# Complexity of Equations over Sets of Natural Numbers

@article{Je2009ComplexityOE, title={Complexity of Equations over Sets of Natural Numbers}, author={Artur Jeż and Alexander Okhotin}, journal={Theory of Computing Systems}, year={2009}, volume={48}, pages={319-342} }

AbstractSystems of equations of the form Xi=φi(X1,…,Xn) (1≤i≤n) are considered, in which the unknowns are sets of natural numbers. Expressions φi may contain the operations of union, intersection and elementwise addition
$S+T=\{m+n\mid m\in S$
, n∈T}. A system with an EXPTIME-complete least solution is constructed in the paper through a complete arithmetization of EXPTIME-completeness. At the same time, it is established that least solutions of all such systems are in EXPTIME. The general…

## 26 Citations

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Variants of the union and concatenation operations on formal languages are investigated, in which Boolean logic in the definitions is replaced with the operations in the two-element field GF(2) (conjunction and exclusive OR), and a new class of formal grammars based on GF( 2)-operations is defined.

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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.

## References

SHOWING 1-10 OF 29 REFERENCES

### Equivalence Problems for Circuits over Sets of Natural Numbers

- Mathematics, Computer ScienceTheory of Computing Systems
- 2008

This work gives a systematic characterization of the complexity of equivalence problems over sets of natural numbers and provides an improved upper bound for the case of {∪,∩,−,+,×}-circuits.

### The Complexity of Membership Problems for Circuits over Sets of Positive Numbers

- MathematicsFCT
- 2007

It is shown that the membership problem for the general case and for (∪∩,+,×) is PSPACE-complete, whereas it is NEXPTIME-hard if one allows 0, and several other cases are resolved.

### Unresolved systems of language equations: Expressive power and decision problems

- Computer ScienceTheor. Comput. Sci.
- 2005

### Integer circuit evaluation is PSPACE-complete

- MathematicsProceedings 15th Annual IEEE Conference on Computational Complexity
- 2000

The integer circuit problem is PSPACE-complete, resolving an open problem posed by P. McKenzie, H. Vollmer, and K. W. Wagner (2000).

### Word Problems and Membership Problems on Compressed Words

- Mathematics, Computer ScienceSIAM J. Comput.
- 2006

A fixed deterministic context-free language with a PSPACE-complete compressed membership problem for finitely presented monoids and completeness results for complexity classes in the range from P to EXPSPACE are obtained.

### Conjunctive Grammars over a Unary Alphabet: Undecidability and Unbounded Growth

- Computer ScienceTheory of Computing Systems
- 2008

The results imply undecidability of a number of decision problems of unary conjunctive grammars, as well as non-existence of a recursive function bounding the growth rate of the generated languages.

### Conjunctive Grammars Can Generate Non-regular Unary Languages

- LinguisticsDevelopments in Language Theory
- 2007

A negative answer is given, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language \(\{ a^{4^{n}} : n \in \mathbb{N} \}\).

### On the recognition of context-free languages

- Computer ScienceSymposium on Computation Theory
- 1984

The first result states that cfl's can be recognized on a cube-connected computer or on a perfect-shuffle computer in log2n time using n6 processors and it can be viewed as an application of parallel algorithms to the design of efficient sequential algorithms.