Xishun Zhao

Theory and Applications of Satisfiability Testing – SAT 2008

Modelling Max-CSP as Partial Max-SAT.- A Preprocessor for Max-SAT Solvers.- A Generalized Framework for Conflict Analysis.- Adaptive Restart Strategies for Conflict Driven SAT Solvers.- New Results on the Phase Transition for Random Quantified Boolean Formulas.- Designing an Efficient Hardware Implication Accelerator for SAT Solving.- Attacking Bivium Using SAT Solvers.- SAT Modulo the Theory of Linear Arithmetic: Exact, Inexact and Commercial Solvers.- Random Instances of W[2]-Complete Problems: Thresholds, Complexity, and Algorithms.- Complexity and Algorithms for Well-Structured k-SAT Instances.- A Decision-Making Procedure for Resolution-Based SAT-Solvers.- Online Estimation of SAT Solving Runtime.- A Max-SAT Inference-Based Pre-processing for Max-Clique.- SAT, UNSAT and Coloring.- Computation of Renameable Horn Backdoors.- A New Bound for an NP-Hard Subclass of 3-SAT Using Backdoors.- Improvements to Hybrid Incremental SAT Algorithms.- Searching for Autarkies to Trim Unsatisfiable Clause Sets.- Nenofex: Expanding NNF for QBF Solving.- SAT(ID): Satisfiability of Propositional Logic Extended with Inductive Definitions.- Towards More Effective Unsatisfiability-Based Maximum Satisfiability Algorithms.- A CNF Class Generalizing Exact Linear Formulas.- How Many Conflicts Does It Need to Be Unsatisfiable?.- Speeding-Up Non-clausal Local Search for Propositional Satisfiability with Clause Learning.- Local Restarts.- Regular and General Resolution: An Improved Separation.- Finding Guaranteed MUSes Fast.

On the structure of some classes of minimal unsatisfiable formulas

We investigate classes of minimal unsatisfiable formulas which are closed under splitting. For marginal formulas the equivalence to some natural classes of formulas is proved. Further, we show that maximal formulas are closely related to the so-called hitting formulas. That are formulas for which any two clauses contain a pair of complementary literals.

Linear CNF formulas and satisfiability

In this paper, we study linear CNF formulas generalizing linear hypergraphs under combinatorial and complexity theoretical aspects w.r.t. SAT. We establish NP-completeness of SAT for the unrestricted linear formula class, and we show the equivalence of NP-completeness of restricted uniform linear formula classes w.r.t. SAT and the existence of unsatisfiable uniform linear witness formulas. On that basis we prove NP-completeness of SAT for uniform linear classes in a resolution-based manner by constructing large-sized formulas. Interested in small witness formulas, we exhibit some combinatorial features of linear hypergraphs closely related to latin squares and finite projective planes helping to construct rather dense, and significantly smaller unsatisfiable k-uniform linear formulas, at least for the cases k=3,4.

On models for quantified Boolean formulas

A quantified Boolean formula is true, if for any existentially quantified variable there exists a Boolean function depending on the preceding universal variables, such that substituting the existential variables by the Boolean functions results in a true formula. We call a satisfying set of Boolean functions a model. In this paper, we investigate for various classes of quantified Boolean formulas and various classes of Boolean functions the problem whether a model exists. Furthermore, for these classes the complexity of the model checking problem – whether a set of Boolean functions is a model for a formula – will be shown. Finally, for classes of Boolean functions we establish some characterizations in terms of quantified Boolean formulas which have such a model. For example, roughly speaking any satisfiable quantified Boolean Horn formula can be satisfied by monomials and vice versa.

On Boolean Models for Quantified Boolean Horn Formulas

For a Quantified Boolean Formula \(({\it QBF })\) Φ=Qφ, an assignment is a function \(\cal M\) that maps each existentially quantified variable of Φ to a Boolean function, where φ is a propositional formula and Q is a linear ordering of quantifiers on the variables of Φ. An assignment \(\cal M\) is said to be proper, if for each existentially quantified variable y i , the associated Boolean function f i does not depend upon the universally quantified variables whose quantifiers in Q succeed the quantifier of y i . An assignment \(\cal M\) is said to be a model for Φ, if it is proper and the formula \(\phi^{\cal M}\) is a tautology, where \(\phi^{\cal M}\) is the formula obtained from φ by substituting f i for each existentially quantified variable y i . We show that any true quantified Horn formula has a Boolean model consisting of monotone monomials and constant functions only; conversely, if a QBF has such a model then it contains a clause–subformula in \({\it QHORN }\cap {\it SAT }\).

Boolean Functions as Models for Quantified Boolean Formulas

In this paper, we introduce the notion of models for quantified Boolean formulas. For various classes of quantified Boolean formulas and various classes of Boolean functions, we investigate the problem of determining whether a model exists. Furthermore, we show for these classes the complexity of the model checking problem, which is to check whether a given set of Boolean functions is a model for a formula. For classes of Boolean functions, we establish some characterizations in terms of classes of quantified Boolean formulas that have such a model.

Model-equivalent reductions

In this paper, the notions of polynomial–time model equivalent reduction and polynomial–space model equivalent reduction are introduced in order to investigate in a subtle way the expressive power of different theories. We compare according to these notions some classes of propositional formulas and quantified Boolean formulas. Our results show that classes of theories with the same complexity might have different representation strength under some conjectures which are widely believed to be true in computation complexity theory.

Resolution and Expressiveness of Subclasses of Quantified Boolean Formulas and Circuits

We present an extension of Q-Unit resolution for formulas that are not completely in clausal form. This b-unit resolution is applied to different classes of quantified Boolean formulas in which the existential and universal variables satisfy the Horn property. These formulas are transformed into propositional equivalents consisting of only polynomially many subformulas. We obtain compact encodings as Boolean circuits and show that both representations have the same expressive power.

Equivalence models for quantified boolean formulas

In this paper, the notion of equivalence models for quantified Boolean formulas with free variables is introduced. The computational complexity of the equivalence model checking problem is investigated in the general case and in some restricted cases. We also establish a connection between the structure of some quantified Boolean formulas and the structure of models.

Transformations into normal forms for quantified circuits

We consider the extension of Boolean circuits to quantified Boolean circuits by adding universal and existential quantifier nodes with semantics adopted from quantified Boolean formulas (QBF). The concept allows not only prenex representations of the form ∀x1∃x1...∀xn∃yn c where c is an ordinary Boolean circuit with inputs x1, ..., xn, y1, ..., yn. We also consider more general non-prenex normal forms with quantifiers inside the circuit as in non-prenex QBF, including circuits in which an input variable may occur both free and bound. We discuss the expressive power of these classes of circuits and establish polynomialtime equivalence-preserving transformations between many of them. Additional polynomial-time transformations show that various classes of quantified circuits have the same expressive power as quantified Boolean formulas and Boolean functions represented as finite sequences of nested definitions (NBF). In particular, universal quantification can be simulated efficiently by circuits containing only existential quantifiers if overlapping scopes of variables are allowed.