## Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs (Extended Abstract) (2007)

Venue: | IN: PROC. 12TH WORKSHOP ON LOGIC, LANGUAGE, INFORMATION AND COMPUTATION |

Citations: | 8 - 3 self |

### BibTeX

@INPROCEEDINGS{Dawar07choicelesspolynomial,

author = {Anuj Dawar and David Richerby and Benjamin Rossman},

title = {Choiceless polynomial time, counting and the Cai–Fürer–Immerman graphs (Extended Abstract)},

booktitle = {IN: PROC. 12TH WORKSHOP ON LOGIC, LANGUAGE, INFORMATION AND COMPUTATION},

year = {2007},

pages = {13--26},

publisher = {}

}

### OpenURL

### Abstract

We consider Choiceless Polynomial Time ( ˜ CPT), a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Fürer and Immerman to separate fixed-point logic with counting (IFP + C) from P. This settles a question posed by Blass et al. The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank of sets used, even in ˜ CPT(Card), an extension of ˜ CPT with counting.

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Citation Context ... [1], who showed that a class of structures is recognizable by a nondeterministic machine in polynomial time if, and only if, it is definable in existential second-order logic. Immerman [2] and Vardi =-=[3]-=- independently showed that the polynomial-time computable properties of linearly ordered structures are definable in the fixedpoint logic IFP (in this paper we will always use inflationary fixed-point... |

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Citation Context ...∞ω. There is, however, a single polynomial-time Turing machine that is able to distinguish the members of every pair. The logics we refer to are reviewed in Section 2 below and the reader may consult =-=[7,8]-=- for the proofs of the stated results and a more thorough account of the problem of providing a logical characterization of P. Since the result of Cai et al., some logics have been proposed that exten... |

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Citation Context ... work of Fagin [1], who showed that a class of structures is recognizable by a nondeterministic machine in polynomial time if, and only if, it is definable in existential second-order logic. Immerman =-=[2]-=- and Vardi [3] independently showed that the polynomial-time computable properties of linearly ordered structures are definable in the fixedpoint logic IFP (in this paper we will always use inflationa... |

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Citation Context ...IFP is too weak to express all polynomial-time properties of unordered structures. In particular, there is no formula of IFP that asserts that the size of a structure is even. Cai, Fürer and Immerman =-=[6]-=- showed that IFP + C, the extension of IFP with counting terms, is still too weak to express all polynomial-time properties, though it can easily express evenness and similar cardinality properties. T... |

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Citation Context ...uctures I and J and a logic L, we write I ≡L J to indicate that no L-formula distinguishes I and J. For finite I and J, I ≡Lk J if, and only if, I ≡Lk∞ω J [15] and I ≡Ck J if, and only if, I ≡Ck ∞ω J =-=[16]-=-. Thus, to establish that a Boolean query is not definable in Cω ∞ω (and, a fortiori, not definable in IFP + C) it suffices to exhibit structures Ik and Jk for each k, that are distinguished by the qu... |

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Citation Context ...nstruct. It remains open whether this logic captures all polynomialtime computations. In an attempt to understand the role of choice in defining polynomial-time properties, Blass, Gurevich and Shelah =-=[12]-=- define choiceless polynomial time, ˜ CPT, a ‘logic’ based on a machine model, which they describe as an attempt to characterize what can be done without introducing choice. They show that this logic ... |

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Citation Context ...∞ω. There is, however, a single polynomial-time Turing machine that is able to distinguish the members of every pair. The logics we refer to are reviewed in Section 2 below and the reader may consult =-=[7,8]-=- for the proofs of the stated results and a more thorough account of the problem of providing a logical characterization of P. Since the result of Cai et al., some logics have been proposed that exten... |

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Citation Context ...efinable in the fixedpoint logic IFP (in this paper we will always use inflationary fixed-point logic, though we could equally well take least fixed-point logic, as it has equivalent expressive power =-=[4,5]-=-). IFP is too weak to express all polynomial-time properties of unordered structures. In particular, there is no formula of IFP that asserts that the size of a structure is even. Cai, Fürer and Immerm... |

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Citation Context ... + (I). 6 Fixed-point logics In this section, we give some results on the fixed-point definability of queries defined by ˜ CPT(Card) programs and vice-versa. It is known from the work of Blass et al. =-=[13]-=- that IFP + C cannot express every query definable in ˜CPT(Card), and this also follows from the previous section of the present paper, since IFP + C cannot determine the parity of CFI graphs. However... |

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Citation Context ...orward but is problematic because it is possible that a machine might accept some codings of a structure and reject others. In contrast, ˜ CPT programs are based on Gurevich’s abstract state machines =-=[18]-=-, so are parallel and operate directly on relational structures. Given an input structure I of vocabulary σ, a ˜ CPT program operates on hereditarily finite sets over |I|, with the elements of |I| vie... |

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Citation Context ...efinable in the fixedpoint logic IFP (in this paper we will always use inflationary fixed-point logic, though we could equally well take least fixed-point logic, as it has equivalent expressive power =-=[4,5]-=-). IFP is too weak to express all polynomial-time properties of unordered structures. In particular, there is no formula of IFP that asserts that the size of a structure is even. Cai, Fürer and Immerm... |

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Citation Context ... some logics have been proposed that extend the power of IFP + C while still defining only polynomial-time properties. A significant one is the logic with specified symmetric choice of Gire and Hoang =-=[9]-=- (see also [10,11] for an analysis of symmetric choice) which relies on a choice construct. It remains open whether this logic captures all polynomialtime computations. In an attempt to understand the... |

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(Show Context)
Citation Context ...ave been proposed that extend the power of IFP + C while still defining only polynomial-time properties. A significant one is the logic with specified symmetric choice of Gire and Hoang [9] (see also =-=[10,11]-=- for an analysis of symmetric choice) which relies on a choice construct. It remains open whether this logic captures all polynomialtime computations. In an attempt to understand the role of choice in... |

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(Show Context)
Citation Context ...e is a program ¯ Π of ˜ CPT(Card) that, on input (I, ā), determines whether (I, ā) � ϕ. Further, ¯ Π uses only sets of rank at most 1 and uses counting if, and only if, ϕ does. PROOF. It follows from =-=[19]-=- that we may assume ϕ to be of the form ∃λ∃y (ifp X, ¯x ψ) (λ . . . λy . . . y), where ψ ∈ FO(#), λ is a number variable and y a point variable. We may also assume that every quantifier in ψ uses a fr... |