Choiceless polynomial time

Blass, Andreas; Gurevich, Yuri; Shelah, Saharon

doi:10.1016/s0168-0072(99)00005-6

Cited by 90 publications

(171 citation statements)

References 16 publications

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“…As a consequence, we solve an open problem of [BGS99] concluding that there is a formula of fixed-point logic with counting which defines the size of a maximum b-matching in a graph. This is one demonstration of the power of the ellipsoid method and linear optimisation that can be brought to bear even in the setting of logical definability.…”

Section: Resultsmentioning

confidence: 99%

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Anderson

Dawar

Holm

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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We establish the expressibility in fixed-point logic with counting (FPC) of a number of natural polynomial-time problems. In particular, we show that the size of a maximum matching in a graph is definable in FPC.This settles an open problem first posed by Blass, Gurevich and Shelah [BGS99], who asked whether the existence of perfect matchings in general graphs could be determined in the more powerful formalism of choiceless polynomial time with counting. Our result is established by showing that the ellipsoid method for solving linear programs can be implemented in FPC. This allows us to prove that linear programs can be optimised in FPC if the corresponding separation oracle problem can be defined in FPC. On the way to defining a suitable separation oracle for the maximum matching problem, we provide FPC formulas defining maximum flows and canonical minimum cuts in capacitated graphs.

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Section: Resultsmentioning

confidence: 99%

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Anderson

Dawar

Holm

2013

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science

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“…I remember a meeting on finite model theory in Oberwolfach in 1998, just a few months before the honorary doctorate ceremony, where he gave a talk on Choiceless Polynomial Time [8], a very expressive database query language in which only polynomial-time queries can be expressed. The language is nice because it borrows its high expressivity from set theory in a natural way, and also because it is based on Gurevich's Abstract State Machines (ASM [25]) .…”

Section: Choiceless Polynomial Timementioning

confidence: 99%

Database Theory, Yuri, and Me

Bussche

2010

Fields of Logic and Computation

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Abstract. Yuri Gurevich made many varied and deep contributions to logic for computer science. Logic provides also the theoretical foundation of database systems. Hence, it is almost unavoidable that Gurevich made some great contributions to database theory. We discuss some of these contributions, and, along the way, present some personal anecdotes connected to Yuri and the author. We also describe the honorary doctorate awarded to Gurevich by Hasselt University (then called Limburgs Universitair Centrum) in 1998.Dedicated to Yuri Gurevich, the "man with a plan", on his 70th birthday. Database TheoryThe theory of database systems is a very broad field of theoretical computer science, concerned with the theoretical design and analysis of all data management aspects of computer science. One can get a good idea of the current research in this field by looking at the proceedings of the two main conferences in the area: the International Conference on Database Theory, and the ACM Symposium on Principles of Database Systems. As data management research in general follows the rapid changes in computing and software technology, database theory can appear quite trendy to the outsider. Nevertheless there are also timeless topics such as the theory of database queries, to which Yuri Gurevich has made a number of fundamental contributions.An in-depth treatment of database theory until the early 1990s can be found in the book of Abiteboul, Hull and Vianu [1]; Yuri Gurevich appears nine times in the bibliography.A relational database schema is a finite relational vocabulary, i.e., a finite set of relation names with associated arities. Instead of numbering the columns of a relation with numbers, as usual in mathematical logic, in database theory it is also customary to name the columns with attributes. In that case the arity is replaced by a finite set of attributes (called a relation scheme). A database instance over some schema is a finite relational structure over that schema, i.e., an assignment of a concrete, finite, relation content to each of the relation names. So, if R is a relation name of arity k and D is a database, then D(R) is a finite subset of U k , where U is some universe of data elements. The idea is that the contents of a database can be updated frequently, hence the term "instance". We will often drop this term, however, and simply talk about a "database".

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“…This model was already proposed in [4,Subsection 4.8] as worthy of further study. The present paper contains the first results of that study.…”

Section: Introductionmentioning

confidence: 97%

“…The proof of this involved exceptionally simple instances of the bipartite matching problem. In the traditional picture of bipartite matching, where the input consists of a set of boys, a set of girls, and a (symmetric) "willing to marry" relation between them, the instances used in [4] can be described as follows. First suppose there are 2n boys and 2n girls, divided into two gangs of n boys and n girls each; a boy and a girl are willing to marry if and only if they belong to the same gang.…”

Section: Introductionmentioning

confidence: 99%

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On polynomial time computation over unordered structures

2002

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This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting. Without padding, they remain in polynomial time but appear not to be in choiceless polynomial time plus counting. Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded. Finally, we describe another plausible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting.

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Choiceless polynomial time

Cited by 90 publications

References 16 publications

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Maximum Matching and Linear Programming in Fixed-Point Logic with Counting

Database Theory, Yuri, and Me

On polynomial time computation over unordered structures

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