# FLoC 2018

### Selected Talks

Dependency Quantified Boolean Formulas

 ​QBF PSPACE-complete DQBF Dependencies explicitly specified NEXPTIME-complete

​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

Rabe: "DQBF can encode existential quantification over functions"

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

Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

See original slides

PySAT: A Python Toolkit for
Prototyping with SAT Oracles

• Python toolkit for working with multiple SAT solvers
• Its MAXSAT implementation won the MAXSAT competition

By krr

• 907