Introduction
Satisfiability Problems
Open Problems
Satisfiability of Algebraic Circuits over Sets of
Natural Numbers
Christian Glaßer, Christian Reitwießner, Stephen Travers,
Matthias Waldherr
Department of Computer Science
University of Würzburg, Germany
FSTTCS 2007, New Delhi, India
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Previous Work
Most important papers on algebraic circuits:
1973 Stockmeyer and Meyer
1984 Wagner
2000 Yang
2003 McKenzie and Wagner
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Definition of Algebraic Circuits
Definition (Algebraic O-Circuit)
Finite, directed,
acyclic graph
Several input gates,
one output gate
Input gates: indegree 0,
label: natural number
Other gates: label from
O ⊆ { , ∪, ∩, +, ×},
-gates: indegree 1,
all other gates: indegree 2
Christian Reitwießner
Example
3
0
3
∪
2
∪
×
+
∩
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
∪
2
∪
×
+
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
{0, 3} ∪
2
∪
×
+
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
{0, 3} ∪
2
∪ {2, 3}
×
+
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
{0, 3} ∪
2
∪ {2, 3}
{0, 6, 9} ×
+
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
{0, 3} ∪
2
∪ {2, 3}
{0, 6, 9} ×
+ {4, 5, 6}
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
{0, 3} ∪
2
∪ {2, 3}
{0, 6, 9} ×
+ {4, 5, 6}
∩ {6}
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Sets Computed by an Algebraic Circuit
Example (Algebraic Circuit and Its Computed Sets)
3
0
3
{0, 3} ∪
2
∪ {2, 3}
{0, 6, 9} ×
+ {4, 5, 6}
∩ {6}
{0, 1, 2, 3, 4, 5, 7, 8, . . . }
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
A More Sophisticated Example For a Circuit
Example (More Sophisticated Circuit)
0
1
∪
×
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
A More Sophisticated Example For a Circuit
Example (More Sophisticated Circuit)
0
1
∪ {0, 1}
×
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
A More Sophisticated Example For a Circuit
Example (More Sophisticated Circuit)
0
1
∪ {0, 1}
{2, 3, 4, . . . }
×
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
A More Sophisticated Example For a Circuit
Example (More Sophisticated Circuit)
0
1
∪ {0, 1}
{2, 3, 4, . . . }
× {x | x is a composite number ≥ 2}
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
A More Sophisticated Example For a Circuit
Example (More Sophisticated Circuit)
0
1
∪ {0, 1}
{2, 3, 4, . . . }
× {x | x is a composite number ≥ 2}
{0, 1} ∪ {x | x is prime}
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
A More Sophisticated Example For a Circuit
Example (More Sophisticated Circuit)
0
1
∪ {0, 1}
{2, 3, 4, . . . }
× {x | x is a composite number ≥ 2}
{0, 1} ∪ {x | x is prime}
∩ {x | x is prime}
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Membership Problems for Algebraic Circuits
Definition (Membership Problems)
Given a circuit C and a number b ∈ N, is b ∈ I(C)?
MC(O) := {(C, b) | C is an O-circuit, b ∈ N and b ∈ I(C)}
Different problems with different complexities for different
subsets O ⊆ { , ∪, ∩, +, ×}.
Extensive study by McKenzie and Wagner in 2003.
Complexity ranges from NL to NEXPTIME.
Major open problem: Unknown if MC( , ∪, ∩, +, ×)
(i.e. the general problem) is decidable or not.
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Complexity of the General Membership Problem
A terminating algorithm for MC( , ∪, ∩, +, ×) would solve
Goldbach’s conjecture (and many other number-theoretic
problems):
Goldbach’s Conjecture (1742)
Every even integer greater than 2
can be written as the sum of two primes.
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Complexity of the General Membership Problem
Example (Circuit for Goldbach’s Conjecture)
PRIMES
sums of
two primes
+
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Complexity of the General Membership Problem
Example (Circuit for Goldbach’s Conjecture)
PRIMES
sums of
two primes
2
1
∪
+
×
Christian Reitwießner
0
even numbers
greater than two
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Complexity of the General Membership Problem
Example (Circuit for Goldbach’s Conjecture)
2
PRIMES
sums of
two primes
1
∪
+
×
G.C. counter-examples
0
even numbers
greater than two
∩
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Algebraic Circuits
Complexity Issues
Complexity of the General Membership Problem
Example (Circuit for Goldbach’s Conjecture)
2
PRIMES
sums of
two primes
1
∪
+
×
G.C. counter-examples
0
even numbers
greater than two
∩
G.C. holds iff 0 ∈
/
Christian Reitwießner
×
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Definition
Results
Satisfiability Problems for Algebraic Circuits
Satisfiability Problems
showing MC( , ∪, ∩, +, ×) undecidable seems out of reach
but it could be done for a generalization of
MC( , ∪, ∩, +, ×)
our approach: introduction of variables
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Definition
Results
Definition of Satisfiability Problems
Definition (Satisfiability Problems)
SC(O) := {(C, b) | C is an O-circuit with some unlabeled
input gates (x1 , x2 , . . . , xn ), b ∈ N and there
is an assignment (a1 , a2 , . . . , an ) ∈ Nn
of these inputs such that
b ∈ I(C(a1 , a2 , . . . , an ))}
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Definition
Results
Example for Satisfiability
Example (Satisfiability of an Algebraic Circuit)
2
×
C :=
+
Is (C, 11) ∈ SC({+, ×})?
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Definition
Results
Example for Satisfiability
Example (Satisfiability of an Algebraic Circuit)
2
3
×
C :=
{11} +
Is (C, 11) ∈ SC({+, ×})? Yes!
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Definition
Results
Undecidability of SC(∩, +, ×)
Theorem
SC(∩, +, ×) is undecidable (and thus also SC( , ∪, ∩, +, ×)).
Proof Idea.
Reduction from Diophantine Equations.
Minor obstacle to overcome:
Circuits cannot use negative numbers.
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Definition
Results
Complexities of the Satisfiability Problems
∪
∪
∪
∪
∪
∪
∪
∪
∪
∪
∪
∪
O
∩
∩
∩
∩
∩
∩
∩
∩
+×
+
×
+×
+
×
+×
+
×
∩+×
∩+
∩ ×
∩
+×
+
×
Lower Bound Upper Bound
undecidable
PSPACE
PSPACE
PSPACE
NP
NP
undecidable
PSPACE
PSPACE
PSPACE
NEXP
P
P
PSPACE
PSPACE
NP
NP
NP
NP
NL
NL
undecidable
NP
NP
NP
NP
NL
NL
NP
NP
NP
NP
NL
UP ∩ coUP
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers
Introduction
Satisfiability Problems
Open Problems
Open Problems
Open Problems
Is SC( , ∪, ∩, +) decidable or not?
Exact complexity of SC(×) (connections to factorization)
And of course the decidability/undecidability of
MC( , ∪, ∩, +, ×)
Christian Reitwießner
Satisfiability of Algebraic Circuits over Sets of Natural Numbers