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Choiceless Computation and Symmetry
, 2010
"... Many natural problems in computer science concern structures like graphs where elements are not inherently ordered. In contrast, Turing machines and other common models of computation operate on strings. While graphs may be encoded as strings (via an adjacency matrix), the encoding imposes a linear ..."
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Many natural problems in computer science concern structures like graphs where elements are not inherently ordered. In contrast, Turing machines and other common models of computation operate on strings. While graphs may be encoded as strings (via an adjacency matrix), the encoding imposes a linear order on vertices. This enables a Turing machine operating on encodings of graphs to choose an arbitrary element from any nonempty set of vertices at low cost (the Augmenting Paths algorithm for Bipartite Matching being an example of the power of choice). However, the outcome of a computation is liable to depend on the external linear order (i.e., the choice of encoding). Moreover, isomorphisminvariance/encodingindependence is an undecidable property of Turing machines. This trouble with encodings led Blass, Gurevich and Shelah [3] to propose a model of computation known as BGS machines that operate directly on structures. BGS machines preserve symmetry at every step in a computation, sacrificing the ability to make arbitrary choices between indistinguishable elements of the input structure (hence “choiceless computation”). Blass et al. also introduced a complexity class CPT+C (Choiceless Polynomial Time with Counting) defined in terms of polynomially bounded BGS machines. While every property finite structures in CPT+C is polynomialtime computable in the usual sense, it is open whether conversely every isomorphisminvariant property in P belongs to CPT+C. In this paper we give evidence that CPT+C = P by proving the separation of the corresponding classes of function problems. Specifically, we show that there is an isomorphisminvariant polynomialtime computable function problem on finite vector spaces (“given a finite vector space V, output the set of hyperplanes in V ”) that is not computable by any CPT+C program. In addition, we give a new simplified proof of the Support Theorem, which is a key step in the result of [3] that a weak version of CPT+C absent counting cannot decide the parity of sets. 1
Axiomatization of Database Transformations
"... Abstract. In database theory, we seek a general computation model for database transformations as an umbrella for queries and updates. However, the literature shows that queries and updates have different flavours, and the completeness standards for query languages cannot be naturally extended to up ..."
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Abstract. In database theory, we seek a general computation model for database transformations as an umbrella for queries and updates. However, the literature shows that queries and updates have different flavours, and the completeness standards for query languages cannot be naturally extended to updates. This motivates the question whether we can use ASMs to characterize database transformations over complex value structures in a unified computation model. In this paper we start examining the differences between database transformations and algorithms that give rise to the notion of Abstract Database Transformation Machine (ADTM), which captures computations involved in database transformations over complex value structures, and encompasses not only queries but also updates. We show that ADTMs can behaviourally simulate any database transformation over complex value structures. 1
Logician in the land of OS: Abstract State Machines in Microsoft
 PROCEEDINGS OF LICS 2001, IEEE COMPUTER SOCIETY, SILVER SPRING, MD
, 2001
"... Analysis of foundational problems like "What is computation?" leads to a sketch of the paradigm of abstract state machines (ASMs). This is followed by a brief discussion on ASMs applications. Then we present some theoretical problems that bridge between the traditional LICS themes and abst ..."
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Analysis of foundational problems like "What is computation?" leads to a sketch of the paradigm of abstract state machines (ASMs). This is followed by a brief discussion on ASMs applications. Then we present some theoretical problems that bridge between the traditional LICS themes and abstract state machines.
Program Schemes, Queues, the Recursive Spectrum and ZeroOne Laws
"... We prove that a very basic class of program schemes augmented with access to a queue and an additional numeric universe within which counting is permitted, so that the resulting class is denoted NPSQ+ (1), is such that the class of problems accepted by these program schemes is exactly the class of r ..."
