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}
}
  • H. Breunig
  • Published in FCT 27 August 2007
  • Mathematics
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… 
Equivalence Problems for Circuits over Sets of Natural Numbers
TLDR
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
TLDR
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
  • Titus Dose
  • Mathematics, Computer Science
    Electron. Colloquium Comput. Complex.
  • 2018
TLDR
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
TLDR
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
Constraint Satisfaction Problems around Skolem Arithmetic
TLDR
This work studies interactions between Skolem Arithmetic and certain classes of Constraint Satisfaction Problems (CSPs) and proves the decidability of SkoleM Arithmetic.
Multiobjective Optimization and Language Equations
TLDR
A general technique is developed that helps to avoid an obstacle that is often hindering in multiobjective approximation: the problem of combining two solutions such that the new solution is balanced in all objectives and also mostly retains the structure of the original solutions.
Circuit satisfiability and constraint satisfaction around Skolem Arithmetic
  • Theor. Comput. Sci.
  • 2017

References

SHOWING 1-5 OF 5 REFERENCES
Word problems requiring exponential time(Preliminary Report)
TLDR
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
TLDR
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.
Mathematical Foundations of Computer Science 1984
Introduction to Circuit Complexity
  • H. Vollmer
  • Computer Science
    Texts in Theoretical Computer Science An EATCS Series
  • 1999