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

@article{McKenzie2007TheCO, title={The Complexity of Membership Problems for Circuits Over Sets of Natural Numbers}, author={Pierre McKenzie and Klaus W. Wagner}, journal={computational complexity}, year={2007}, volume={16}, pages={211-244} }

Abstract.The problem of testing membership in the subset of the natural numbers produced at the output gate of a {$$\bigcup, \bigcap, ^-, +, \times$$} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {$$\bigcup, \bigcap, +, \times$$} is shown NEXPTIME-complete, the cases {$$\bigcup, \bigcap, ^-, \times$$ }, {$$\bigcup, \bigcap, \times$$}, {$$\bigcup, \bigcap, +$$} are shown PSPACE-complete, the case {$$\bigcup…

## 43 Citations

### Equations over Sets of Natural Numbers with Addition Only

- MathematicsSTACS
- 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 $S=\ensuremath{ \{ n \: | \: 16n+13 \in T \} }$.

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

- MathematicsTheory 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.

### On equations over sets of integers

- MathematicsSTACS
- 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.

### 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.

### 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.

### Satisfiability of algebraic circuits over sets of natural numbers

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2007

### Unambiguous Conjunctive Grammars over a One-Letter Alphabet

- Computer ScienceDevelopments in Language Theory
- 2013

It is demonstrated that unambiguous conjunctive grammars over a unary alphabet Σ = {a} have non-trivial expressive power, and that their basic properties are undecidable. The key result is that for…

### 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.

### Complexity of solutions of equations over sets of natural numbers

- MathematicsSTACS
- 2008

The general membership problem for these equations is proved to be EXPTIME-complete and it is established that least solutions of all such systems are in EXPTime.

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

- Mathematics, Computer ScienceTheory 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.

## References

SHOWING 1-10 OF 41 REFERENCES

### Finite Moniods: From Word to Circuit Evaluation

- MathematicsSIAM J. Comput.
- 1997

It is shown that circuit evaluation in the case of any nonsolvable monoid is $P$ complete, while circuits over solvable monoids can be evaluated in $DET \subseteq NC^2$.

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

- MathematicsSIGA
- 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.

### An Optimal Parallel Algorithm for Formula Evaluation

- Computer ScienceSIAM J. Comput.
- 1992

A new approach to Buss’s NC 1 algorithm for evaluation of Boolean formulas is presented and is used to solve the more general problem of evaluating arithmetic formulas by using arithmetic circuits.

### Isolation, Matching, and Counting Uniform and Nonuniform Upper Bounds

- Computer ScienceJ. Comput. Syst. Sci.
- 1999

It is shown that the perfect matching problem is in the complexity class SPL (in the nonuniform setting), and if there are problems in DSPACE(n) requiring exponential-size circuits, then all of the results hold in the uniform setting.

### 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).

### Uniform constant-depth threshold circuits for division and iterated multiplication

- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 2002

### 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.

### Uniform circuits for division: consequences and problems

- Computer Science, MathematicsProceedings 16th Annual IEEE Conference on Computational Complexity
- 2001

Tight bounds are obtained on the uniformity required for division, by showing that division is complete for the complexity class FOM+POW obtained by augmenting FOM with a predicate for powering modulo small primes, and it is shown that, under a well-known number-theoretic conjecture, POW lies in FOM.

### Computing Algebraic Formulas Using a Constant Number of Registers

- Computer Science, MathematicsSIAM J. Comput.
- 1992

It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by algebraic straight-line programs that use only three registers, which is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas.

### Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1998