## Computing with Infinitary Logic (1995)

Venue: | Theoretical Computer Science |

Citations: | 9 - 6 self |

### BibTeX

@ARTICLE{Abiteboul95computingwith,

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

title = {Computing with Infinitary Logic},

journal = {Theoretical Computer Science},

year = {1995},

volume = {149},

pages = {101--128}

}

### Years of Citing Articles

### OpenURL

### Abstract

Most recursive extensions of relational calculus converge around two central classes of queries: fixpoint and while. Infinitary logic (with finitely many variables) is a very powerful extension of these languages which provides an elegant unifying formalism for a wide variety of query languages. However, neither the syntax nor the semantics of infinitary logic are effective, and its connection to practical query languages has been largely unexplored. We relate infinitary logic to another powerful extension of fixpoint and while, called relational machine, which highlights the computational style of these languages. Relational machines capture the kind of computation occurring when a query language is embedded in a host programming language, as in C+SQL. The main result of this paper is that relational machines correspond to the natural effective fragment of infinitary logic. Other well-known query languages are related to infinitary logic using syntactic restrictions formula...

### Citations

10980 |
Computers and Intractability – A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...nistic. Certain problems, however, seem to defy description by such iterations. As an example, consider nonuniversality of finite automata over a binary alphabet, which is known to be pspace-complete =-=[GJ79]-=-. That is, we are given a finite automaton over the alphabet f0; 1g and we want to know whether there is a word rejected by the automaton. An instance of this problem can be viewed as a structure (S; ... |

3847 |
Introduction to automata theory, languages, and computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...icted query languages, yielding characterizations in language-theoretic terms, is considered in Section 4. 2 Preliminaries We assume basic knowledge of databases [K90, U88] and formal language theory =-=[HU79]-=-. In databases, only finite structures are considered. Most traditional query languages are based on first-order logic without function symbols (here FO). The simplicity of Codd's algebraization of FO... |

838 |
Theory of Recursive Functions and Effective Computability
- ROGERS
- 1967
(Show Context)
Citation Context ...mulas yield the analog of the arithmetic hierarchy defined using relational machines. The arithmetic hierarchy based on relational machines is defined in analogy to the classical arithmetic hierarchy =-=[Ro67]-=-. More precisely, add oracles for sets of structures accepted by relational machines to obtain the first level of the hierarchy; next, add oracles on the set of structures that can thereby be accepted... |

527 |
Complexity of relational query languages
- Vardi
- 1982
(Show Context)
Citation Context ...omial time. For instance, the transitive closure of relation E is given by the fixpoint querysT '(x; y; T ) for ' = E(x; y)s9z(E(x; z)sT (z; y)): Fixpoint expresses exactly ptime on ordered databases =-=[I86, V82]-=-. It cannot, however, express the evenness 2 query. The while language was originally introduced in [CH82] in a procedural form. FO is extended with (i) sorted relational variables (X; Y; :::), (ii) a... |

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

243 | Structure and complexity of relational queries
- Chandra, Harel
- 1982
(Show Context)
Citation Context ...eneric Machine). The term relational machine was first used in [AVV92]. 2 Finally, we examine the connection between infinitary logic and well-known restricted query languages like fixpoint and while =-=[CH82]-=-, the most well-studied extensions of FO, which motivated defining both infinitary logic and relational machines. We describe syntactic restrictions of infinitary logic corresponding to such languages... |

191 |
Admissible Sets and Structures
- Barwise
- 1975
(Show Context)
Citation Context ...i k in the syntax tree of '. Remark: The above definition is the analogue of the definition of the Church-Kleene fragment L CK ! 1 ;! , which is the syntactically effective fragment of L ! 1 ;! ; see =-=[Ba75]-=-. It turns out that syntactic effectiveness does not guarantee semantic effectiveness. Indeed, we will see that the r.e. formulas yield the analog of the arithmetic hierarchy defined using relational ... |

127 | On uniformity within NC
- Barrington, Immerman, et al.
- 1990
(Show Context)
Citation Context ...considered. Most traditional query languages are based on first-order logic without function symbols (here FO). The simplicity of Codd's algebraization of FO and the fact that FO is in (uniform) AC 0 =-=[BIS90]-=- (and, thus, in a reasonable sense, takes constant parallel time) explain the appeal of FO as a query language. In this section, we define the infinitary logics L1! and L ! 1! [Ba77] and briefly prese... |

118 |
Generic computation and its complexity
- Abiteboul, Vianu
- 1991
(Show Context)
Citation Context ...anguage and the database, e.g., by including dynamically generated queries, and addressable relations, without increasing the expressive power of the model. 1 The relational machine was introduced in =-=[AV91]-=- as GM loose (Loosely-coupled Generic Machine). The term relational machine was first used in [AVV92]. 2 Finally, we examine the connection between infinitary logic and well-known restricted query lan... |

