DeepAI

# QRAT+: Generalizing QRAT by a More Powerful QBF Redundancy Property

The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction operation to eliminate universal literals from clauses. We apply QUP to an abstraction of the QBF where certain universal quantifiers are converted into existential ones. This way, we obtain a generalization of QRAT we call QRAT+. The redundancy property in QRAT+ based on QUP is more powerful than the one in QRAT based on UP. We report on proof theoretical improvements and experimental results to illustrate the benefits of QRAT+ for QBF preprocessing.

• 7 publications
• 5 publications
04/29/2019

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## Code Repositories

### qratpreplus

QRATPre+, a preprocessor for quantified Boolean formulas.

## 1 Introduction

In practical applications of propositional logic satisfiability (SAT), it is necessary to establish correctness guarantees on the results produced by SAT solvers by proof checking [7]. The DRAT (deletion resolution asymmetric tautology) [22] approach has become state of the art to generate and check propositional proofs.

The logic of quantified Boolean formulas (QBF) extends propositional logic by existential and universal quantification of the propositional variables. Despite the PSPACE-completeness of QBF satisfiability checking, QBF technology is relevant in practice due to the potential succinctness of QBF encodings [4].

DRAT has been lifted to QBF to obtain the (quantified RAT) proof system [8, 10]. allows to represent and check (un)satisfiability proofs of QBFs and compute Skolem function certificates of satisfiable QBFs. The system simulates virtually all inference rules applied in state of the art QBF reasoning tools, such as Q-resolution [15] including its variant long-distance Q-resolution [13, 24], and expansion of universal variables [3].

A proof of a QBF in prenex CNF consists of a sequence of inference steps that rewrite the QBF by adding and deleting clauses and universal literals that have the redundancy property. Informally, checking whether a clause has amounts to checking whether all possible resolvents of on a literal (under certain restrictions) are propositionally implied by the quantifier-free CNF part of the QBF. The principle of redundancy checking by inspecting resolvents originates from the RAT property in propositional logic [12] and was generalized to first-order logic in terms of implication modulo resolution [14]. Instead of a complete (and thus computationally hard) propositional implication check on a resolvent, the system relies on an incomplete check by propositional unit propagation (UP). Thereby, it is checked whether UP can derive the empty clause from the CNF augmented by the negated resolvent. Hence redundancy checking in is unaware of the quantifier structure, which is entirely ignored in UP.

We generalize redundancy checking in by making it aware of the quantifier structure of a QBF. To this end, we check the redundancy of resolvents based on QBF specific UP (QUP). It extends UP by the universal reduction (UR) operation [15] and is a polynomial-time procedure like UP. UR is central in resolution based QBF calculi [1, 15] as it shortens individual clauses by eliminating universal literals depending on the quantifier structure. We apply QUP to abstractions of the QBF where certain universal quantifiers are converted into existential ones. The purpose of abstractions is that if a resolvent is found redundant by QUP on the abstraction, then it is also redundant in the original QBF.

Our contributions are as follows: (1) by applying QUP and QBF abstractions instead of UP, we obtain a generalization of the system which we call . In contrast to , redundancy checking in is aware of the quantifier structure of a QBF. We show that (2) the redundancy property in based on QUP is more powerful than the one in based on UP. can detect redundancies which cannot. As a formal foundation, we introduce (3) a theory of QBF abstractions used in . Redundancy elimination by or can lead to (4) exponentially shorter proofs in certain resolution based QBF calculi, which we point out by a concrete example. Note that here we do not study the power of or as proof systems themselves, but the impact of redundancy elimination. Finally, we report on experimental results (5) to illustrate the benefits of redundancy elimination by and for QBF preprocessing. Our implementation of and for preprocessing is the first one reported in the literature.

## 2 Preliminaries

We consider QBFs in prenex conjunctive normal form (PCNF) with a quantifier prefix and a quantifier free CNF not containing tautological clauses. The prefix consists of quantifier blocks , where are blocks (i.e., sets) of propositional variables and are quantifiers. We have , and . The CNF is defined precisely over the variables in so that all variables are quantified, i.e., is closed. The quantifier of literal is if the variable of appears in . The set of variables in a clause is . A literal is existential if and universal if . If and , then iff . We extend the ordering to an arbitrary but fixed ordering on the variables in every block .

