# The Complexity of Membership Problems for Circuits over Sets of Integers

@inproceedings{Travers2004TheCO, title={The Complexity of Membership Problems for Circuits over Sets of Integers}, author={Stephen D. Travers}, booktitle={MFCS}, year={2004} }

We investigate the complexity of membership problems for \(\{\cup,\cap,\!\bar{\quad},+,\times\}\)-circuits computing sets of integers. These problems are a natural modification of the membership problems for circuits computing sets of natural numbers studied by McKenzie and Wagner (2003). We show that there are several membership problems for which the complexity in the case of integers differs significantly from the case of the natural numbers: Testing membership in the subset of integers… Expand

#### Topics from this paper

#### 23 Citations

Emptiness Problems for Integer Circuits

- Mathematics, Computer Science
- MFCS
- 2017

It turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity. Expand

On equations over sets of integers

- Mathematics, Computer Science
- STACS
- 2010

The class of sets representable by unique solutions of equations using the operations of union and addition using ultimately periodic constants is exactly the class of hyper-arithmetical sets. Expand

Equivalence Problems for Circuits over Sets of Natural Numbers

- Mathematics, Computer Science
- Theory 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. Expand

Representing Hyper-arithmetical Sets by Equations over Sets of Integers

- Computer Science, Mathematics
- Theory of Computing Systems
- 2011

The results settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups, over sets of natural numbers equipped with union, addition and subtraction. Expand

Balance Problems for Integer Circuits

- Computer Science, Mathematics
- Electron. Colloquium Comput. Complex.
- 2018

The work shows that the balance problem for {−, ·}-circuits is undecidable which is the first natural problem for integer circuits or related constraint satisfaction problems that admits only one arithmetic operation and is proven to be Undecidable. Expand

Evaluation of Circuits Over Nilpotent and Polycyclic Groups

- Mathematics, Computer Science
- Algorithmica
- 2017

It is shown that the compressed word problem for every finitely generated nilpotent group is in $${\mathsf {DET}} \subseteq {\mathsf{NC}}^2$$DET⊆NC2, and within the larger class of polycyclic groups the authors find examples where the compressedWord problem is at least as hard as polynomial identity testing for skew arithmetic circuits. Expand

Polynomial-Space Decidable Membership Problems for Recurrent Systems over Sets of Natural Numbers

- Mathematics, Computer Science
- Theory of Computing Systems
- 2007

The set of operations from which functions are built and the impact on the complexity of the membership problems for special finite recurrent systems for PSPACE-decidable membership problems are focused on and completeness results for the complexity classes NL, NP and PSPACE are shown. Expand

Least and greatest solutions of equations over sets of integers

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2016

It is demonstrated that greatest solutions of such equations represent exactly the Σ 1 1 -sets in the analytical hierarchy, and all those sets can already be represented by systems in the resolved form X i = ? i ( X 1, ?, X n ) . Expand

Complexity of Equations over Sets of Natural Numbers

- Mathematics, Computer Science
- 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. Expand

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

- Computer Science, Mathematics
- CiE
- 2016

We study interactions between Skolem Arithmetic and certain classes of Circuit Satisfiability and Constraint Satisfaction Problems (CSPs). We revisit results of Glaser et al. [16] in the context of… Expand

#### References

SHOWING 1-10 OF 11 REFERENCES

Integer circuit evaluation is PSPACE-complete

- Mathematics, Computer Science
- Proceedings 15th Annual IEEE Conference on Computational Complexity
- 2000

A polynomial-time algorithm is shown that reduces QBP (quantified Boolean formula) problem to the integer circuit problem, which complements the result of K. Wagner (1984) to show that IC problem is PSPACEcomplete. Expand

Word problems requiring exponential time(Preliminary Report)

- Computer Science
- STOC
- 1973

A number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines are considered. Expand

The monotone and planar circuit value problems are log space complete for P

- Computer Science, Mathematics
- SIGA
- 1977

It is shown that Ladner's simulation of Turing mac]hines by boolean circuits seems to require an "adequate" set of gates, such as AND and NOT, but the same simulation is possible with monotone circuits using AND and OR gates only. Expand

The complexity of algorithmic problems on succinct instances

- Computer Science
- 1992

Two powerful lemmas quantifying exactly this increase of complexity are presented to show that previous results in the area can be interpreted assufficient conditions for completeness in the logarithmic time and polynomial time counting hierarchies. Expand

The twenty-fourth Fermat number is composite

- Computer Science, Mathematics
- Math. Comput.
- 2003

The rigorous Pepin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors, and it was shown by machine proof that F24 = 2224 + 1 is composite. Expand

The method of forced enumeration for nondeterministic automata

- Computer Science
- Acta Informatica
- 2004

SummaryEvery family of languages, recognized by nondeterministic L(n) tape-bounded Turing machines, where L(n)≥logn, is closed under complement. As a special case, the family of context-sensitive… Expand

Nondeterministic Space is Closed Under Complementation

- Mathematics, Computer Science
- SIAM J. Comput.
- 1988

It immediately follows that the context-sensitive languages are closed under complementation, thus settling a question raised by Kuroda. Expand

An Introduction to the Theory of Numbers

- Mathematics, Philosophy
- 1938

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,… Expand

Making computation count: arithmetic circuits in the nineties

- Computer Science
- SIGA
- 1997

This issue's expert guest column is by Eric Allender, who has just taken over the Structural Complexity Column in the Bulletin of the EATCS, and points out that, while the journal does not require the "alpha" citation style, it does strongly advise authors to use that style. Expand

Maze Recognizing Automata and Nondeterministic Tape Complexity

- Computer Science
- J. Comput. Syst. Sci.
- 1973

A new device called a maze recognizing automaton is introduced that accepts precisely the threadable mazes and can be simulated by a deterministic L(n)-tape bounded Turing machine, provided L( n)>=log"2n". Expand