84 | Computable queries for relational databases
- Chandra, Harel
- 1980
(Show Context)
Citation Context ...he relational store, whose arity can vary throughout the computation. The resulting device is called untyped relational machine. This is in the spirit of the first query-complete language proposed in =-=[CH80]-=-. An equivalent typed version of these machines has been defined in [AV91] under the name Generic Machine (GM). A generic machine compensates for the typedness by allowing the transfer of data between... |

80 |
Fixed-point extensions of first-order logic
- Gurevich, Shelah
- 1986
(Show Context)
Citation Context ...ause of the positiveness restriction) in polynomial time. The equivalence between the definition using positive formulas and the definition by inflationary iteration of arbitrary formulas is shown in =-=[GS85]. Recall t-=-hat while and partial fixpoint logic are equivalent. A procedural analog to fixpoint can be obtained by using the while language with a "cumulative" semantics for the assignment (instead of ... |

64 |
On Moschovakis closure ordinals
- Barwise
- 1977
(Show Context)
Citation Context ... in finite-model theory is infinitary logic, an extension of first-order logic with infinite conjunctions and disjunctions [BF85]. Infinitary logic with finitely many variables (denoted L ! 1! ) (see =-=[Ba77]-=-) provides an elegant framework for studying important phenomena such as 0-1 laws [KV90b] and expressive power of query languages [ACY91, KV90a]. While infinitary logic is an elegant theoretical tool,... |

56 | Infinitary logic and inductive definability over finite structures
- Dawar, Lindell, et al.
- 1995
(Show Context)
Citation Context ...ult shown for the full L k 1! in [KV92]. It is also shown in [KV92] that every formula of L k 1! is equivalent to a countable disjunction of FO l -formulas, for some lsk. Furthermore, it follows from =-=[DLW91]-=- that this holds also for l = k. To complete the picture, we next make a connection between the (syntactically) r.e. L ! 1! -formulas and relational machines. We show that the formulas correspond to t... |

52 | Computing with first-order logic
- Abiteboul, Vianu
- 1995
(Show Context)
Citation Context ...airs [I ; /] where I is an instance and / is in A, and output I if I j= /. (ii) ) (i). Suppose that S = models(') is r.e., where ' 2 L k 1! . Since ' is in L k 1! , we can use the normal form theorem =-=[AV92]-=-, i.e., rewrite ' as f ffi / where: ffl f is a fixpoint query whose output is an ordered structure and ffl / is obtained from ' by replacing each FO k -formula by a formula working on the ordered outp... |

52 | On the expressive power of Datalog: tools and a case study - Kolaitis, Vardi - 1995 |

37 | Fixpoint logics, relational machines, and computational conlplexity
- Abiteboul, Vardi, et al.
- 1992
(Show Context)
Citation Context ...dels. Thus the word model refers to finite models only. 2 The relational machine was introduced in [AV91b] as GM loose (Loosely-coupled Generic Machine). The term relational machine was first used in =-=[AVV92]-=-. interfaces between the host programming language and the database, e.g., by including dynamically generated queries and addressable relations, without increasing the expressive power of the model. F... |

33 |
The decision problem for the probabilities of higher-order properties
- Kolaitis, Vardi
- 1987
(Show Context)
Citation Context ...he evenness query on a set. An elegant proof uses the fact that all properties expressible by while have a 0-1 law, i.e. the asymptotic probability that the property holds exists and is either 0 or 1 =-=[KV87]-=-. It is of interest whether the 0-1 law holds for more powerful languages. Infinitary logic with finitely many variables is a very powerful extension of while that still has a 0-1 law [KV90b]. This pr... |

33 |
A.K.: Programming primitives for database languages
- Chandra
- 1981
(Show Context)
Citation Context ...riables. Note that while programs may not terminate and that they are in pspace. Indeed, while expresses pspace on ordered databases [V82], but cannot express the evenness query 3 on an unordered set =-=[C81]-=-. Fixpoint Logics The main construct of fixpoint logics, called a fixpoint operator, allows the iteration of FO formulas up to a fixpoint. This in effect allows defining relations inductively, using F... |

32 | On Datalog vs polynomial time - Afrati, Cosmadakis, et al. - 1995 |

19 |
0-1 laws for infinitary logics
- Kolaitis, Vardi
- 1992
(Show Context)
Citation Context ...nite conjunctions and disjunctions [BF85]. Infinitary logic with finitely many variables (denoted L ! 1! ) (see [Ba77]) provides an elegant framework for studying important phenomena such as 0-1 laws =-=[KV90b]-=- and expressive power of query languages [ACY91, KV90a]. While infinitary logic is an elegant theoretical tool, infinitary logic formulas have a non-effective syntax, and can define non-computable que... |

17 |
Elements of relational database theory Handbook of theoretical computer science (vol. B): formal models and semantics:1073–1156
- Kanellakis
- 1990
(Show Context)
Citation Context ...al machines. The connection with restricted query languages, yielding characterizations in language-theoretic terms, is considered in Section 4. 2 Preliminaries We assume basic knowledge of databases =-=[K90, U88]-=- and formal language theory [HU79]. In databases, only finite structures are considered. Most traditional query languages are based on first-order logic without function symbols (here FO). The simplic... |

