Satisfiability of Algebraic Circuits over Sets of ... - Semantic Scholar
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
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