An assignment maps the variables of a QBF to truth constants (true) or (false). Assignment is complete if it assigns every variable in , otherwise is partial. By we denote under , where each occurrence of variable in is replaced by and is removed from the prefix of , followed by propositional simplifications on . We consider as a set of literals such that, for some variable , if and if .

An assignment tree [10] of a QBF is a complete binary tree of depth where the internal (non-leaf) nodes of each level are associated with a variable of . An internal node is universal (existential) if it is associated with a universal (existential) variable. The order of variables along every path in respects the extended order of the prefix of . An internal node associated with variable has two outgoing edges pointing to its children: one labelled with and another one labelled with , denoting the assignment of to false and true, respectively. Each path in from the root to an internal node (leaf) represents a partial (complete) assignment. A leaf at the end of is labelled by , i.e., the value of under . An internal node associated with an existential (universal) variable is labelled with iff one (both) of its children is (are) labelled with . The QBF is satisfiable (unsatisfiable) iff the root of is labelled with ().

Given a QBF and its assignment tree , a subtree of is a pre-model [10] of if (1) the root of is the root of , (2) for every universal node in both children are in , and (3) for every existential node in exactly one of its children is in . A pre-model of is a model [10] of , denoted by , if each node in is labelled with . A QBF is satisfiable iff it has a model. Given a QBF and one of its models , is a rooted subtree of () if has the same root as and the leaves of are a subset of the leaves of .

We consider CNFs defined over a set of variables without an explicit quantifier prefix. A model of a CNF is a model of the QBF which consists only of the single path . We write if is a model of . For CNFs and , is implied by () if, for all , it holds that if then . Two CNFs and are equivalent (), iff and . We define notation to explicitly refer to QBF models. For QBFs and , is implied by () if, for all , it holds that if then . QBFs and are equivalent () iff and , and satisfiability equivalent () iff is satisfiable whenever is satisfiable. Satisfiability equivalence of CNFs is defined analogously and denoted by the same symbol ’’.

## 3 The Original QRAT Proof System

Before we generalize , we recapitulate the original proof system [10] and emphasize that redundancy checking in is unaware of quantifier structures.

###### Definition 1 ([10])

The outer clause of clause on literal with respect to prefix is the clause .

The outer clause of on contains only literals that are smaller than or equal to in the variable ordering of prefix , excluding .

###### Definition 2 ([10])

Let be a clause with and be a clause with occurring in QBF . The outer resolvent of with on with respect to is the clause .

###### Example 1

Given with and , we have , , , and . Computing outer resolvents is asymmetric since .

###### Definition 3 ([10])

Clause has property (quantified implied outer resolvent) on literal with respect to QBF iff for all with .

Property relies on checking whether every possible outer resolvent of some clause on a literal is redundant by checking if is propositionally implied by the quantifier-free CNF of the given QBF . If has on literal then, depending on whether is existential or universal and side conditions, either is redundant and can be removed from QBF or is redundant and can be removed from , respectively, resulting in a satisfiability-equivalent QBF.

###### Theorem 3.1 ([10])

Given a QBF and a clause with on an existential literal with respect to QBF where . Then .

###### Theorem 3.2 ([10])

Given a QBF and where has on a universal literal with respect to . Let with . Then .

Note that in Theorems 3.1 and 3.2 clause is actually removed from the QBF for the check whether has on a literal. Checking propositional implication () as in Definition 3 is co-NP hard and hence intractable. Therefore, in practice a polynomial-time incomplete implication check based on propositional unit propagation (UP) is applied. The use of UP is central in the proof system.

###### Definition 4 (propositional unit propagation, UP)

For a CNF and clause , let denote the fact that propositional unit propagation (UP) applied to produces the empty clause, where is the conjunction of the negation of all the literals in . If then we write to denote that can be derived from by UP (since ).

###### Definition 5 ([10])

Clause has property (asymmetric tautology) with respect to a CNF iff .

is a propositional clause redundancy property that is used in the proof system to check whether outer resolvents are redundant, thereby replacing propositional implication () in Definition 3 by unit propagation () as follows.