16 |
Algorithmic procedures, generalized Turing algorithms and elementary recursion theory, in: Logic Colloquium
- Friedman
- 1969
(Show Context)
Citation Context ...es were investigated in [AV91b] (under the name "loosely coupled generic machine") and in [AV, AVV92]. Computing devices in the spirit of relational machines had already been investigated by=-= Friedman [Fri71]-=- and Leivant [Lei89], but with a different focus: Friedman's emphasis was on generalizing recursion theory, whereas Leivant's was on logic characterizations of computational complexity on ordered (or ... |

13 |
Finite-model theory—a personal perspective
- Fagin
- 1990
(Show Context)
Citation Context ... to these results, we obtain interesting normal forms for infinitary logic formulas. 1 1 Introduction Recently, there has been considerable interaction between database theory and finite-model theory =-=[F90]-=-. A particularly useful formalism developed in finite-model theory is infinitary logic, an extension of first-order logic with infinite conjunctions and disjunctions [BF85]. Infinitary logic with fini... |

6 |
Monotonic use of space and computational complexity over abstract structures
- Leivant
- 1989
(Show Context)
Citation Context ... in [AV91b] (under the name "loosely coupled generic machine") and in [AV, AVV92]. Computing devices in the spirit of relational machines had already been investigated by Friedman [Fri71] an=-=d Leivant [Lei89]-=-, but with a different focus: Friedman's emphasis was on generalizing recursion theory, whereas Leivant's was on logic characterizations of computational complexity on ordered (or enumerated) structur... |

5 |
Datalog Extensions for Database Updates and Queries
- Abiteboul, Vianu
- 1991
(Show Context)
Citation Context ...ile some condition (e.g., X = ;) holds. An alternative definition of while based on partial fixpoint logic is 2 The evenness query on a set S is the query even(S)= true iff jSj is even. 3 proposed in =-=[AV88]-=-. While expresses pspace on ordered databases [V82], but, like fixpoint, cannot express the evenness query on an unordered set. Infinitary Logic As stated above, while cannot compute the evenness quer... |

5 |
Fixpoint logics, relational machines, and computational complexity
- Abiteboul, Vardi, et al.
- 1997
(Show Context)
Citation Context ...s, without increasing the expressive power of the model. 1 The relational machine was introduced in [AV91] as GM loose (Loosely-coupled Generic Machine). The term relational machine was first used in =-=[AVV92]-=-. 2 Finally, we examine the connection between infinitary logic and well-known restricted query languages like fixpoint and while [CH82], the most well-studied extensions of FO, which motivated defini... |

4 |
Infinitary logics and 0-1
- Kolaitis, Vardi
- 1992
(Show Context)
Citation Context ...nite conjunctions and disjunctions [BF85]. Infinitary logic with finitely many variables (denoted L ! 1! ) (see [Ba77]) provides an elegant framework for studying important phenomena such as 0-1 laws =-=[KV92a]-=- and expressive power of query languages [ACY91, KV90a]. While infinitary logic is an elegant theoretical tool, infinitary logic formulas have a non-effective syntax, and can define non-computable que... |

3 | R.: Inferring Structure
- Nestorov, Abiteboul, et al.
- 1997
(Show Context)
Citation Context ...c characterizations of computational complexity on ordered (or enumerated) structures. In contrast, the focus on unordered structures lies at the core of the investigations in [AV91b, AV, AVV92]. See =-=[AV93]-=- for more details on the relationship between these devices. A normal form We informally describe a powerful normal form for the while language and for relational machines, shown in [AV91b, AV], which... |

2 |
Fixpoint vs. infinitary logic in finite-model theory, to appear
- Kolaitis, Vardi
- 1992
(Show Context)
Citation Context ... we obtain a relational machine accepting models('). 2 We note that the normal form provided by (iii) above can be viewed as the effective counterpart of a similar result shown for the full L k 1! in =-=[KV92]-=-. It is also shown in [KV92] that every formula of L k 1! is equivalent to a countable disjunction of FO l -formulas, for some lsk. Furthermore, it follows from [DLW91] that this holds also for l = k.... |

1 |
Finite-state Datalog automata and relational languages, unpublished manuscript
- Shemesh, Francez
- 1988
(Show Context)
Citation Context ...tactic fragments of infinitary logic. 4 The crux of the approach is to characterize the shape of desired formulas in terms of languages (such as regular languages). 4 We note that Shemesh and Francez =-=[SF88] attempted-=- to provide a language-theoretic characterization of Datalog queries. Their approach, which is quite different than ours, is in terms of what they call "Datalog automata" and it does not com... |

1 |
The Power of the Reflective
- Abiteboul, Papadimitriou, et al.
- 1994
(Show Context)
Citation Context ... the model either [AV]. As will be seen in Section 4, the expressive power is increased if queries generated by the machine can use an unbounded number of variables (such a machine is investigated in =-=[APV94]-=-). Finally, a third extension allows addressable relations, i.e. the relational store has a variable, unbounded number of relations of bounded arity. This extension does not affect expressive power ei... |