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We prove that a very basic class of program schemes augmented with access to a queue and an additional numeric universe within which counting is permitted, so that the resulting class is denoted NPSQ+ (1), is such that the class of problems accepted by these program schemes is exactly the class of recursively solvable problems. The class of problems accepted by the program schemes of the class NPSQ(1) where only access to a queue, and not the additional numeric universe, is allowed is exactly the class of recursively solvable problems that are closed under extensions. We dene an innite hierarchy of classes of program schemes for which NPSQ(1) is the rst class and the union of the classes of which is the class NPSQ. We show that the class of problems accepted by the program schemes of NPSQ has a zeroone law and is the union of the classes of problems dened by the sentences of all vectorized Lindstrom logics formed using operators whose corresponding problems are recursively solvab...
An Infinite Hierarchy in a Class of PolynomialTime Program Schemes
"... We dene a class of program schemes RFDPS constructed around notions of forallloops, repeatloops, arrays and ifthenelse instructions, and which take nite structures as inputs, and we examine the class of problems, denoted RFDPS also, accepted by such program schemes. The class of program schemes ..."
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We dene a class of program schemes RFDPS constructed around notions of forallloops, repeatloops, arrays and ifthenelse instructions, and which take nite structures as inputs, and we examine the class of problems, denoted RFDPS also, accepted by such program schemes. The class of program schemes RFDPS is a logic, in Gurevich's sense, in that: every program scheme accepts an isomorphismclosed class of nite structures; we can recursively check whether a given nite structure is accepted by a given program scheme; and we can recursively enumerate the program schemes of RFDPS. We show that the class of problems RFDPS properly contains the class of problems denable in inductive xedpoint logic (for example, the wellknown problem Parity is in RFDPS) and that there is a strict, innite hierarchy of classes of problems within RFDPS (the union of which is RFDPS) parameterized by the depth of nesting of forallloops in our program schemes. This is the rst strict, innite hierarchy in ...
FixedPoint Definability and Polynomial Time on Chordal Graphs and Line Graphs
, 2009
"... The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In pa ..."
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The question of whether there is a logic that captures polynomial time was formulated by Yuri Gurevich in 1988. It is still wide open and regarded as one of the main open problems in finite model theory and database theory. Partial results have been obtained for specific classes of structures. In particular, it is known that fixedpoint logic with counting captures polynomial time on all classes of graphs with excluded minors. The introductory part of this paper is a short survey of the stateoftheart in the quest for a logic capturing polynomial time. The main part of the paper is concerned with classes of graphs defined by excluding induced subgraphs. Two of the most fundamental such classes are the class of chordal graphs and the class of line graphs. We prove that capturing polynomial time on either of these classes is as hard as capturing it on the class of all graphs. In particular, this implies that fixedpoint logic with counting does not capture polynomial time on these classes. Then we prove that fixedpoint logic with counting does capture polynomial time on the class of all graphs that are both chordal and line graphs.
Maximum matching and linear programming in fixedpoint logic with counting. Forthcoming
"... We establish the expressibility in fixedpoint logic with counting (FPC) of a number of natural polynomialtime 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 as ..."
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We establish the expressibility in fixedpoint logic with counting (FPC) of a number of natural polynomialtime 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.
An ASM Macro Language for Sets
, 1998
"... In the paper I introduce a macro language which allows to use in Gurevich's Abstract State Machines (ASMs) directly the set notation. I define families of sets, a language of set terms (union, intersection, instances of families, Cartesian products), their semantics if they appear in transitio ..."
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In the paper I introduce a macro language which allows to use in Gurevich's Abstract State Machines (ASMs) directly the set notation. I define families of sets, a language of set terms (union, intersection, instances of families, Cartesian products), their semantics if they appear in transition rules (extension of familyinstances, vary over set terms, assignments of set terms to familyinstances), and their semantics in boolean terms like setinclusion and elementof relation. The semantics is given in terms of ASMrules. The idea of this macro language is to allow to manipulate sets directly without changing the semantics of ASMs [Gur95]. An integration of sets in the semantics of ASMs has been formalized in [BGS97]. The presented macros have shown to be very useful in the specification of SQL [DiF97]. In ASMs the state is an algebra which has one carrier set, the so called SuperUniverse. Subsets of the super universe, so called universes, are represented by their characteristic f...