###### Definition 6 ([10])

Clause has property (quantified resolution asymmetric tautology) on literal with respect to QBF iff, for all with , the outer resolvent has with respect to CNF .

###### Example 2

Consider with and from Example 1. does not have with respect to CNF , but has on with respect to QBF since has with respect to CNF .

is a restriction of , i.e., a clause that has also has but not necessarily vice versa. Therefore, the soundness of removing redundant clauses and literals based on follows right from Theorems 3.1 and 3.2.

Based on the redundancy property, the proof system [10] consists of rewrite rules to eliminate redundant clauses, denoted by , to add redundant clauses, denoted by , and to eliminate redundant universal literals, denoted by . In a satisfaction proof (refutation), a QBF is reduced to the empty formula (respectively, to a formula containing the empty clause) by applying the rewrite rules. The proof systems has an additional rule to eliminate universal literals by extended universal reduction (). We do not present because it is not affected by our generalization of , which we define in the following. Observe that and (and hence also ) are based on propositional implication () and unit propagation (), i.e., the quantifier structure of the given QBF is not exploited.

## 4 QRAT+: A More Powerful QBF Redundancy Property

We make redundancy checking of outer resolvents in aware of the quantifier structure of a QBF. To this end, we generalize and by replacing propositional implication () and unit propagation () by QBF implication and QBF unit propagation, respectively. Thereby, we obtain a more general and more powerful notion of the redundancy property, which we call .

First, in Proposition 2 we point out a property of (Definition 3) which is due to the following result from related work [20]: if we attach a quantifier prefix to equivalent CNFs and , then the resulting QBFs are equivalent.

###### Proposition 1 ([20])

Given CNFs and such that and a quantifier prefix defined precisely over . If then .

###### Proposition 2

If clause has on literal with respect to QBF , then for all with .

###### Proof

Since has on literal with respect to QBF , by Definition 3 we have for all with , and further also . Then by Proposition 1. ∎

By Proposition 2 any outer resolvent of some clause that has with respect to some QBF is redundant in the sense that it can be added to the QBF in an equivalence preserving way (), i.e., is implied by the QBF (). This is the central characteristic of our generalization of . We develop a redundancy property used in which allows to, e.g., remove a clause from a QBF in a satisfiability preserving way (like in , cf. Theorem 3.1.) if all respective outer resolvents of are implied by the QBF . Since checking QBF implication is intractable just like checking propositional implication in , in practice we apply a polynomial-time incomplete QBF implication check based on QBF unit propagation.

In the following, we develop a theoretical framework of abstractions of QBFs that underlies our generalization of . Abstractions are crucial for the soundness of checking QBF implication by QBF unit propagation.

###### Definition 7 (nesting levels, prefix/QBF abstraction)

Let be a QBF with prefix . For a clause , is the set of nesting levels in .111In general, clauses are always (implicitly) interpreted under a quantifier prefix . The abstraction of with respect to with produces the abstracted prefix for and otherwise . The abstraction of with respect to with produces the abstracted QBF with prefix .

###### Example 3

Given the QBF with prefix . We have , , .

In an abstracted QBF universal variables from blocks smaller than or equal to are converted into existential ones. If the original QBF has a model , then all nodes in associated to universal variables must be labelled with , in particular the universal variables that are existential in . Hence, for all models of , every model of is a subtree of .

###### Proposition 3

Given a QBF with prefix and for some arbitrary with . For all and we have that if and is a pre-model of , then .

###### Proof

By induction on . The base case is trivial.

As induction hypothesis (IH), assume that the claim holds for some with , i.e., for all and we have that if and is a pre-model of , then . Consider for , which is an abstraction of . We have to show that, for all and we have that if and is a pre-model of , then . We distinguish cases by the type of in the abstracted prefix of .

If then . Since , the claim holds for by IH.

