Circuit satisfiability problem
In theoretical computer science, the Circuit Satisfiability Problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true.[1] In other words, it asks whether the inputs to a given Boolean Circuit can be consistently set to 1 or 0 such that the circuit outputs 1. If that is the case, the circuit is called satisfiable. Otherwise, the circuit is called unsatisfiable. For instance, in the figure above, the circuit on the left can be satisfied by setting both inputs to be 1, but the circuit on the right is unsatisfiable.
CircuitSAT is closely related to Boolean satisfiability problem (SAT), and likewise, has been proven to be NP-complete.[2] In fact, it is a prototypical NP-complete problem; the Cook–Levin theorem is sometimes proved on CircuitSAT instead of on the SAT and then reduced to the other satisfiability problems to prove their NP-completeness.[1][3]
Properties
The satisfiability of a circuit containing m arbitrary binary gates can be decided in time .[4]
Variants
Reduction
Arithmetization is to reduce a CircuitSAT problem into a set of arithmetic constrains. Currently, in the literature, there are two direction of reduction: reduce to a combination satisfiability problem or reduce to an algebraic satisfiability problem. An example of such reduction from boolean circuit to polynomial is presented here, and they are called Zhegalkin_polynomial.
The Tseitin transformation
There is a straightforward reduction from CircuitSAT to SAT, known as the Tseitin transformation. The transformation is especially easy to describe if the circuit is wholly constructed from 2-input NAND gates (a functionally-complete set of Boolean operators): assign every net in the circuit a variable, then for each NAND gate, construct the conjunctive normal form clauses (v1 ∨ v3) ∧ (v2 ∨ v3) ∧ (¬v1 ∨ ¬v2 ∨ ¬v3) where v1 and v2 are the inputs to the NAND gate and v3 is the output. These clauses completely describe the relationship between the three variables. Conjoining the clauses from all the gates with an additional clause constraining the circuit's output variable to be true completes the reduction; an assignment of the variables satisfying all of the constraints exists if and only if the original circuit is satisfiable, and any solution is a solution to the original problem of finding inputs that make the circuit output 1.[1][5] (The converse, that SAT is reducible to CircuitSAT, is even easier—we simply rewrite the Boolean formula as a circuit and solve that.)
See Also
- Circuit Value Problem
- Structured Circuit Satisfibility
- Satisfiability problem
References
- ^ a b c David Mix Barrington and Alexis Maciel (July 5, 2000). "Lecture 7: NP-Complete Problems" (PDF).
- ^ Luca Trevisan (November 29, 2001). "Notes for Lecture 23: NP-completeness of Circuit-SAT" (PDF).
- ^ See also, for example, the informal proof given in Scott Aaronson's lecture notes from his course Quantum Computing Since Democritus.
- ^ Sergey Nurk (December 1, 2009). "An O(2^{0.4058m}) upper bound for Circuit SAT".
- ^ Marques-Silva, João P. and Luís Guerra e Silva (1999). "Algorithms for Satisfiability in Combinational Circuits Based on Backtrack Search and Recursive Learning" (PDF).