## Computing With First-Order Logic (1995)

Citations: | 52 - 13 self |

### BibTeX

@MISC{Abiteboul95computingwith,

author = {Serge Abiteboul and Victor Vianu},

title = {Computing With First-Order Logic},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study two important extensions of first-order logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...

### Citations

4052 |
Introduction to Automata Theory, Languages and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...e q is in GM loose j PT IME \Gamma while. Consider (i). Clearly, while ` GM loose j PSPACE . Consider a language L in expspace and not in pspace. Such a language exists by the space hierarchy theorem =-=[HU79]-=-. We use the same technique as in (iii) to obtain a query in GM loose j PSPACE \Gamma while. Part (ii) is proven in the same way, using a language L in exptime and not in ptime, which exists by the ti... |

1589 | Foundations of Databases - Abiteboul, Hull, et al. - 1995 |

538 |
The complexity of relational query languages
- Vardi
- 1982
(Show Context)
Citation Context ...tions on the database, such as the presence or absence of a total order on the domain. Indeed, with such an order, fixpoint expresses exactly ptime [Imm86, Var82] and while expresses precisely pspace =-=[Var82]-=-. On the other hand, both languages collapse to FO on sets, i.e. the set of queries they express on inputs consisting of unordered sets coincides with the FO queries on such inputs. This shows, in par... |

471 |
Optimal implementation of conjunctive queries in relational databases
- Chandra, P
- 1977
(Show Context)
Citation Context ...uery. For instance, we will consider query ' (for ' as above) assuming that x comes first in the default order. Conjunctive Queries We will also use an important subset of FO. The conjunctive queries =-=[CM77]-=- are FO queries whose formulas are in prenex form 3 and (a) use only existential quantifiers, and (b) use only the connectives(i.e., do not use , : or !). For example, 9z(R(x; z)sR(z; y)); 9y(P (x; y)... |

298 | Introduction to Automata Theory - Hopcroft, Motwani, et al. - 2000 |

296 |
Object identity as a query language primitive
- Abiteboul, Kanellakis
- 1989
(Show Context)
Citation Context ...ously defined (computationally complete) query languages from [CH80, AV90] coincide respectively with gen-ptime and gen-pspace. Similar results can be obtained for the object-oriented language IQL of =-=[AK89]-=-. Indeed, more recently, in [DV91], an object-oriented model of database parallel computation using methods is investigated. It is shown that the ptime and pspace complexity classes defined using that... |

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

248 | Structure and complexity of relational queries
- Chandra, Harel
- 1982
(Show Context)
Citation Context ...ed in polynomial time. For while, each iteration is destructive, so while ` pspace. In particular, while queries may not converge. We describe fixpoint and while next. The fixpoint queries (fixpoint) =-=[CH82] are-=- constructed using the first-order constructors as in FO together with a fixpoint operator (��). The fixpoint operator binds a predicate symbol R that is free and that appears only positively (i.e... |

156 | Datalog extensions for database queries and updates - Abiteboul, Vianu - 1991 |

155 | Logic and the Challenge of Computer Science - Gurevich - 1988 |

128 | Multivalued dependencies and a new normal form for relational databases - FAGIN - 1977 |

122 |
Generic computation and its complexity
- Abiteboul, Vianu
- 1991
(Show Context)
Citation Context ...d relational polynomial space, respectively. This matches the intuition that, in some natural but restricted sense, fixpoint and while represent the polynomial time and space generic computations. In =-=[AV91a]-=- we argue that there is a mismatch between Turing complexity and the actual hardness of database queries. A typical example is the query even on a set, which has low Turing complexity but is not expre... |

116 | Computable queries for relational data bases - CHANDRA, HAREL - 1980 |

109 | Elements of Relational Database Theory - Kanellakis - 1990 |

109 | Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen - Pólya - 1937 |

84 | Computable queries for relational databases
- Chandra, Harel
- 1980
(Show Context)
Citation Context ...m the input, stays within pspace. We next consider briefly complete languages. To break the space barrier, (i) or (ii) must be relaxed. The first complete language, while unsorted , was introduced in =-=[CH80]-=-, and is based on relaxing (i). It is obtained by using an unsorted algebra, and allowing in while, unsorted variables, i.e. variables standing for relations whose arity is not fixed. The possibility ... |

