# 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

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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

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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

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- 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
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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

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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.
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### 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
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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.

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