## On winning strategies in Ehrenfeucht-Fraïssé games (1997)

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Venue: | Theoretical Computer Science |

Citations: | 14 - 3 self |

### BibTeX

@ARTICLE{Arora97onwinning,

author = {Sanjeev Arora and Ronald Fagin},

title = {On winning strategies in Ehrenfeucht-Fraïssé games},

journal = {Theoretical Computer Science},

year = {1997},

volume = {174}

}

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### Abstract

We present a powerful and versatile new sufficient condition for the second player (the

### Citations

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Computers and Intractability: a Guide to the Theory of NP-completeness
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Citation Context ... monadic NP. The \Sigma 1 1 sentences constructed in Examples 5.2 and 5.3 are monadic, and so 3-colorability and nonreachability are in monadic NP. Note that 3-colorability is an NP-complete property =-=[GJ79]-=-. Thus, monadic NP includes NP-complete properties. The \Sigma 1 1 sentence constructed in Example 5.1 is not monadic (since ! is a binary relation symbol). Indeed, Tur'an [Tur84] has shown that Hamil... |

612 |
A Mathematical Introduction to Logic
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Citation Context ...i-Fagin games First-order logic allows for quantification only over members of the universe, and not over sets of members of the universe, or more generally, over relations; for details, see Enderton =-=[End72]-=- or Shoenfield [Sho67]. When we pass from first-order logic to second-order logic, we allow quantification over relations. In particular, a \Sigma 1 1 sentence is a sentence of the form 9A 1 :::9A k /... |

537 | The complexity of relational query languages - Vardi - 1982 |

513 | Graphs and hypergraphs - Berge - 1976 |

276 |
Generalized first-order spectra and polynomial-time recognizable sets
- Fagin
- 1974
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Citation Context ...ptive complexity of a problem is the complexity of describing the problem in some logical formalism [Imm89]. The two complexities are sometimes related. This was first discovered by Fagin, who showed =-=[Fag74]-=- that the complexity class NP coincides with the class of properties of finite structures expressible in existential second-order logic, otherwise known as \Sigma 1 1 . Consequently, NP=co-NP if and o... |

273 | Relational queries computable in polynomial time - Immerman - 1986 |

246 | Structure and Complexity of Relational Queries
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- 1982
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Citation Context ...ssed in fixpoint logic with a built-in successor relation (or a built-in linear order). Allowing successor is crucial in this case, since evenness is not definable in fixpoint logic without successor =-=[CH82]-=-. Finally, built-in relations can be viewed as adding an element of nonuniformity to the class, and thus changing it somewhat (this is analogous to the way circuit-based computational complexity diffe... |

218 |
Mathematical Logic
- Shoenfield
- 1967
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Citation Context ...der logic allows for quantification only over members of the universe, and not over sets of members of the universe, or more generally, over relations; for details, see Enderton [End72] or Shoenfield =-=[Sho67]-=-. When we pass from first-order logic to second-order logic, we allow quantification over relations. In particular, a \Sigma 1 1 sentence is a sentence of the form 9A 1 :::9A k /, where / is first-ord... |

161 | An application of games to the completeness problem for formalized theories - Ehrenfeucht - 1961 |

129 | On local and nonlocal properties - Gaifman - 1982 |

85 |
On monadic np vs. monadic co-np
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- 1995
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Citation Context ...fication (i.e., over relations of arbitrary arity), allow quantification only over sets. Since quantifying over sets corresponds to quantification over monadic relations, Fagin, Stockmeyer, and Vardi =-=[FSV95]-=- term these restricted classes as monadic NP and monadic co-NP respectively, a terminology that has since gained acceptance. Note that, in spite of its seemingly restrictive syntax, monadic NP contain... |

82 | Sur quelques classifications des systèmes de relations, Publ. Scient. de l’Université d’Alger - Fräıssé - 1954 |

79 | Degrees of acyclicity for hypergraphs and relational database schemes
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- 1983
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Citation Context ...cles of length 2, this definition corresponds to Berge's notion [Ber76] of a cycle in a hypergraph. (There are various other notions of a cycle in a hypergraph that are not equivalent to Berge's; see =-=[Fag83]-=-.) We note that if A is a structure over a language with a single binary relation, then its hypergraph is an ordinary undirected graph, and the concept of distance and cycle are the familiar ones. We ... |

