Circuits over sets of natural numbers


Circuits over sets of natural numbers

Circuits over natural numbers is a mathematical model used in studying computational complexity theory. It is a special case of circuit, the object is a labeled directed acyclic graph the nodes of which evaluate to sets of natural numbers, the leaves are finite sets, and the gates are set operations or arithmetic operations.

As an algorithmic problem, the possible question are to find if a given natural number is an element is in the output node or if two circuits compute the same set. The decidability is still an open question, but there are results on restriction of those circuits. Finding answers to some questions about this model could serve as a proof to many important mathematical conjectures, like Goldbach's conjecture.

Contents

Formal definition

An natural number circuit is a circuit, i.e. a labelled directed acyclic graph of in-degree at most 2. The nodes of in-degree 0, the leaves, are finite sets of natural numbers, the labels of the nodes of in-degree 1 are −, where \overline{A}=\{x\in\mathbb{N}|x\not\in A\} and the labels of the nodes of in-degree 2 are +, × and ∪, where A+B=\{a+b|a\in A, b\in B\}, A\times B=\{a\times b|a\in A, b\in B\} and ∪ and ∩ with the usual set meaning.

The subset of circuits which do not use all of the possible labels are also studied.

Algorithmic problems

One can ask:

  • Is a given number n a member of the output node.
  • Is the output node empty, does it contain a specific element, is it equal to \mathbb{N}?
  • Is one node is a subset of another.

For circuits which use all the labels, all these problems are equivalent. They are all equivalent if we can use any labels for the gates.

Proof

The first problem is reducible to the second one, by taking the intersection of the output gate and n. Indeed the new output get will be empty if and only if n was not an element of the former output gate.

The first problem is reducible to the third one, by asking if the node n is a subset of the output node.

The second problem is reducible to the first one, it suffices to multiply the output gate by 0, then 0 will be in the output gate if and only if the former output gate were not empty.

The third problem is reducible to the second one, checking if A is a subset of B is equivalent to ask if there is an element in A\cap\overline{B}.

Restrictions

Let O be a subset of {∪,∩,−,+,×}, then we call MC(O) the problem of finding if a natural number is inside the output gate of a circuit the gates' labels of which are in O, and MF(O) the same problem with the added constraint that the circuit must be a tree.

Examples

  • The set of numbers greater than n is \overline{\{0,\dots,n\}}. In particular \mathbb{N}=\overline{\{\}}?
  • The set of prime numbers with 0 and 1, PRIME' is the complement of the numbers who are multiple of 2 natural numbers greater than 2, so it is \overline{\overline{\{0,1\}}\times \overline{\{0,1\}}}
  • The set of prime numbers, PRIME is then PRIME'\cap \overline{\{0,1\}}, the elements of PRIME' greater than 2.
  • The set of EVEN numbers is \mathbb{N}\times \{2\}.

Goldbach's conjecture asks if there is an even number greater than 2 which is not the sum of two prime numbers, it is natural to rephrase this question by asking if there is an element in EVEN\cap \overline{\{2\}}\cap \overline{PRIMES+PRIMES}.

Quickly growing set

One difficulty come from the fact that the complement of a finite set is infinite, and computer has got only a finite memory. But even without complementation, one can create double exponential number. Let E_0=\{2\}, E_{i+1}=E_i\times E_i, then one can easily prove by induction on i that E_i=\{2^{2^i}\}, indeed E_0=\{2\}=\{2^1\}=\{2^{2^0}\} and by induction E_{i+1}=E_i\times E_i=\{2^{2^i}\}\times\{2^{2^i}\}=\{(2^{2^i})^2\}=\{2^{2^i\times2}\}=\{2^{2^{i+1}}\}.

And even double exponential—sized sets: let S_0=\{0,1,2\}, S_{i+1}=(S_i\times S_i)+S_i, then \{x|0<x<2^{2^i}\}\subset S_i, i.e. Si contains the 2^{2^i} firsts number. Once again this can be proved by induction on i, it is true for S0 by definition and let x\in\{x|0<x<2^{2^{i+1}}\}, dividing x by 2^{2^i} we see it can be written as x=2^{2^i}\times d+r where d,r< 2^{2^i}, and by induction, 2^{2^i}, d and r are in Si, so indeed x\in (S_i \times S_i)+ S_i.

Those examples explains why addition and multiplication are enough to create problem of high complexity.

Complexity results

Membership problem

The membership problem ask if, given an element n and a circuit, if n is in the output gate of the circuit.

When the class of authorized gate is restricted, the membership problem lay inside well known complexity classes.

Complexity
O MC(O) MF(O)
∪,∩,−,+,× NEXPTIME-hard

Decidable with an oracle for the halting problem

PSPACE-hard
∪,∩,+,× NEXPTIME-complete NP-complete
∪,+,× PSPACE-complete NP-complete
∩,+,× P-hard, in co-R LOGCFL
+,× P-complete L-complete
∪,∩,−,+ PSPACE-complete PSPACE-complete
∪,∩,+ PSPACE-complete NP-complete
∪,+ NP-complete NP-complete
∩,+ L-complete
+ L-complete
∪,∩,−,× PSPACE-complete PSPACE-complete
∪,∩,× PSPACE-complete NP-complete
∪,× NP-complete NP-complete
∩,× P L-complete
× NL-complete L-complete
∪,∩,− P-complete NC1-complete
∪,∩ P-complete L-complete
NL-complete L-complete
NL-complete L-complete

Equivalence problem

The equivalence problem ask if, given two gates of a circuits, they evaluate to the same set.

When the class of authorized gate is restricted, the equivalence problem lay inside well known complexity classes[1]. We call EC(O) and EF(O) the problem of equivalence over circuits and formulae the gate of which are in O.

Complexity
O EC(O) EF(O)
∪,∩,−,+,× NEXPTIME-hard

Decidable with an oracle for the halting problem

PSPACE-hard

Decidable with an oracle for the halting problem

∪,∩,+,× NEXPTIME-hard, in coNEXPNP ΠP2-complete
∪,+,× NEXPTIME-hard, in coNEXPNP ΠP2-complete
∩,+,× P-hard, in BPP L-hard, in LOGCFL
+,× L-hard, in LOGCFL P-hard, in coRP
∪,∩,−,+ PSPACE-complete PSPACE-complete
∪,∩,+ PSPACE-complete ΠP2-complete
∪,+ ΠP2-complete ΠP2-complete
∩,+ coL-complete
+ L-complete
∪,∩,−,× PSPACE-complete PSPACE-complete
∪,∩,× PSPACE-complete ΠP2-complete
∪,× ΠP2-complete ΠP2-complete
∩,× coP L-complete
× P L-complete
∪,∩,− P-complete L-complete
∪,∩ P-complete L-complete
NL-complete L-complete
NL-complete L-complete

References

  1. ^ Christian Glaßer, Katrin Herr, Christian Reitwießner, Stephen Travers and Matthias Waldherr (2007), "Equivalence Problems for Circuits over Sets of Natural Numbers", Lecture Notes in Computer Science (Berlin / Heidelberg: Springer) Volume 4649/2007: 127–138, doi:10.1007/978-3-540-74510-5, ISBN 978-3-540-74509-9, http://www.springerlink.com/content/c007kk787054v746/ 

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