If then, towards a contradiction, assume that, for some and , and is a pre-model of , but . Then the root of is labelled with , and in particular the nodes of all the variables which are existential in with respect to are also labelled with . These existential variables appear along a single branch in , i.e., is a partial assignment of the variables in . Since and in , the root of is labelled with since there is the branch containing the variables in whose nodes are labelled with in . Hence , which is a contradiction to IH. Therefore, we conclude that . ∎

If an abstraction is unsatisfiable then also the original QBF is unsatisfiable due to Proposition 3. We generalize Proposition 1 from CNFs to QBFs and their abstractions. Note that the full abstraction for of a QBF is a CNF, i.e., it does not contain any universal variables.

###### Lemma 1

Let and be QBFs with the same prefix . Then for all , if then .

###### Proof

By induction on up to . The base case is trivial.

As induction hypothesis (IH), assume that the claim holds for some with , i.e., if then . Let and consider and , which are abstractions of and . We have and . We show that if then , and hence also by IH. Assume that . We distinguish cases by the type of in . If then , and hence .

If , then towards a contradiction, assume that but . Then there exists such that but . Since there exists a pre-model of such that the root of is labelled with , and in particular the nodes of all the variables which are existential in with respect to (and universal with respect to ) are also labelled with . These existential variables appear along a single branch in , i.e., is a partial assignment of the variables in . Therefore we have . Since and , we have by Proposition 3, which contradicts the assumption that . ∎

The converse of Lemma 1 does not hold. From the equivalence of two QBFs and we cannot conclude that the abstractions and are equivalent. In our generalization of the system we check whether an outer resolvent of some clause is implied () by an abstraction of the given QBF. If so then by Lemma 1 the outer resolvent is also implied by the original QBF. Below we prove that this condition is sufficient for the soundness of redundancy removal in . To check QBF implication in an incomplete way and in polynomial time, in practice we apply QBF unit propagation, which is an extension of propositional unit propagation, to abstractions of the given QBF.

###### Definition 8 (universal reduction, UR [15])

Given a QBF and a non-tautological clause , universal reduction (UR) of produces the clause .

###### Definition 9 (QBF unit propagation, QUP)

QBF unit propagation (QUP) extends UP (Definition 4) by applications of UR. For a QBF and a clause , let denote the fact that QUP applied to produces the empty clause, where is the conjunction of the negation of all the literals in . If and additionally then we write to denote that can be derived from by QUP.

In contrast to UP (Definition 4), deriving the empty clause by QUP by propagating on a QBF is not sufficient to conclude that is implied by .

###### Example 4

Given the QBF with prefix and CNF and the clause . We have since propagating produces , which is reduced to by UR. However, since is satisfiable whereas is unsatisfiable. Note that .

To correctly apply QUP for checking whether some clause (e.g., an outer resolvent) is implied by a QBF and thus avoid the problem illustrated in Example 4, we carry out QUP on a suitable abstraction of with respect to . Let be the maximum nesting level of variables that appear in . We show that if QUP derives the empty clause from the abstraction augmented by the negated clause , i.e., , then we can safely conclude that is implied by the original QBF, i.e., . This approach extends failed literal detection for QBF preprocessing [16].

###### Lemma 2

Let be a QBF with prefix and a clause such that . If then .

###### Proof

By contradiction, assume but . Then there is a path such that . Since and , the QBF is unsatisfiable and in particular . Since , we have and hence , which is a contradiction. ∎

###### Lemma 3

Let be a QBF, a clause, and . If then .

###### Proof

The claim follows from Lemma 2 since all variables that appear in are existentially quantified in in the leftmost quantifier block. ∎

###### Lemma 4

Let be a QBF, a clause, and . If then .

###### Proof

By Lemma 3 and Lemma 1. ∎

Lemma 4 provides us with the necessary theoretical foundation to lift (Definition 5) from UP, which is applied to CNFs, to QUP, which is applied to suitable abstractions of QBFs. The abstractions are constructed depending on the maximum nesting level of variables in the clause we want to check.

###### Definition 10 (QAT)

Let be a QBF, a clause, and Clause has property (quantified asymmetric tautology) with respect to iff .

As an immediate consequence from the definition of QUP (Definition 9) and Lemma 3, we can conclude that a clause has with respect to a QBF if QUP derives the empty clause from the suitable abstraction of with respect to (i.e., ). Further, if has then we have by Lemma 4, i.e., is implied by the given QBF .

