A1 Refereed original research article in a scientific journal
ON EQUATIONS OVER SETS OF NUMBERS AND THEIR LIMITATIONS
Authors: Lehtinen T, Okhotin A
Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD
Publication year: 2011
Journal: International Journal of Foundations of Computer Science
Journal name in source: INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
Journal acronym: INT J FOUND COMPUT S
Number in series: 2
Volume: 22
Issue: 2
First page : 377
Last page: 393
Number of pages: 17
ISSN: 0129-0541
DOI: https://doi.org/10.1142/S012905411100809X
Abstract
Systems of equations of the form X = Y + Z and X C, in which the unknowns sets of natural numbers, "+" denotes elementwise sum of sets S + T = {m + n vertical bar m is an element of S, n is an element of T}, and C is an ultimately periodic constant, have recently been proved to be computationally universal (Jet, Okhotin, "Equations over sets of natural numbers with addition only", STACS 2009). This paper establishes some limitations of such systems. A class of sets of numbers that cannot be represented by unique, least or greatest solutions of systems of this form is defined, and a particular set in this class is constructed. The argument is then extended to equations over sets of integers.
Systems of equations of the form X = Y + Z and X C, in which the unknowns sets of natural numbers, "+" denotes elementwise sum of sets S + T = {m + n vertical bar m is an element of S, n is an element of T}, and C is an ultimately periodic constant, have recently been proved to be computationally universal (Jet, Okhotin, "Equations over sets of natural numbers with addition only", STACS 2009). This paper establishes some limitations of such systems. A class of sets of numbers that cannot be represented by unique, least or greatest solutions of systems of this form is defined, and a particular set in this class is constructed. The argument is then extended to equations over sets of integers.
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