Spec is an extension of Datalog with negation; the language of ConSQL is a generalization of SQL, and thus the l... |

12 | The impossibility of an algorithm for the decision problem for finite domains - Trahtenbrot - 1950 |

10 | Guarded fixed point logic and the monadic theory of trees - Grädel |

10 |
Relational queries computable in polytime
- Immerman
- 1982
(Show Context)
Citation Context ...is reduced to evaluating FO(ID) formula on an expanded structure. By the equivalence of FO(ID) with FO(LFP), his can be done in polynomial time of the size of the structure when formula is fixed (by (=-=Immerman, 1982-=-; Vardi, 1982; Livchak, 1982)) and in exponential time when the formula is a part of the input by (Vardi, 1982). In the second case, the size of the expansion predicates can be exponential in the size... |

8 | Grounding for model expansion in k-guarded formulas, 2006. Short presentation at LICS’06
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(Show Context)
Citation Context ..., 1998). A fragment which is most interesting to us is GFk (Gottlob et al., 2001), where the conjunction of up to k atoms may act as a guard. That fragment was used to achieve efficient grounding in (=-=Patterson, Liu, Ternovska, & Gupta, 2007-=-). In (?), it was shown that GFk is a natural abstraction of the type system of ESSENCE, and that all quantification in that language is guarded in the sense of GFk. This fragment also naturally refle... |

7 |
Systems of bounded arithmetic from descriptive complexity
- Kolokolova
- 2004
(Show Context)
Citation Context ... induction is possible to simulate in ∃SO, to model such properties, a “guess and check” approach is needed. For transitive closure, for example, such modeling is a highly non-trivial task, see e.g. (=-=Kolokolova, 2004-=-) for details. 1.3 Logics for MX The languages used in the Constraints community generally do not have recursion. On the other hand, most logic programming languages (the languages of ASP, Datalog LIT... |

7 | Satisfiability checking for PC(ID
- Mariën, Mitra, et al.
- 2005
(Show Context)
Citation Context ...icity, causality and temporal reasoning (Denecker & Ternovska, 2004b, 2007), and as such is a good knowledge representation language. Two native solvers for the propositional fragment were developed (=-=Mariën, Mitra, Denecker, & Bruynooghe, 2005-=-) and (Pelov & Ternovska, 2005). East and Truszczynski’s system ASPPS (East & Truszczynski, 2006) is a system for solving NP-search problems whose modeling language is a language of propositional sche... |

6 | Model expansion as a framework for modelling and solving search problems
- Mitchell, Ternovska, et al.
- 2006
(Show Context)
Citation Context ...y of the formula, whether there is an expansion of this structure that satisfies the formula. Recent development of a problem-solving paradigm based on model expansion by (Mitchell & Ternovska, 2005; =-=Mitchell, Ternovska, Hach, & Mohebali, 2006-=-) posed the question of complexity of this problem for logics used in the paradigm. We discuss the complexity of the model expansion problem for a number of logics, alongside that of satisfiability an... |

5 | Reducing inductive definitions to propositional satisfiability
- Pelov, Ternovska
- 2005
(Show Context)
Citation Context ...& Ternovska, 2004b, 2007), and as such is a good knowledge representation language. Two native solvers for the propositional fragment were developed (Mariën, Mitra, Denecker, & Bruynooghe, 2005) and (=-=Pelov & Ternovska, 2005-=-). East and Truszczynski’s system ASPPS (East & Truszczynski, 2006) is a system for solving NP-search problems whose modeling language is a language of propositional schemata with rule-based notation ... |

4 | Complexity of expanding a finite structure and related tasks
- Kolokolova, Liu, et al.
- 2006
(Show Context)
Citation Context ...aper. We present these results in the context of the complexity of the closely related, but more well-studied, tasks of finite satisfiability and model checking. This work extends an earlier version (=-=Kolokolova, Liu, Mitchell, & Ternovska, 2006-=-). 1.1 Model Expansion, Constraint Programming, and Descriptive Complexity A number of somewhat independent lines of work, under labels such as “constraint modeling languages” or “declarative programm... |

2 |
Languages for polynomial-time queries. In Computer-based modeling and optimization of heat-power and electrochemical objects
- Livchak
- 1982
(Show Context)
Citation Context ...D) formula on an expanded structure. By the equivalence of FO(ID) with FO(LFP), his can be done in polynomial time of the size of the structure when formula is fixed (by (Immerman, 1982; Vardi, 1982; =-=Livchak, 1982-=-)) and in exponential time when the formula is a part of the input by (Vardi, 1982). In the second case, the size of the expansion predicates can be exponential in the size of the structure (since the... |

1 | 16 Expansion and Expressiveness of FO(ID) and Other Logics Cadoli - M, Mancini - 2002 |