###### Example 5

Given the QBF with and . Clause has with respect to with since is still universal in the abstraction. By QUP clause becomes unit and clause becomes empty by UR. However, clause does not have since is treated as an existential variable in UP, hence clause does not become empty by UR.

In contrast to , is aware of quantifier structures in QBFs as shown in Example 5. We now generalize to by replacing by . Similarly, we generalize to by replacing propositional implication () and equivalence (Proposition 1), by QBF implication and equivalence (Lemma 4).

###### Definition 11 (QRAT+)

Clause has property on literal with respect to QBF iff, for all with , the outer resolvent has with respect to QBF .

###### Definition 12 (QIOR+)

Clause has property on literal with respect to QBF iff for all with .

If a clause has then it also has . Moreover, due to Proposition 2, if a clause has then it also has . Hence and indeed are generalizations of and , which are strict, as we argue below. The soundness of removing redundant clauses and universal literals based on (and on ) can be proved by the same arguments as original , which we outline in the following. We refer to the appendix for full proofs.

###### Definition 13 (prefix/suffix assignment [10])

For a QBF and a complete assignment in the assignment tree of , the partial prefix and suffix assignments of with respect to variable , denoted by and , respectively, are defined as and .

For a variable from block of a QBF, Definition 13 allows us to split a complete assignment into three parts , where the prefix assignment assigns variables (excluding ) from blocks smaller than or equal to , is a literal of , and the suffix assignment assigns variables from blocks larger than .

Prefix and suffix assignments are important for proving the soundness of satisfiability-preserving redundancy removal by (and ). Soundness is proved by showing that certain paths in a model of a QBF can safely be modified based on prefix and suffix assignments, as stated in the following.

###### Lemma 5 (cf. Lemma 6 in [10])

Given a clause with with respect to QBF on literal with . Let be a model of and be a path in . If then for all with .

###### Proof (sketch, see appendix)

Let be a clause with and . By Definition 12, we have for all with . The rest of the proof considers a path in and works in the same way as the proof of Lemma 6 in [10]. ∎

###### Theorem 4.1

Given a QBF and a clause with on an existential literal with respect to QBF where . Then .

###### Proof (sketch, see appendix)

The proof relies on Lemma 5 and works in the same way as the proof of Theorem 7 in [10]. A model of is obtained from a model of by flipping the assignment of variable on a path in to satisfy clause . All with are satisfied by such modified . ∎

###### Theorem 4.2

Given a QBF and where has on a universal literal with respect to . Let with . Then .

###### Proof (sketch, see appendix)

The proof relies on Lemma 5 and works in the same way as the proof of Theorem 8 in [10]. A model of is obtained from a model of by modifying the subtree under the node associated to variable . Suffix assignments of some paths in are used to construct modified paths in under which clause is satisfied. All with are still satisfied after such modifications. ∎

Analogously to the proof system that is based on the redundancy property (Definition 6), we obtain the proof system based on property (Definition 11). The system consists of rewrite rules , , and to eliminate or add redundant clauses, and to eliminate redundant universal literals. On a conceptual level, these rules in are similar to their respective counterparts in the system. The extended universal reduction rule is the same in the and systems. In contrast to , is aware of quantifier structures of QBFs because it relies on the QBF specific property and QUP instead of on propositional and UP.

The system has the same desirable properties as the original system. simulates virtually all inference rules applied in QBF reasoning tools and it is based on redundancy property that can be checked in polynomial time by QUP. Further, allows to represent proofs in the same proof format as . However, proof checking, i.e., checking whether a clause listed in the proof has on a literal, must be adapted to the use of QBF abstractions and QUP. Consequently, the available proof checker QRATtrim [10] cannot be used out of the box to check proofs.

Notably, Skolem functions can be extracted from proofs of satisfiable QBFs in the same way as in (consequence of Theorem 4.1, cf. Corollaries 26 and 27 in [10]). Hence like , can be integrated in complete QBF workflows that include preprocessing, solving, and Skolem function extraction [5].

## 5 Exemplifying the Power of QRAT+

In the following, we point out that the system is more powerful than in terms of redundancy detection. In particular, we show that the rules and in the