TY - JOUR

T1 - On equations over sets of numbers and their limitations

AU - Lehtinen, Tommi

AU - Okhotin, Alexander

PY - 2011/2

Y1 - 2011/2

N2 - Systems of equations of the form X = Y + Z and X = C, in which the unknowns are sets of natural numbers, "+" denotes elementwise sum of sets S + T = {m + n | m ∈ S, n ∈ T}, and C is an ultimately periodic constant, have recently been proved to be computationally universal (Jeż, 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.

AB - Systems of equations of the form X = Y + Z and X = C, in which the unknowns are sets of natural numbers, "+" denotes elementwise sum of sets S + T = {m + n | m ∈ S, n ∈ T}, and C is an ultimately periodic constant, have recently been proved to be computationally universal (Jeż, 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.

KW - Language equations

KW - unary languages

UR - http://www.scopus.com/inward/record.url?scp=79952139667&partnerID=8YFLogxK

U2 - 10.1142/S012905411100809X

DO - 10.1142/S012905411100809X

M3 - Article

AN - SCOPUS:79952139667

VL - 22

SP - 377

EP - 393

JO - International Journal of Foundations of Computer Science

JF - International Journal of Foundations of Computer Science

SN - 0129-0541

IS - 2

ER -