### qratpreplus

QRATPre+, a preprocessor for quantified Boolean formulas.

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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.

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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.

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 ’’.

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

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 .

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 .

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

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.

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

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.

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 ).

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.

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 .

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.

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.

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

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

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.

Let be a QBF with prefix .
For a clause , is the set of
*nesting levels* in .^{1}^{1}1In 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 .

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 .

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

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.

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

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.

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

*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 .

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].

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

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. ∎

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

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

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

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.

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 .

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).

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

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.

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.

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 .

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

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

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.

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