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Subclasses of Binary NP
, 1996
"... Binary NP consists of all sets of finite structures which are expressible in existential second order logic with second order quantification restricted to relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of rela ..."
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Binary NP consists of all sets of finite structures which are expressible in existential second order logic with second order quantification restricted to relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of relations. We consider mainly three types of classes of relations: unary functions, order relations and graphs with degree bounds. We show that many of these restrictions have the same expressive power and establish a 4level strict hierarchy, represented by sets, permutations, unary functions and arbitrary binary relations, respectively.
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 ...
Descriptive Complexity, Lower Bounds and Linear Time
 In Proceedings of the 12th International Workshop on Computer Science Logic (CSLâ€™98
, 1998
"... This paper surveys two related lines of research: ffl Logical characterizations of (nondeterministic) linear time complexity classes, and ffl nonexpressibility results concerning sublogics of existential secondorder logic. Starting from Fagin's fundamental work there has been steady progress in ..."
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This paper surveys two related lines of research: ffl Logical characterizations of (nondeterministic) linear time complexity classes, and ffl nonexpressibility results concerning sublogics of existential secondorder logic. Starting from Fagin's fundamental work there has been steady progress in both fields with the effect that the weakest logics that are used in characterizations of linear time complexity classes are closely related to the strongest logics for which inexpressibility proofs for concrete problems have been obtained. The paper sketches these developments and highlights their connections as well as the obstacles that prevent us from closing the remaining gap between both kinds of logics. 1 Introduction The theory of computational complexity is quite successful in classifying computational problems with respect to their intrinsic consumption of ressources. Unfortunately it is until now much less successful in proving that the complexity classes that are used for these...
Algebraic and Logical Characterizations of Deterministic Linear Time Classes
 In Proc. 14th Symposium on Theoretical Aspects of Computer Science STACS 97
, 1996
"... In this paper an algebraic characterization of the class DLIN of functions that can be computed in linear time by a deterministic RAM using only numbers of linear size is given. This class was introduced by Grandjean, who showed that it is robust and contains most computational problems that are usu ..."
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In this paper an algebraic characterization of the class DLIN of functions that can be computed in linear time by a deterministic RAM using only numbers of linear size is given. This class was introduced by Grandjean, who showed that it is robust and contains most computational problems that are usually considered to be solvable in deterministic linear time. The characterization is in terms of a recursion scheme for unary functions. A variation of this recursion scheme characterizes DLINEAR, the class which allows polynomially large numbers. A second variation defines a class that still contains DTIME(n), the class of functions that are computable in linear time on a Turing machine. From these algebraic characterizations, logical characterizations of DLIN and DLINEAR as well as complete problems (under DTIME(n) reductions) are derived. 1 Introduction Although deterministic linear time is a frequently used notion in the theory of algorithms it still does not have a universally accept...
Computation of Prime Implicants using Matrix and Paths
"... In this paper, an efficient algorithm to compute the set of prime implicants of a prepositional formula in Conjunctive Normal Form (CNF) is presented. The proposed algorithm uses a concept of representing the formula as a binary matrix and computing paths through the matrix as implicants. The algori ..."
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In this paper, an efficient algorithm to compute the set of prime implicants of a prepositional formula in Conjunctive Normal Form (CNF) is presented. The proposed algorithm uses a concept of representing the formula as a binary matrix and computing paths through the matrix as implicants. The algorithm finds the prime implicants as the prime paths using the divideandconquer technique. The proposed algorithm can be used for knowledge compilation, Clause Maintenance Systems where the knowledge base is prepositional formulae. Moreover, the algorithm is easily adaptable to the incremental mode of computation where an earlier formula is updated by a set of clauses.