FLoC 2018

Dependency Quantified Boolean Formulas

​Every \(\exists\) depends on all \(\forall\) before it.

Dependency Quantified Boolean Formulas

\(\forall x_1...\forall x_n \exists y_1(D_{y1}).. y_n(D_{yn}) : \phi \)

with \(D_{yi} \subseteq \{x_i, ..., x_n\}\)

General Form:

Dependency Quantified Boolean Formulas

Example:

\(\forall x_1, x_2, x_3 \exists y_1 (x_1,x_2), y_2(x_1,x_3):\phi\)

\(\forall x_1\)

\(\forall x_2\)

\(\forall x_3\)

\(\exists y_1\)

\(\exists y_2\)

\(S_{y_1}(t,t)\)

\(S_{y_1}(t,t)\)

\(S_{y_1}(t,f)\)

\(S_{y_1}(t,f)\)

\(S_{y_2}(t,t)\)

\(S_{y_2}(t,f)\)

\(S_{y_2}(t,t)\)

\(S_{y_2}(t,f)\)

Dependency Quantified Boolean Formulas

Example:

\(\forall x_1, x_2, x_3 \exists y_1 (x_1,x_2), y_2(x_1,x_3):\phi\)

\(\forall x_1\)

\(\forall x_2\)

\(\forall x_3\)

\(\exists y_1\)

\(\exists y_2\)

Dependency Quantified Boolean Formulas

Example application: Partial Equivalence Checking

\(\forall x_1 x_2 \exists y_1 y_2\)     \(\forall x_1 \exists y_1 \forall x_2 \exists y_2\)     \(\forall x_1 \exists y_2 \forall x_2 \exists y_1\)
Linearizations cannot express the problem

Dependency Quantified Boolean Formulas

Solved using:

• Incomplete Approximations: implicit depencies \(\supseteq\) explicit
• Search + Skolem clauses
• Universal Expansion
  • until QBF: Eliminate \(x_i\) from incomparable dependency sets (MAXSAT)
  • until SAT: (potentially) exponential increase of variables
• (information fork resolution)

Dependency Quantified Boolean Formulas

More expressive/harder problems

Can model \[\forall x : (\forall y : \phi(x,y)) \land (\forall z : \psi(x,z))\]

without loss of information/smaller search space