On polynomial time computation over unordered structures

Blass, Andreas; Gurevich, Yuri; Shelah, Saharon

doi:10.2178/jsl/1190150152

Cited by 59 publications

(67 citation statements)

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“…While the definability of the first follows also from a result by Blass, Gurevich, and Shelah [6] and is not strictly new, the second strengthens it and is new; see the section on related work for more on this.…”

Section: Consequencesmentioning

confidence: 80%

“…Despite the negative results from [7], the expressive power of these logics is still the object of study. Somewhat unexpectedly, it was shown in [6] that the property of having a perfect matching in bipartite graphs is expressible in the uniform version of C k ∞ω called IFP + C. Here we revisit matchings in bipartite graphs and consider the more general problems of st-flows in networks with unit capacities. Our results show that the existence of such flows with prescribed values is expressible in C 3 ∞ω .…”

Section: Related Workmentioning

confidence: 97%

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Sherali-Adams relaxations and indistinguishability in counting logics

Atserias

Maneva

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

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Abstract. Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity P T AP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali-Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known color-refinement heuristic for graph isomorphism called the Weisfeiler-Lehman algorithm, or, equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications in both finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer, and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to Ω(n) levels, where n is the number of vertices in the graph.Key words. first-order logic, counting quantifiers, linear programming, Weisfeiler-Lehman algorithm, graph isomorphism, combinatorial optimization AMS subject classifications. 68Q17, 68Q19, 52B12, 05C60, 05C72, 03C80 DOI. 10.1137/120867834 1. Introduction. Let A and B be the adjacency matrices of two labeled graphs on {1, . . . , n}. The two graphs being isomorphic is equivalent to the existence of a permutation matrix P for which the relation P T AP = B holds. Multiplying both sides by P gives the equivalent condition AP = PB. At this point a linear programming relaxation suggests itself: relax the condition that P is a permutation matrix to a doubly stochastic matrix. How much coarser is this than actual isomorphism?The concept of fractional isomorphism as defined in the preceding paragraph falls within the framework of linear programming relaxations of combinatorial problems. Other types of relaxations of isomorphism include the color-refinement method called the Weisfeiler-Lehman (WL) algorithm. In this algorithm the vertices of the graphs are classified according to their degree, then according to the multiset of degrees of their neighbors, and so on until a fixed point is achieved. If the two graphs get partitions with different parameters, the graphs are definitely not isomorphic. As it turns out, fractional isomorphism and color refinement yield one and the same relaxation: it was shown by Ramana, Scheinerman, and Ullman [31] that two graphs

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Section: Consequencesmentioning

confidence: 80%

Section: Related Workmentioning

confidence: 97%

Sherali-Adams relaxations and indistinguishability in counting logics

Atserias

Maneva

2012

Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

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“…Parallel ASMs have been used for studying the complexity of algorithms over unordered structures. See [10,37].…”

Section: Resultsmentioning

confidence: 99%

The Generic Model of Computation

Dershowitz

2012

Electron. Proc. Theor. Comput. Sci.

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Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein. BackgroundAbstract state machines (ASMs), invented by Yuri Gurevich [24], constitute a most general model of computation, one that can operate on any desired level of abstraction of data structures and native operations. All (ordinary) models of computation are instances of this one generic paradigm. Here, we give an overview of the foundational considerations underlying the model (cobbled together primarily from [18,3,12]). 1 Programs (of the sequential, non-interactive variety) in this formalism are built from three components:• There are generalized assignments f (s 1 , . . . , s n ) := t, where f is any function symbol (in the vocabulary of the program) and the s i and t are arbitrary terms (in that vocabulary).• Statements may be prefaced by a conditional test, if C then P or if C then P else Q, where C is a propositional combination of equalities between terms.• Program statements may be composed in parallel, following the keyword do, short for do in parallel.An ASM program describes a single transition step; its statements are executed repeatedly, as a unit, until no assignments have their conditions enabled. (Additional constructs beyond these are needed for interaction and large-scale parallelism, which are not dealt with here.) As a simple example, consider the program shown as Algorithm 1, describing a version of selection sort, where F(0), . . . , F(n − 1) contain values to be sorted, F being a unary function symbol. Initially, n ≥ 1 is the quantity of values to be sorted, i is set to 0, and j to 1. The brackets indicate statements that are executed in parallel. The program proceeds by repeatedly modifying the values of i and j, as well as of locations in F, referring to terms F(i) and F( j). When all conditions fail, that is, when j = n and i + 1 = n, the values in F have been sorted vis-à-vis the black-box relation ">". The program halts, as there is nothing left to do. (Declarations and initializations for program constants and variables are not shown.)This sorting program is not partial to any particular representation of the natural numbers 1, 2, etc., which are being used to index F. Whether an implementation uses natural language, or decimal numbers, 1 For a video lecture of Gurevich's on this subject, see http://www.youtube.com/v/7XfA5EhH7Bc. Generic Model of ComputationAlgorithm 1 An abstract-state-machine program for sorting.Algorithm 2 An abstract-state-machine program for bisection search.or binary strings is immaterial, as long as addition behaves as expected (and equality and disequality, too). Furthermore, the program will work regardless of the domain from which...

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“…We note that extensions of choiceless polynomial time are still being actively researched in connection with the QPTIME problem [9,16,15].…”

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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On polynomial time computation over unordered structures

Cited by 59 publications

References 20 publications

Sherali-Adams relaxations and indistinguishability in counting logics

Sherali-Adams relaxations and indistinguishability in counting logics

The Generic Model of Computation

Database Theory, Yuri, and Me

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