# Representing Hyper-arithmetical Sets by Equations over Sets of Integers

@article{Je2012RepresentingHS,
title={Representing Hyper-arithmetical Sets by Equations over Sets of Integers},
author={Artur Jeż and Alexander Okhotin},
journal={Theory of Computing Systems},
year={2012},
volume={51},
pages={196-228}
}
• Published 1 August 2012
• Mathematics
• Theory of Computing Systems
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, defined as S+T={m+n∣m∈S,n∈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…
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## References

SHOWING 1-10 OF 31 REFERENCES

• Mathematics
STACS
• 2009
It is shown that for every recursive (r.e.) set $S \subseteq \mathbb{N}$ there exists a system with a unique (least, greatest) solution containing a component $T$ with $S=\ensuremath{ \{ n \: | \: 16n+13 \in T \} }$.
• Mathematics
Theory of Computing Systems
• 2009
The general membership problem for equations of the form Xi=φi (X1,…,Xn) (1≤i≤n) is proved to be EXPTIME-complete, and it is established that least solutions of all such systems are in EXPTime.
• Mathematics
ICALP
• 2008
Systems of equations of the form φ j (X 1 , ..., X n ) = i¾? j (X 1 , ..., X n ) with $1 \leqslant j \leqslant m$ are considered, in which the unknowns X i are sets of natural numbers, while the
• Mathematics
computational complexity
• 2007
The problem of testing membership in the subset of the natural numbers produced at the output gate of a combinational circuit is shown to capture a wide range of complexity classes, and results extend in nontrivial ways past work by Stockmeyer and Meyer, Wagner, Wagner and Yang.
• Artur Jeż
• Linguistics
Developments in Language Theory
• 2007
A negative answer is given, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language $$\{ a^{4^{n}} : n \in \mathbb{N} \}$$.
• Computer Science
Theory of Computing Systems
• 2011
The compressed membership problem for one-nonterminal conjunctive grammars over {a} is proved to be EXPTIME-complete; the same problem for the context-free grammar is decidable in NLOGSPACE, but becomes NP-complete if the grammar is compressed as well.