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

@inproceedings{Breunig2007TheCO, title={The Complexity of Membership Problems for Circuits over Sets of Positive Numbers}, author={Hans Georg Breunig}, booktitle={FCT}, year={2007} }

We investigate the problems of testing membership in the subset of the positive numbers produced at the output of (∪∩,-,+,×) combinational circuits. These problems are a natural modification of those studied by McKenzie and Wagner (2003), where circuits computed sets of natural numbers. It turns out that the missing 0 has strong implications, not only because 0 can be used to test for emptiness. We show that the membership problem for the general case and for (∪∩,+,×) is PSPACE-complete…

## 12 Citations

Equivalence Problems for Circuits over Sets of Natural Numbers

- Mathematics, Computer ScienceCSR
- 2007

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.

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.

Emptiness Problems for Integer Circuits

- MathematicsMFCS
- 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.

Balance Problems for Integer Circuits

- Mathematics, Computer ScienceElectron. 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.

Complexity of Equations over Sets of Natural Numbers

- MathematicsTheory 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.

Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

- Mathematics, Computer ScienceCiE
- 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…

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2017

Constraint Satisfaction Problems around Skolem Arithmetic

- Mathematics, Computer ScienceArXiv
- 2015

This work studies interactions between Skolem Arithmetic and certain classes of Constraint Satisfaction Problems (CSPs) and proves the decidability of SkoleM Arithmetic.

## References

SHOWING 1-9 OF 9 REFERENCES

Equivalence Problems for Circuits over Sets of Natural Numbers

- Mathematics, Computer ScienceCSR
- 2007

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 integers

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2006

Integer circuit evaluation is PSPACE-complete

- MathematicsProceedings 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.

Word problems requiring exponential time(Preliminary Report)

- Computer Science, MathematicsSTOC
- 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.

The Complexity of Membership Problems for Circuits Over Sets of Natural Numbers

- Mathematicscomputational 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.

Reducing the complexity of reductions

- Mathematicscomputational complexity
- 2001

The Berman—Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions, and it is shown that for any class of complete sets closed under uniform TC0-computable many-one reductions, the following three theorems hold.

Introduction to Circuit Complexity

- Computer ScienceTexts in Theoretical Computer Science An EATCS Series
- 1999

Mathematical Foundations of Computer Science 1984

- Computer ScienceLecture Notes in Computer Science
- 1984