82 |
Fixed-point extensions of firstorder logic
- Gurevich, Shelah
- 1986
(Show Context)
Citation Context ...or instance, the transitive closure of relation Q is given by the fixpoint query: ��R'(x; y; R) for ' as above. The same expressive power can be achieved using an inflationary fixpoint logic (FO+I=-=FP) [GS86]-=-. The inflationary fixpoint operator (IFP) no longer requires that R occur positively in '(R), and convergence is instead guaranteed in polynomial time by cumulating iterations of '(R) up to a fixpoin... |

73 | clause queries and generalizations - Horn - 1985 |

58 |
Procedural Languages for Database Queries and Updates
- Abiteboul, Vianu
(Show Context)
Citation Context ...xed. The possibility to count using the arity of intermediary results yields the desired computational power. Another complete language, while invent , is based on relaxing (ii), and was presented in =-=[AV90]-=-. The expressive power is obtained by having a constructor (new) allowing the introduction of new domain elements in the database. For instance, for a binary relation R, the following program creates ... |

57 |
Theory of Database Queries
- Chandra
- 1988
(Show Context)
Citation Context ... produced in the analysis phase by its corresponding integer. By applying the normal form to the while queries, we show that fixpoint = while iff ptime = pspace. We thereby answer an open question of =-=[Cha88]-=-. Surprisingly, the result also applies to the ptime fragment of while (i.e., whilej PT IME ): we prove that ptime = pspace iff fixpoint = whilej PT IME . Note that this reduces the separation of ptim... |

56 | Infinitary logic and inductive definability over finite structures
- Dawar, Lindell, et al.
- 1995
(Show Context)
Citation Context ... k . From Theorem 3.10 it then follows that equivalence of m-tuples (msk) with respect to L k 1! is definable in fixpoint. The definability of equivalence with respect to L k 1! is also considered in =-=[DLW91]-=-, where several results of the present paper and of [AV91a] are presented within the framework of L k 1! . We will discuss again infinitary logic later in the paper. We finally consider the problem of... |

51 | Languages which capture complexity classes
- Immerman
- 1983
(Show Context)
Citation Context ...t FO has a very simple algebraization involving basic operations on relations (see Preliminaries), which can be implemented very efficiently. In particular, note that FO queries are in (uniform) AC 0 =-=[Imm87b]-=-, and thus, in a reasonable sense, take constant parallel time. GM loose is of special interest because it subsumes most query languages which have emerged as central, including the fixpoint and while... |

41 | Expressibility as a Complexity Measure: Results and Directions - Immerman - 1987 |

38 | Fixpoint logics, relational machines, and computational complexity
- Abiteboul, Vardi, et al.
- 1997
(Show Context)
Citation Context ...relational machine consisting of a Turing Machine (TM) component loosely coupled with a relational store. The machine is called loosely coupled Generic Machine - GM loose (in follow-up papers such as =-=[AVV92b]-=-, the machine is simply called relational machine). The other extensions augment while with integer variables and the ability to perform arithmetic. This yields two languages, while + and while ++ , w... |

36 |
0-1 laws and decision problems for fragments of second-order logic
- Kolaitis, Vardi
- 1990
(Show Context)
Citation Context ... infinitary logic with k variables, denoted L k 1! . The logic allows infinite conjunctions and disjunctions, as long as just k variables are used. It has been pointed out by P. Kolaitis and M. Vardi =-=[KV90]-=- that each formula in that logic can be expressed using a (possibly infinite) disjunction of (possibly infinite) conjunctions of formulas in FO k . Thus, two tuples are equivalent with respect to L k ... |

33 |
A.K.: Programming primitives for database languages
- Chandra
- 1981
(Show Context)
Citation Context ...( ~ t) is a formula. The While Queries The while language also provides an inductive definition of relations up to a fixpoint. It was originally introduced in a procedural form, under the names LE in =-=[Cha81]-=-, and RQ in [CH82]. A is extended with (i) sorted relational variables (X; Y; :::), (ii) assignment of FO queries to variables, and (iii) a while construct allowing to iterate a program while some fir... |

