eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2010-03-09
477
488
10.4230/LIPIcs.STACS.2010.2478
article
On Equations over Sets of Integers
Jez, Artur
Okhotin, Alexander
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 $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and with ultimately periodic constants is exactly the class of hyper-arithmetical sets.
Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction $S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}$. Testing whether a given system has a solution is $\Sigma^1_1$-complete 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol005-stacs2010/LIPIcs.STACS.2010.2478/LIPIcs.STACS.2010.2478.pdf
Language equations
computability
arithmetical hierarchy
hyper-arithmetical hierarchy