## Fixpoint Logics, Relational Machines, and Computational Complexity (1993)

Venue: | In Structure and Complexity |

Citations: | 38 - 5 self |

### BibTeX

@INPROCEEDINGS{Abiteboul93fixpointlogics,,

author = {Serge Abiteboul and Moshe Y. Vardi and Victor Vianu},

title = {Fixpoint Logics, Relational Machines, and Computational Complexity},

booktitle = {In Structure and Complexity},

year = {1993},

pages = {156--168}

}

### Years of Citing Articles

### OpenURL

### Abstract

We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1st-order operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic -- while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...

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Citation Context ...terministic or alternating. Indeed, part of this research was motivated by the question whether PSPACE-complete problems such as "quantified Boolean formulas" or "nonuniversality for fi=-=nite automata" [GJ79]-=- can be described in noninflationary fixpoint logic. By augmenting fixpoint logic with nondeterministic or alternating iteration we find that the connection between fixpoint logic and computational co... |

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Citation Context ... noninflationary operators \Phi 1 and \Phi 2 corresponding to the next input symbol being 0 or 1. The problem, however, does not seem to be expressible by a deterministic iteration. Savitch's Theorem =-=[Sav80]-=- tells us how to convert the nondeterministic polynomial-space algorithm into a deterministic polynomial-space algorithm, but this construction assume a built-in order and does not seem to be describa... |

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Citation Context ...scriptive complexity. This intimate connection was first discovered by Fagin, who showed that the complexity class NP coincides with the class of properties expressible in existential 2nd-order logic =-=[Fag74]-=- (cf. [JS74]). Another demonstration of this connection was shown by Immermanand Vardi, who discovered tight relationships between the complexity class P and inflationary fixpoint logic [Imm86, Var82]... |

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Citation Context ...logic can describe P-complete problems, there are some very easy problems in P that are not expressible in inflationary fixpoint logic (e.g., checking whether the cardinality of the structure is even =-=[CH82]-=-). It is only when we assume built-in order that we get that P coincides with the class of properties expressible in inflationary fixpoint logic [Imm86, Var82]. Similarly, it is only when we assume a ... |

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Citation Context ... [Var82] (cf. [AV89]). 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between "logic and complexity", was then proclaimed by=-= Immerman [Imm87b]-=-, and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. Although the relationsh... |

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Citation Context ...e heart of our framework is the view of fixpoint logic as 1st-order logic augmented with iteration. While traditionally fixpoint logic was viewed as the extension of 1st-order logic by recursion (cf. =-=[Mos74]-=-), iteration proved to be a more general extension to 1st-order logic than recursion [AV89, GS86, Lei90]. In both inflationary and noninflationary fixpoint logics, iteration is applied in its simplest... |

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Citation Context ... 5 Actually, it is shown in [AV90] that there is no loss of generality in restricting attention to converging 1st-order operators. Noninflationary fixpoint logic was introduced by Abiteboul and Vianu =-=[AV91a]-=- (who called it partial fixpoint logic). In particular, they observed that noninflationary fixpoint logic coincides with the query language RQL introduced in [CH82] and studied further in [Var82]. It ... |

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Citation Context ...6, GSS90, Lei91]. 2 and descriptive complexity. (See also [IL90] for a discussion of the order issue ?from another perspective.) The order issue was partially overcome recently by Abiteboul and Vianu =-=[AV91], who-=- showed that indeed P=PSPACE if and only if inflationary and noninflationary fixpoint logics have the same expressive power. The crux of their result is a clever description of a certain "interna... |

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Citation Context ...hat are augmented by a relational store and the ability to perform relational operations on that store. This idea of extending 1st-order logic with a general computational capability was suggested in =-=[CH80]-=- and pursued in [AV91]. 3 Unlike Turing machines, which operate on encodings of problems, relational machines operate directly on the underlying mathematical structures. For Turing machines, the most ... |

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Citation Context ...tion rule for positive formulas. It is easy to see that IFP is at least as expressive as positive fixpoint logic. Gurevich and Shelah showed that in fact the two logics have the same expressive power =-=[GS86]-=- (see also [Lei90]). The complexity-theoretic aspects of IFP were studied in [CH82, Imm86, Var82]. (These papers actually focused on positive fixpoint logic, but, as observed above, positive fixpoint ... |

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Citation Context ...onnection between finite-model theory and complexity. The connection between logic and complexity has also a proof-theoretic aspect; see [Bus86, GSS90, Lei91]. 2 and descriptive complexity. (See also =-=[IL90]-=- for a discussion of the order issue ?from another perspective.) The order issue was partially overcome recently by Abiteboul and Vianu [AV91], who showed that indeed P=PSPACE if and only if inflation... |

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Citation Context ...ems that cannot be solved in polynomial time, but also because it contains problems that cannot be solved in relational polynomial time even though they can be solved in standard polynomial time. See =-=[DLW91] for -=-related results. 6 The equivalence of Pr = PSPACEr " P and P=PSPACE. was shown in [AV91]. 11 4.2 Relational Machines and Fixpoint Logics Fixpoint logics involve iterations of 1st-order formulas. ... |