27 | Finite model theory - a personal perspective - Fagin - 1993 |

27 | Descriptive characterizations of computational complexity - Leivant - 1989 |

20 | Finite-model theory --- a personal perspective - Fagin - 1990 |

12 | Queries are easier than you thought (probably
- Abiteboul, Compton, et al.
- 1992
(Show Context)
Citation Context ...ewed as a form of optimization. To evaluate the potential for optimization, it is important to have information on the likelihood that size k;F (I) is much smaller than jI j. This issue is studied in =-=[ACV92]-=-, where it is shown that for each k there exists a constant M such that size k;F (I)sM with asymptotic probability 1. This indicates that the potential for optimization using the normal form is high. ... |

10 |
Axiomatic recursive function theory
- Friedman
- 1971
(Show Context)
Citation Context ...ne GM loose is similar in spirit to several devices previously proposed, which operate directly on structures rather than on encodings of structures. Such machines have been introduced by H. Friedman =-=[Fri71]-=- and later by D. Leivant [Lei89a, Lei89b]. The device introduced in [Fri71], the FAP, is reminiscent of program schemes. The emphasis in [Fri71] is on extending recursion theory to FAPs. In [Lei89a, L... |

9 | Computing with Infinitary Logic
- Abiteboul, Vardi, et al.
- 1995
(Show Context)
Citation Context ...y it expresses always exists and is 0 or 1 [KV90]. As a consequence, GM loose also has a 0/1 law, so it cannot compute even. The connection between GM loose and infinitary logic is further studied in =-=[AVV92a]-=-; it is shown there that GM loose corresponds precisely to a natural effective fragment of L ! 1! . Specifically, GM loose expresses exactly the properties definable by sentences in L ! 1! whose sets ... |

9 |
An analysis of fixed-point queries on binary trees
- Lindell
- 1991
(Show Context)
Citation Context ...me = pspace. To prove the result, we will use the fact that certain fixpoint computations on inputs with a high degree of regularity can be achieved very efficiently. This uses a technique of Lindell =-=[Lin90]-=- and is based on encoding words in a relational structure. Suppose we wish to encode a word w over f0; 1g in a relational structure. This can be easily done as follows. Suppose w = x 1 :::x d . We use... |

8 | The power of methods with parallel semantics
- Denningho, Vianu
- 1991
(Show Context)
Citation Context ...plete) query languages from [CH80, AV90] coincide respectively with gen-ptime and gen-pspace. Similar results can be obtained for the object-oriented language IQL of [AK89]. Indeed, more recently, in =-=[DV91]-=-, an object-oriented model of database parallel computation using methods is investigated. It is shown that the ptime and pspace complexity classes defined using that model essentially coincide once a... |

8 | The number of finite relational structures - Fagin - 1977 |

6 |
The power of reflective relational machines
- Abiteboul, Papadimitriou, et al.
- 1994
(Show Context)
Citation Context ...er of GM loose . A more powerful device results if the bound on the number of variables in dynamically generated queries is removed. Such a device, called reflective relational machine, is studied in =-=[APV94]-=-. 5.5 Polynomial Time and Space Restrictions of GM loose We consider here polynomial time and space restrictions of GM loose . We consider the relation of fixpoint and while to the ptime and pspace re... |

6 | Monotonic use of space and computational complexity over abstract structures - Leivant - 1989 |

3 | Optimizing Active Databases using the Split Technique
- Abiteboul, Gelder
- 1992
(Show Context)
Citation Context ...h asymptotic probability 1. This indicates that the potential for optimization using the normal form is high. Practical optimization algorithms based on the normal form are developed and evaluated in =-=[AG92]-=-. The results of [ACV92] also shed additional light on the connection between the type index # k (I) and size k;F (I). Indeed, it is known that finite structures are rigid (they have abs 2 (R 2 OE ) 1... |