72 | Reachability is harder for directed than for undirected graphs
- Ajtai, Fagin
- 1990
(Show Context)
Citation Context ...ant to develop tools for proving that the duplicator wins. Several such conditions have been identified. Among these are (a) a formulation of secondorder Ehrenfeucht-Fraiss'e games by Ajtai and Fagin =-=[AF90]-=-, for which it seems easier to prove that the duplicator has a winning strategy; (b) a sufficient condition (due essentially to Hanf [Han65]) for the duplicator to have a winning strategy; and (c) the... |

66 |
Model-theoretic methods in the study of elementary logic
- Hanf
- 1965
(Show Context)
Citation Context ...of secondorder Ehrenfeucht-Fraiss'e games by Ajtai and Fagin [AF90], for which it seems easier to prove that the duplicator has a winning strategy; (b) a sufficient condition (due essentially to Hanf =-=[Han65]-=-) for the duplicator to have a winning strategy; and (c) the idea of playing EhrenfeuchtFra iss'e games over random structures. Techniques (a) and (c) were used by Ajtai and Fagin [AF90], and all thre... |

59 |
Monadic generalized spectra. Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik 21:89--96
- Fagin
- 1975
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Citation Context ...ving separation results: Ehrenfeucht-Fraiss'e games (see Section 3). Fagin showed that \Sigma 1 1 6= \Pi 1 1 if and only if such a separation can be proven via second-order Ehrenfeucht-Fraiss'e games =-=[Fag75]-=-. In the same paper, Fagin also suggested that partial progress on the \Sigma 1 1 6= \Pi 1 1 question could be made by restricting the expressive power of these classes: instead of allowing second-ord... |

47 | Descriptive and Computational Complexity
- Immerman
- 1998
(Show Context)
Citation Context ...mount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of a problem is the complexity of describing the problem in some logical formalism =-=[Imm89]-=-. The two complexities are sometimes related. This was first discovered by Fagin, who showed [Fag74] that the complexity class NP coincides with the class of properties of finite structures expressibl... |

23 |
Second-order and inductive definability on finite structures. Zeitschr. f. math. Logik und Grundlagen d
- Rougemont
- 1987
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Citation Context ...is not in monadic NP, and (2) they show that connectivity is not in monadic NP in the presence of a large class of built-in relations (this was previously known only for a built-in successor relation =-=[dR87]-=-; we discuss built-in relations shortly). In this paper we give a new sufficient condition for the duplicator to have a winning strategy. We hope that this condition will prove useful and intuitive. W... |

20 | Graphs and Hypergraphs (North-Holland - Berge - 1983 |

19 |
Bounds on the propagation of selection into logic programs
- BEERI, KANELLAKIS, et al.
- 1987
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Citation Context ...esting problem because it is known in some senses to be a more difficult problem to deal with than connectivity. For instance, a surprising result of Kanellakis (private communication, 1986; see also =-=[BKBR87]-=-) says that undirected reachability (where we consider only undirected graphs) is in monadic NP. (In contrast, undirected connectivity is not in monadic NP, as mentioned above.) Furthermore, Cosmadaki... |

18 | Graph connectivity and monadic NP
- Schwentick
- 1994
(Show Context)
Citation Context ...expressibility results in the presence of certain built-in relations. We summarize in Section 9. Other related work. A related development (independent of this paper) is a recent result by Schwentick =-=[Sch94]-=-. He gives another sufficient condition for the duplicator to have a winning strategy, and uses it to show that connectivity is not in monadic NP in the presence of an even larger class of built-in re... |

13 |
On the definability of properties of finite graphs
- Turán
- 1984
(Show Context)
Citation Context ... NP-complete property [GJ79]. Thus, monadic NP includes NP-complete properties. The \Sigma 1 1 sentence constructed in Example 5.1 is not monadic (since ! is a binary relation symbol). Indeed, Tur'an =-=[Tur84]-=- has shown that Hamiltonicity is not in monadic NP; in fact, he showed the stronger result that Hamiltonicity cannot be defined by a monadic second-order sentence (where we allow arbitrary quantificat... |

11 |
Logical reducibility and monadic NP
- Cosmadakis
- 1993
(Show Context)
Citation Context ...(It is arguable, though, whether this makes the proof simpler.) There are several reasons for interest in connectivity and directed reachability, the two problems considered in this paper. Cosmadakis =-=[Cos93]-=- has shown that connectivity reduces (via a very weak kind of reduction) to a host of other problems, including non-3-colorability. So the fact that connectivity is not in monadic NP implies that thes... |

5 | Rougemont Second-order and inductive definability on finite structures, Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik - de - 1987 |

1 | Degrees of acyclicity for hypergraphs and relational detabase schemes - Fagin - 1983 |