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Fixpoint Logics, Relational Machines, and Computational Complexity
 In Structure and Complexity
, 1993
"... We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have t ..."
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Cited by 36 (5 self)
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We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic  while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...
Computing on Structures
"... this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathemat ..."
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Cited by 3 (1 self)
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this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathematicians such as Tarski [Tar86], and Harvey Friedman [Fri71] (see Section 5). It has come up more recently in the context of databases, where devices computing on structures model more acurately database computation carried out against an abstract interface hiding the internal representation of data. Thus, the primary benefit of studying devices and languages computing on structures is that they clarify issues which are obscured in classical devices such as Turing machines. For example, they yield new notions of complexity, quite different from classical computational complexity. They reflect more acurately the actual complexity of computation, which, like database computation, cannot take advantage of encodings of structures. An example is provided by the query even on a set
Padding and the Expressive Power of Existential SecondOrder Logics
 Proceedings of the Annual Conference of the European Association for Computer Science Logic, Lecture Notes in Computer Science
, 1997
"... Padding techniques are wellknown from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential secondorder logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of ..."
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Cited by 2 (1 self)
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Padding techniques are wellknown from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential secondorder logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula ' in the presence of padding, if ' is able to distinguish between (sufficiently) padded versions of graphs from A and padded versions of graphs that are not in A. From results of Lynch [Lyn82, Lyn92] it can be easily concluded that (essentially) every NP set of graphs is weakly expressible by an existential monadic secondorder (Mon\Sigma 1 1 ) formula with polynomial padding and builtin addition. In particular, NP 6= coNP if and only if there is a coNPset of graphs that is not weakly expressible by a Mon\Sigma 1 1 formula in the presence of addition, even if polynomial padding is allowed. In some sense, this implies that Mon\Sigma ...
Connection Matrices for MSOLdefinable Structural Invariants
"... Abstract. Connection matrices of graph parameters were first introduced by M. Freedman, L. Lovász and A. Schrijver (2007) to study the question which graph parameters can be represented as counting functions of weighted homomorphisms. The rows and columns of a connection matrix M(f, □) of a graph pa ..."
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Abstract. Connection matrices of graph parameters were first introduced by M. Freedman, L. Lovász and A. Schrijver (2007) to study the question which graph parameters can be represented as counting functions of weighted homomorphisms. The rows and columns of a connection matrix M(f, □) of a graph parameter f and a binary operation □ are indexed by all finite (labeled) graphs Gi and the entry at (Gi, Gj) is given by the value of f(Gi□Gj). Connection matrices turned out to be a very powerful tool for studying graph parameters in general. B. Godlin, T. Kotek and J.A. Makowsky (2008) noticed that connection matrices can be defined for general relational structures and binary operations between them, and for general structural parameters. They proved that for structural parameters f definable in Monadic Second Order Logic, (MSOL) and binary operations compatible with MSOL, the connection matrix M(f, □) has always finite rank. In this talk we discuss several applications of this Finite Rank Theorem, and outline ideas for further research. 1