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COMPLEXITY OF SOLUTIONS OF EQUATIONS OVER SETS OF NATURAL NUMBERS
, 2008
"... Systems of equations over sets of natural numbers (or, equivalently, language equations over a oneletter alphabet) of the form Xi = ϕi(X1,..., Xn) (1 � i � n) are considered. Expressions ϕi may contain the operations of union, intersection and pairwise sum A+B = {x+y  x ∈ A, y ∈ B}. A system with ..."
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Cited by 6 (2 self)
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Systems of equations over sets of natural numbers (or, equivalently, language equations over a oneletter alphabet) of the form Xi = ϕi(X1,..., Xn) (1 � i � n) are considered. Expressions ϕi may contain the operations of union, intersection and pairwise sum A+B = {x+y  x ∈ A, y ∈ B}. A system with an EXPTIMEcomplete least solution is constructed, and it is established that least solutions of all such systems are in EXPTIME. The general membership problem for these equations is proved to be EXPTIMEcomplete.
EQUATIONS OVER SETS OF NATURAL NUMBERS WITH ADDITION ONLY
, 2009
"... Systems of equations of the form X = Y Z and X = C are considered, in which the unknowns are sets of natural numbers, “+ ” denotes pairwise sum of sets S+T = {m + n  m ∈ S, n ∈ T}, and C is an ultimately periodic constant. It is shown that such systems are computationally universal, in the sense t ..."
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Cited by 2 (2 self)
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Systems of equations of the form X = Y Z and X = C are considered, in which the unknowns are sets of natural numbers, “+ ” denotes pairwise sum of sets S+T = {m + n  m ∈ S, n ∈ T}, and C is an ultimately periodic constant. It is shown that such systems are computationally universal, in the sense that for every recursive (r.e., cor.e.) set S ⊆ N there exists a system with a unique (least, greatest) solution containing a component T with S = {n  16n + 13 ∈ T}. This implies undecidability of basic properties of these equations. All results also apply to language equations over a oneletter alphabet with concatenation and regular constants.
Authors: Authors' previous work (Year
 In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science
, 2005
"... where The zetadimension of a set A of positive integers is Dimζ(A) = inf{s  ζA(s) < ∞}, ζA(s) = � n −s. Zetadimension serves as a fractal dimension on Z + that extends naturally and usefully to discrete lattices such as Z d, where d is a positive integer. This paper reviews the origins of zeta ..."
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Cited by 1 (0 self)
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where The zetadimension of a set A of positive integers is Dimζ(A) = inf{s  ζA(s) < ∞}, ζA(s) = � n −s. Zetadimension serves as a fractal dimension on Z + that extends naturally and usefully to discrete lattices such as Z d, where d is a positive integer. This paper reviews the origins of zetadimension (which date to the eighteenth and nineteenth centuries) and develops its basic theory, with particular attention to its relationship with algorithmic information theory. New results presented include a gale characterization of zetadimension and a theorem on the zetadimensions of pointwise sums and products of sets of positive integers. 1
Functions Definable by Arithmetic Circuits
"... Abstract. An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed graph specifying a cascade of arithmetic and logical operations to be performed on sets of nonnegative integers. In this paper, we consider the definability of functions by means of arithmetic circuits. We prove two n ..."
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Cited by 1 (1 self)
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Abstract. An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed graph specifying a cascade of arithmetic and logical operations to be performed on sets of nonnegative integers. In this paper, we consider the definability of functions by means of arithmetic circuits. We prove two negative results: the first shows, roughly, that a function is not circuitdefinable if it has an infinite range and sublinear growth; the second shows, roughly, that a function is not circuitdefinable if it has a finite range and fails to converge on certain ‘sparse ’ chains under inclusion. We observe that various functions of interest fall under these descriptions.
Functions Definable by Numerical SetExpressions
"... Abstract. A numerical setexpression is a term specifying a cascade of arithmetic and logical operations to be performed on sets of nonnegative integers. If these operations are confined to the usual Boolean operations together with the result of lifting addition to the level of sets, we speak of a ..."
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Abstract. A numerical setexpression is a term specifying a cascade of arithmetic and logical operations to be performed on sets of nonnegative integers. If these operations are confined to the usual Boolean operations together with the result of lifting addition to the level of sets, we speak of additive circuits. If they are confined to the usual Boolean operations together with the result of lifting addition and multiplication to the level of sets, we speak of arithmetic circuits. In this paper, we investigate the definability of sets and functions by means of additive and arithmetic circuits, occasionally augmented with additional operations.
Complex algebras of arithmetic
, 2009
"... An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural number ..."
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An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper we investigate the algebraic structure of complex algebras of natural numbers and make some observations regarding the complexity of various theories of such algebras.
www.cosc.brocku.ca Functions Definable by Arithmetic Circuits
, 2009
"... Abstract. An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed graph specifying a cascade of arithmetic and logical operations to be performed on sets of nonnegative integers. In this paper, we consider the definability of functions by means of arithmetic circuits. We prove two n ..."
Abstract
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Abstract. An arithmetic circuit (McKenzie and Wagner [6]) is a labelled, directed graph specifying a cascade of arithmetic and logical operations to be performed on sets of nonnegative integers. In this paper, we consider the definability of functions by means of arithmetic circuits. We prove two negative results: the first shows, roughly, that a function is not circuitdefinable if it has an infinite range and sublinear growth; the second shows, roughly, that a function is not circuitdefinable if it has a finite range and fails to converge on certain ‘sparse ’ chains under inclusion. We observe that various functions of interest fall under these descriptions.
COMPLEXITY OF SOLUTIONS OF EQUATIONS OVER SETS OF NATURAL NUMBERS
, 2008
"... Abstract. Systems of equations over sets of natural numbers (or, equivalently, language equations over a oneletter alphabet) of the form Xi = ϕi(X1,..., Xn) (1 � i � n) are considered. Expressions ϕi may contain the operations of union, intersection and pairwise sum A+B = {x+y  x ∈ A, y ∈ B}. A sy ..."
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Abstract. Systems of equations over sets of natural numbers (or, equivalently, language equations over a oneletter alphabet) of the form Xi = ϕi(X1,..., Xn) (1 � i � n) are considered. Expressions ϕi may contain the operations of union, intersection and pairwise sum A+B = {x+y  x ∈ A, y ∈ B}. A system with an EXPTIMEcomplete least solution is constructed, and it is established that least solutions of all such systems are in EXPTIME. The general membership problem for these equations is proved to be EXPTIMEcomplete. 1.
www.stacsconf.org ON EQUATIONS OVER SETS OF INTEGERS
"... Abstract. Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S +T = {m+nm ∈ S, n ∈ T} and with ultimately periodic constants is exactly the class of hyper ..."
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Abstract. Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition S +T = {m+nm ∈ S, n ∈ T} and with ultimately periodic constants is exactly the class of hyperarithmetical sets. Equations using addition only can represent every hyperarithmetical set under a simple encoding. All hyperarithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction S − · T = {m−nm ∈ S, n ∈ T, m � n}. Testing whether a given system has a solution is Σ 1 1complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups. 1.