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Citation Context ...xity only if it can be calculated by relational machines. The techniques of [AV91b, AV91c] can now be used to show that relational machines can measure k-size. We will use the following result from 8 =-=[AV91c]-=-. Lemma 3.2 : [AV91c] For each input type oe and k ? 0 there exists an IFP formula ' over oe, with 2k free variables x 1 ; :::; x k ; y 1 ; :::; y k such that for each input D of type oe, ' defines a ... |

47 | Descriptive and Computational Complexity
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Citation Context ...ty", was then proclaimed by Immerman [Imm87b], and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2=-= See [Imm89]-=- for a survey. Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While c... |

41 |
Fixpoint extensions of first-order logic and Datalog-like languages
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Citation Context ...nd Vardi, who discovered tight relationships between the complexity class P and inflationary fixpoint logic [Imm86, Var82] and between the class PSPACE and noninflationary fixpoint logic [Var82] (cf. =-=[AV89]). 1 The t-=-ight connection between descriptive and computational complexity, typically referred to as the connection between "logic and complexity", was then proclaimed by Immerman [Imm87b], and studie... |

41 |
Expressibility as a Complexity Measure: Results and Directions
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37 |
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Citation Context ...rder to "fully capture" P and PSPACE, respectively. Nevertheless, Abiteboul and Vianu showed that P=PSPACE if and only if IFP and NFP have the same expressive power [AV91]. 4 Actually, it is=-= shown in [AV90]-=- that there is no loss of generality in restricting attention to converging 1st-order operators. 5 2.3 More Fixpoint Logics IFP and NFP are obtained by iterating inflationary and noninflationary 1st-o... |

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Citation Context ...plexity. This intimate connection was first discovered by Fagin, who showed that the complexity class NP coincides with the class of properties expressible in existential 2nd-order logic [Fag74] (cf. =-=[JS74]-=-). Another demonstration of this connection was shown by Immermanand Vardi, who discovered tight relationships between the complexity class P and inflationary fixpoint logic [Imm86, Var82] and between... |

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Citation Context ... 5 Unlike relational machines, loose generic machines can apply general 1st-order transformations to the relational store. Relational machines are closely related to the 2nd-order pointer machines of =-=[Lei89a]-=-. Pointer machine manipulate their store by means of tagging and untagging operations, while relational machines manipulate their store by means of relational operations. The definition of relational ... |

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Citation Context ...Turing machines because they cannot check that the cardinality of their input is even. The latter follows from the fact that the logic L ! 1! has a 0-1 law, so it cannot express the evenness property =-=[KV90a]-=-. We thus obtain the following: Proposition 4.1: Let \Phi be a polynomially closed set of functions. Then Class r (resource; control; \Phi) ae Class(resource; control; \Phi); for any resource and cont... |

14 | Stable networks and product graphs
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Citation Context ...rmulas. Note that NFP fixpoint logic is an extension of IFP . While IFP formulas can be evaluated in polynomial time, NFP can express PSPACE-complete problems, such as the network convergence problem =-=[Fed81]-=-. Noninflationary fixpoint logic was introduced by Abiteboul and Vianu [AV89] (who called it partial fixpoint logic). In particular, they observed that noninflationary fixpoint logic coincides with th... |

13 |
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Citation Context ...itive formulas. It is easy to see that IFP is at least as expressive as positive fixpoint logic. Gurevich and Shelah showed that in fact the two logics have the same expressive power [GS86] (see also =-=[Lei90]-=-). The complexity-theoretic aspects of IFP were studied in [CH82, Imm86, Var82]. (These papers actually focused on positive fixpoint logic, but, as observed above, positive fixpoint logic and IFP have... |

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Citation Context ...l machines is best understood by viewing them as an effective fragment of a certain infinitary 3 A closely related idea, of generalizing Turing machines to operate on general structures, goes back to =-=[Fri71]-=- and was investigated extensively in [Lei89a, Lei89b]. 3 logic, studied recently in [KV90a, KV90b]. This view yields a precise characterization of the discerning power of relational machines in terms ... |

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Citation Context ... fragment of the logic L ! 1! . This is an infinitary logic with a finite number of variables studied recently in [KV90a, KV90b]. The connection between relational machines and L ! 1! is developed in =-=[AVV92]-=-; it suffices here to say that it yields a characterization of the discerning power of relational machines in terms of certain infinite 2-player pebble games. The k-pebble game between the Spoiler and... |

8 | An algebra and a logic for NC - Compton, Laflamme - 1990 |

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Citation Context ... structures for two reasons. The first reason is technical; questions about logical expressiveness over ordered structures are typically much harder than their unordered counterparts; see for example =-=[dR84]-=-. The second reason is more fundamental; the restriction to ordered structures seems to be a technical device rather than an intrinsic feature. Because of this restriction, the above equivalence provi... |

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Citation Context ...o strings. It can be checked that for a given string w, the size of M (rel(w)) is polynomially related to the size of w. It turns out, however, that using a result of Lindel about the k-size of trees =-=[Lin91]-=-, we can encode strings by structures in way that blows up the size without blowing up the k-size. Proposition 4.6 : For any function f : N ! N there is a mapping rel f from strings to structures such... |

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